5th Year
Maths
Higher Level
Kieran Mills
Algebraic Expressions
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Section 2: Algebraic Expressions
Paper 1 Topics
Section 1. Number
Section 2. Algebraic expressions
Section 3. Algebraic equations
Section 4. Sequences and Series
Section 5. Financial Maths
Section 6. Complex Numbers
Section 7. Functions
Section 8. Differentiation
Section 9. Integration
Section 10: Proof by Induction
Paper 2 Topics
Section 1. Geometry
Section 2. Measurement
Section 3. Trigonometry
Section 4. Co-ordinate Geometry
Section 5. Probability
Section 6. Statistics
Contents
Chapter 3: Working with Algebraic Expressions...............................2
Chapter 4: Polynomial and Rational Expressions..............................18
Chapter 5: Exponentials and Logs.....................................................29
Exercises.............................................................................................39
Answers.................................................................................................58
Leaving Certificate Exam questions
(All the algebra questions are at the end of Section 3)
©The Dublin School of Grinds
Page 1
Mills & Kelly (Power of Maths)
Chapter 3. Working with Algebraic
Expressions
1.
Algebraic Basics
Some important key terms:
3x2 - 2xy + 5 is an expression.
3x2, 2xy and 5 are terms. This expression has three terms.
x and y are variables. x and y can take on different values.
In the term 3x2, 3 is the coefficient of the term.
In the term 2xy, 2 is the coefficient of the term.
5 is a constant.
Example 1: Write down the number of terms, the coefficient of a given term and the
constant term in the following algebraic expressions:
Solution
Number of terms
Coefficient
(a) 4 + x2 + 4xy
xy
(b) 3x2y + 5xy - 3
x2y
(c) 4ab - 3a2b + 5a3b2 - 5
a2b
x2
(d) x3 - 2x2 + 3x
2.
Constant
Combining like terms
Some important points:
1. The order in which you multiply two numbers does not matter.
x × y = y × x = xy = yx
The big 3 and the little 3:
Write out in long hand:
3×2=
=
23 =
=
2.
The coefficient in front of the variable tells you how many times you add the variable to
itself.
3x = x + x + x
5x2 = x2 + x2 + x2 + x2 + x2
©The Dublin School of Grinds
Page 2
Mills & Kelly (Power of Maths)
3.
Combining powers: The little number (power) tells you how many times you multiply a
number by itself.
x3 = x × x × x
x2 × x3 = x × x × x × x × x = x5 = x2 + 3
ap × aq = ap + q
Example 2: Multiply the following terms together to produce a single term
Solution
(a) 4x × 5y =
(b) 5ab × 4ab =
(c) (-2xy)(-5x) =
(d) x2 × x4 =
(e) 3xy3 × -4x2y =
You only add like terms.
2x + 3x = x + x + x + x + x = 5x [2 terms are combined into 1 term]
2a2b + 5a2b - ba2 = 6a2b [3 terms are combined into 1 term]
But 2x + 3y = 2x + 3y [2 terms remain as 2 terms]
Example 3: Simplify the following.
Solution
(a) 2a + 3a + 5a =
(b) 4x - 5x - 7x =
(c) 4x + 3y + 2y - 10x =
(d) 4(a - 2b) - 3(b - 3a) =
=
(e) 3x2y + 4x2y - 11yx2 =
You can only add like terms.
Unlike terms can be multiplied.
4ab + 2ab2 =
4ab × 2ab2 =
©The Dublin School of Grinds
Page 3
Mills & Kelly (Power of Maths)
3.
Multiplying out Brackets
(2 x + 3 y )(2 x − y )
= 2 x(2 x − y ) + 3 y (2 x − y )
= 4 x 2 − 2 xy + 6 yx − 3 y 2
= 4 x 2 − 2 xy + 6 xy − 3 y 2
= 4 x 2 + 4 xy − 3 y 2
∴ (2x + 3y)(2x - y) = 4x2 + 4xy - 3y2
(2x + 3y) and (2x - y) multiply together to give
4x2 + 4xy - 3y2.
(2x + 3y) and (2x - y) are called the factors of
4x2 + 4xy - 3y2.
Example 4: Expand the following.
Solution
(a) (2a + 3b)(2c + d)
Number of terms:
(b) (2a + 3b)(a - 2b)
Number of terms:
Trinomial
(c) (2a + 3b)(2a - 3b)
Number of terms:
Difference of 2 squares
Special situations
1. Difference of two squares: (Difference of two terms) × (Sum of the same two terms)
(2 x − 5 y )(2 x + 5 y )
= 2 x(2 x + 5 y ) − 5 y (2 x + 5 y )
= 4 x 2 + 10 xy − 10 xy − 25 y 2
= 4 x 2 − 25 y 2
= (2 x) 2 − (5 y ) 2
Remember as:
(First - Second)(First + Second) = (First)2 - (Second)2
©The Dublin School of Grinds
Page 4
Mills & Kelly (Power of Maths)
Example 5: Expand the following.
Solution
(a) (3x + y)(3x - y)
2.
(b) (5x + 3y)(5x - 3y)
(c) (ax + by)(ax - by)
Perfect square: Consists of two identical brackets.
(3 x − 2 y ) 2
= (3 x − 2 y )(3 x − 2 y )
= 3 x(3 x − 2 y ) − 2 y (3 x − 2 y )
= 9 x 2 − 6 xy − 6 xy + 4 y 2
= 9 x 2 − 12 xy + 4 y 2
= (3 x) 2 + 2(3 x)(−2 y ) + (−2 y ) 2
Remember as:
(First term + Second term)2 = (First term)2 + 2(First term)(Second term) + (Second term)2
Example 6: Expand the following.
Solution
(a) (3x + y)2
(b) (3x - 4y)2
(c) (ax + by)2
©The Dublin School of Grinds
Page 5
Mills & Kelly (Power of Maths)
3.
Harder multiplication
Example 7: Expand the following.
Solution
(a) (x - 1)(2x + 3)(x + 1)
(b) (x - 2)3
4.
Finding the value of an algebraic expression
To find the value of an algebraic expression, simply fill in the given value(s) of the varible(s).
Example 8: Evaluate 4x2y - 2xy2 + 5xy - 7, if x = 2 and y = -3.
Solution
For you to practice
Exercise 1: page 39
©The Dublin School of Grinds
Page 6
Mills & Kelly (Power of Maths)
2. Binomial Theorem
An expression with two terms is called a binomial. The binomial theorem is a quick way to
multiply out (expand) binomials raised to a power.
Examples: (x + y)6, (2a - b)10
How nCr is calculated
1.
Factorials
Factorials are represented by !
Example: 4! = 4 × 3 × 2 × 1 = 24
Doing factorials on the calculator:
Casio fx-83GT PLUS
Example: 4!
Press 4
Press SHIFT x-1 (This is x!)
Press =
Calculate the following manually and then check the answers on your calculator:
5! =
=
6! =
=
1! =
Write out: n! =
Combinations
We will deal with combinations (C) or selections in greater detail when we do the section on
Probability.
n
Cr =
n!
r !(n - r )!
Formula and Tables Book:
Page 20 (Algebra)
Example: How many way can you select two people from 10 people when forming a
committee?
n
Cr gives you the answer where n = 10, r = 2.
C2 =
10
C2 =
6
C3 =
9
10 !
10 !
10×9×8× 7 × 6×5× 4×3× 2×1 10×9
=
=
=
= 45
2 !(10 - 2)! 2 ! × 8! 2×1×8× 7 × 6×5× 4×3× 2×1
2×1
6×5
= 3×5 = 15
2×1
=
©The Dublin School of Grinds
=
Page 7
Mills & Kelly (Power of Maths)
C4 =
=
10
C3 =
=
12
=
=
Doing combinations on the calculator:
Casio fx-83GT PLUS
Example: 12C3
Press 12
Press SHIFT ÷ (This is nCr)
Press 3
Press =
 n
Alternative way of writing combinations: nCr =  
 r 
2.
Using the Binomial
Formula and Tables Book: Page 20 (Algebra)
 n
 n
 n
 n
 n
( x + y ) n =   x n +   x n-1 y +   x n-2 y 2 + ... +   x n-r y r + ... +   y n
0
1
 2
 r 
n
Example 9: Expand (x + y)5.
Solution
Handy coefficients
Coefficients of (x + y)2: 1, 2, 1
Coefficients of (x + y)3: 1, 3, 3, 1
Coefficients of (x + y)4: 1, 4, 6, 4, 1
Coefficients of (x + y)5:
©The Dublin School of Grinds
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Mills & Kelly (Power of Maths)
Example 10: Expand (2a - b)4.
Solution
Example 11: Expand (3x + 2y)3.
Solution
3.
Picking out terms
General term: (r + 1)st term is nCr (x)n - r (y)r
Example: The fifth term in (p – q)7 has a binomial coefficient of 7C4.
Fifth term = 7C4 (p)3(–q)4 = 35p3q4
Example: The fourth term in (2x + y)8 has a binomial coefficient of 8C3.
Fourth term = 8C3(2x)5(y)3 = 56(32x5)y3 = 1792x5y3
©The Dublin School of Grinds
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Mills & Kelly (Power of Maths)
Example 12: In the expansion of (p + q)7, what is the term with q5?
Solution
Example 13: What is the term with a4 in the expansion of (a − 2b)9?
Solution


2 x + y  , what is the term with y3?

4 
5
Example 14: In
Solution
For you to practice
Exercise 2: page 40
©The Dublin School of Grinds
Page 10
Mills & Kelly (Power of Maths)
3. Factorisation
1.
Factorisation technique 1: HCF
Factorise the following by taking out the highest common factor:
5x + 10y + 15 =
5x2y2 − 10xy =
7x(x − 2y) − 3y(x − 2y) =
2.
Factorisation technique 2: Grouping
Example 15: Factorise ax - bx + ay - by by grouping.
Solution
Example 16: Factorise 3x - 8y - 2 + 12xy by grouping.
Solution
Example 17: Factorise ax - bx - ay + by + a - b by grouping.
Solution
©The Dublin School of Grinds
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Mills & Kelly (Power of Maths)
3.
Factorisation technique 3: Trinomials
Example 18: Factorise 2x2 - 5x - 12.
Solution
Rough work
Rough work
(
)(
)
Example 19: Factorise 4x2 - 13x - 12.
Solution
Rough work
Rough work
(
©The Dublin School of Grinds
)(
Page 12
)
Mills & Kelly (Power of Maths)
4.
Factorisation technique 4: Difference of
two squares
a2 - b2 = (a - b)(a + b)
Remember as: (First)2 – (Second)2 = (First – Second)(First + Second)
25 x 2 - 4 y 2 = (5 x) 2 - (2 y ) 2 = (5 x - 2 y )(5 x + 2 y )
400 x 2 - 49 y 2 =
=
x 2 - a 2b 2 =
=
Example 20: Factorise (x + y)2 - z2.
Solution
5.
Factorisation technique 5: Difference and
Sum of two cubes
a3 + b3 = (a + b)(a2 - ab + b2)
a3 - b3 = (a - b)(a2 + ab + b2)
Remember as:
(First)3 + (Second)3 = (First + Second)((First)2 – (First) × (Second) + (Second)2)
(First)3 – (Second)3 = (First – Second)((First)2 + (First) × (Second) + (Second)2)
x3 + 8 y 3 = ( x)3 + (2 y )3 = ( x + 2 y )( x 2 - 2 xy + 4 y 2 )
8 x3 - 27 =
=
125 x3 - 64a 3b3 =
=
©The Dublin School of Grinds
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Mills & Kelly (Power of Maths)
Example 21: Factorise (a - 1)3 + (a + 1)3.
Solution
6.
Factorisation technique 6: Harder factors
Example 22: Factorise x2 + 2xy + y2 - 1.
Solution
Example 23: Factorise 9a2 - 12ab + 4b2 - 25c2.
Solution
©The Dublin School of Grinds
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Mills & Kelly (Power of Maths)
Example 24: Factorise the following fully:
(a) 1 − 3x − 4x2
(b) 7x2 − 28y2
2
(d) 154x − 50x + 4
(e) (x + 2y)2 − (x – 2y)2
Solution
(c) 2a2 + 6ac − 4ab − 12bc
(f) 15a3 + 120a3b3
For you to practice
Exercise 3: page 41
©The Dublin School of Grinds
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Mills & Kelly (Power of Maths)
4. Algebraic Modelling
Algebraic modelling means translating a problem, stated in words, into an algebraic expression.
Guidelines for modelling
1.
2.
3.
4.
Identify the quantity to be modelled (cost, price, area, length, etc.) and give it a symbol.
Draw a diagram if appropriate (unless given).
Identify the number of variables (one or more) and if appropriate put them on the diagram.
Write the quantity to be modelled in terms of this (these) variable(s).
Example 25: The breadth of a rectangular field is 20 m longer than its length x.
Write down an expression, in terms of x, for:
(a) the breadth b, (b) the perimeter P, (c) the area A.
Solution
Example 26: A running track has two straights and two semicircular ends.
If x m is the length of each straight and y m the radius of each semicircular end,
(a) find an expression in terms of x and y for:
(i) the perimeter P,
(ii) the area A1 of the rectangular region ABCD,
(iii) the total area A2.
(b) If x = 100 m and y = 31∙85 m, find the perimeter to the nearest metre.
(c) If a runner has an average speed of 30 km/h, how long does it take the runner to
complete one full circuit of the track?
xm
Straight
ym
Semicircle
ym
s
A
©The Dublin School of Grinds
C
ym
Semicircle
D
s
ym
Straight
xm
Page 16
B
Mills & Kelly (Power of Maths)
Solution
For you to practice
Exercise 4: page 43
©The Dublin School of Grinds
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Mills & Kelly (Power of Maths)
Chapter 4: Polynomial and Rational Expressions
1.
Polynomial Expressions
Types
1.
Linear expressions
Examples: 2x, 2x + 3
In general: ax + b (Standard form)
2.
Quadratic Expressions
Examples: x2, x2 + 1, 2x2 - 3x + 2
In general: ax2 + bx + c (Standard form)
3.
Cubic expressions
Examples: x3, 3x3 + 1, 3x3 + 2x2 + 3x + 1
In general: ax3 + bx2 + cx + d (Standard form)
Example 1: State what type of expression is shown and write each expression in standard
form.
Solution
Type of polynomial
Standard Form
(a) 2 - 2x
(b) x3 + 3 - 2x
(c) 1 - x2
(d) 3x3 + 3x - 2 + x2
Linear × Linear = Quadratic
Example 2: Multiply (2x + 1)(3x - 5) writing your answer in standard form.
Solution
First × First = First:
×
=
Last × Last = Last:
×
=
©The Dublin School of Grinds
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Mills & Kelly (Power of Maths)
Linear × Quadratic = Cubic
Example 3: Multiply (3x + 1)(4x2 - 2x - 5) writing your answer in standard form.
Solution
First × First = First:
×
=
Last × Last = Last:
×
=
Linear × Linear × Linear = Cubic
Example 4: Multiply (5x + 1)(3x - 2)(x - 1) writing your answer in standard form.
Solution
First × First × First = First:
×
×
=
Last × Last × Last = Last:
×
×
=
©The Dublin School of Grinds
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Mills & Kelly (Power of Maths)
Example 5: Multiply out the following writing your answer in standard form.
(a) (-3x + k)(4x + l)
(b) (2x2 - kx + 3)(x + l)
Solution
2.
Identities
(x - 3)(x - 2) = x2 - 5x + 6
When you multiply out the two brackets on the left you get an identical expression on the right.
This is an identity.
Identities are true for all values of the variable x.
LHS
RHS
(x - 3)(x - 2)
(1 - 3)(1 - 2)
= (-2)(-1)
=2
x2 - 5x + 6
(1)2 - 5(1) + 6
=1-5+6
=2
Be clever. Try x = 3:
(3 - 3)(3 - 2)
= (0)(1)
=0
(3)2 - 5(3) + 6
= 9 - 15 + 6
=0
Be clever. Try x = 2:
(2 - 3)(2 - 2)
= (-1)(0)
=0
(2)2 - 5(2) + 6
= 4 - 10 + 6
=0
Try x = 1:
©The Dublin School of Grinds
Page 20
Mills & Kelly (Power of Maths)
Example 6: If 10x2 - 9x + 2 = (5x - k)(2x - 1), find k.
Solution
Example 7: If x3 + 2x2 + kx + 2 = (x - 1)(x2 + ax - 2), find k and a.
Solution
Method 1 (Lining up)
Method 2 (Choosing values)
©The Dublin School of Grinds
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Mills & Kelly (Power of Maths)
Example 8: If 3x2 + 5x - 1 is a factor of 3x3 + 11x2 + 9x - 2, find the other factor.
Solution
Example 9: If 2x - 1 and x + 2 are factors of 6x3 + x2 - 18x + 8, find the other factor.
Solution
Example 10: If 3x + 2 is a factor of 9x3 - 19x - 10, find the quadratic factor.
Solution
©The Dublin School of Grinds
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Mills & Kelly (Power of Maths)
Example 11: If x2 + px - q is a factor of 2x3 + px2 - 3x + q2, find the values of p and q.
Solution
Example 12: If (2x - 1) is a factor of 2x3 + ax2 - bx + 1, show that a = (2b - 5).
Solution
©The Dublin School of Grinds
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Mills & Kelly (Power of Maths)
Example 13: If 2x2 + bx - 5 is a factor of 2x3 - x2 + kx + 10, find the other factor and b and k.
Solution
Example 14: ax2 + 2 is a factor of 18x3 + kx2 + 6x - 2. Find k and a and the other factor.
Solution
©The Dublin School of Grinds
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Mills & Kelly (Power of Maths)
Example 15: If (x - p)2 is a factor of x3 + qx + r, show: (i) r = 2p3, (ii) q = -3p2.
Solution
Example 16: If x2 - px + 1 is a factor of ax3 + bx + c, show c2 = a(a - b).
Solution
For you to practice
Exercise 5: page 45
©The Dublin School of Grinds
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Mills & Kelly (Power of Maths)
3.
Division of polynomials
Cubic
Cubic
Quadratic
= Linear ,
= Quadratic,
= Linear
Quadratic
Linear
Linear
Always factorise first if you can.
x3 - 8 x 3 - 23 ( x - 2)( x 2 + 2 x + 4)
=
=
= x2 + 2 x + 4
( x - 2)
( x - 2)
x-2
x3 + x 2
=
x +1
=
15 x 2 + 11x -14
=
3x - 2
16 x3 - 28 x 2 + 6 x
=
8x - 2
=
=
=
3
2
Example 17: Simplify 4 x - 4 x - x + 1 .
2 x +1
Solution
For you to practice
Exercise 6: page 46
©The Dublin School of Grinds
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Mills & Kelly (Power of Maths)
4.
Rational Expressions
A rational expression is one expression divided by another.
Addition and subtraction
Technique for adding and subtracting rationals
Find the lowest common denominator (LCD)
1
1
xy + 1
+ 2 = 2
x x y
x y
3
2
4
7
+
, (b) 2
- 2
.
Example 18: Simplify (a)
x -5 5- x
x -1 2 x - 3 x - 5
Solution
Multiplication
Technique for multiplying rationals
Multiply the tops and multiply the bottoms and/or cancel.
-3ab 2 15b 2 c 2 ab 4 c 2 c
×
=
=
5b3c -9a 2b a 2b 4 c a
©The Dublin School of Grinds
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Mills & Kelly (Power of Maths)
a 2 - b2
ab
Example 19: Simplify (a)
×
, (b)
2
+
ab
a
a
b
Solution
 2 x3  x 4 
 +  - .
 x 2  2 x3 
Division (Double-decker fractions)
Technique for multiplying rationals
Multiply above and below by the LCD of all fractions.
9 7
10 + - 2
x x .
Example 20: Simplify
3 2
2+ - 2
x x
Solution
For you to practice
Exercise 7: page 47
©The Dublin School of Grinds
Page 28
Mills & Kelly (Power of Maths)
Chapter 5: Exponentials and Logs
Exponential Expressions (powers)
1.
An exponential expression can be written in the form ap where p ∈ .
a is called the base of the expression.
p is called the power, the index or the exponent of the expression.
1. The multiplication rule
Formula and Tables Book:
Page 21 (Indices and logs)
Remember as: When you multiply two exponential expressions with the same base, you add the
powers.
Rule 1: ap × aq = ap + q
x3 × x5 =
e x+3 × e 2 x+1 =
2 x 6 × 3 x -3 =
(a + 2b)3 × (a + 2b) 4 =
2. The division rule
Rule 2:
ap
= a p -q
aq
Formula and Tables Book:
Page 21 (Indices and logs)
Remember as: When you divide two exponential expressions with the same base, you subtract
the power on the bottom from the power on the top.
x9
=
x4
67
=
65
e 2 x-1
=
e x+2
y5
=
y-2
=
3. The one rule
Rule 3: a0 = 1
20 =
Formula and Tables Book:
Page 21 (Indices and logs)
(a + 2b)0 =
 x 

 =
 2 y 2 
0
4. The power of a power rule
Formula and Tables Book:
Page 21 (Indices and logs)
Remember as: When you put an exponential expression to a power, you multiply the two
powers.
Rule 4: (ap)q = apq
( 65 ) 3 =
( x3 )2 =
( e x -2 ) 4 =
©The Dublin School of Grinds
Page 29
Mills & Kelly (Power of Maths)
5. Powers of products and quotients
p
Formula and Tables Book:
Page 21 (Indices and logs)
a
ap
Rule 5: (ab) p = a p b p and   = p
 b 
b
 xy 
  =
 2 z 
5
(2ab 2 ) 4 =
 a 2 
  =
 b 
4
6. Negative powers
-p
Rule 6: a =
3-2 =
1
1
and - p = a p
p
a
a
3 x -2
=
zy-3
=
1
=
x -3
4-1
=
52
Formula and Tables Book:
Page 21 (Indices and logs)
( xy )-3 =
=
(
)
=
-p
a
The flipping trick:  
 b 




-2

 3 
-  =  4 


-3

 3 x 



=

2

 2 y 






b
=  
 a 
=
p
=
=
7. Non-whole number powers
1
q
q
Rule 7: a = a
Formula and Tables Book:
Page 21 (Indices and logs)
p
4 =
1
27 3 =
-1
4
16
1
1
Other fractional powers: a q = (a p ) q = (a q ) p
1
2
=
2
83 =
=
©The Dublin School of Grinds
3
16 4 =
Page 30
=
=
=
=
Mills & Kelly (Power of Maths)
-2
3
8
Example 21: Simplify (a ) - 1
4 2
Solution
Example 22: Simplify
1
(-5) 2 × 25 2
, ( b ) -3
.
5 × (25)3
(e x ) 2 × (e 2 x-1 )-2
Solution
e
4 x -2
× (e
x +1 -3
, giving your answer in the form eax + b.
)
Example 23: The population P of yeast cells after t hours is given by P = 106 × ( 2 )t .
t
(a) Show that P = 106 × 2 2.
(b) Find the population P1 after n hours.
(c) Find the population P2 after (n + 2) hours.
(d) Find the percentage change in population from n hours to (n + 2) hours.
Solution
Cont...
©The Dublin School of Grinds
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Mills & Kelly (Power of Maths)
For you to practice
Exercise 8: page 48
Surds
2.
A surd expression is an expression involving square roots of variable(s) or numbers that cannot
be simplified into a rational expression.
Examples of surds: 2 , x , x -1
1
a = a2
Simplifying surds
Use the following results:
ab = a b and
Simplify the following surds:
45 =
×
=
50 =
×
=
28 =
×
=
×
16 - 4 x =
2 14 =
a
a
=
b
b
=
=
=
=
©The Dublin School of Grinds
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Mills & Kelly (Power of Maths)
Adding and subtracting surds
Add and subtract like terms only: 5 2 + 3 2 = 8 2
Write in their simplest form:
3+ 5 + 6-2 5 +5 5 =
+
2 5 + 45 - 3 20 =
2 x + x3 =
+
-
+ x2 × x =
=
+
= (
)
Multiplying surds
Write in their simplest form:
2( 3 + 5 2 ) =
x ( x + 3) =
=
(3 + 2 )(4 - 5 2 ) =
=
=
( 3 + 2 )2 =
=
( x + y )2 =
( a + b )( a - b ) =
=
( a - b ) is the conjugate surd of ( a + b ).
Important result: ( a + b )( a - b ) = ( a ) 2 - ( b ) 2 = a - b
( n + 1 - n )( n + 1 + n ) =
=
©The Dublin School of Grinds
=
Page 33
Mills & Kelly (Power of Maths)
Division
When you divide surds, the answer should never have a surd in the denominator.
This process of getting rid of surds on the bottom is called rationalising the denominator.
Write in their simplest form:
5
5
=
×
2
2
=
3
3
=
×
2
2
=
4+ 3 4+ 3
=
×
2
2
=
Example 24: Rationalise the denominators of the following:
x+ y
5
(a )
, ( b)
.
x- y
3- 2
Solution
For you to practice
Exercise 9: page 50
©The Dublin School of Grinds
Page 34
Mills & Kelly (Power of Maths)
Logs
3.
A log is just another way to write a power.
2? = 8 ⇒ 23 = 8
or
log 2 8 = ? ⇒ log 2 8 = ?
2 is the base. log 8 to base 2 equals 3.
In general, loga x is the power to which you must put the base a to get x.
log 3 9 =
, log 3 27 =
, log 2 16 =
, log 4 1 =
, log 4 4 =
Escaping from logs (Hooshing)
log a y = x ⇔ y = a x
log 2 16 = 4 ⇔ 16 = 24
log10 100 = 2 ⇔
=
log 4 16 = 2 ⇔
log 4 1 = 0 ⇔
log 4 4 = 1 ⇔
=
=
log 5 125 = 3 ⇔
=
log 2 12 = -1 ⇔
=
=
Log Rules
1. The addition rule
Rule 1: log a x + log a y = log a ( xy )
Tip: You can only add logs with the same base when applying this rule.
log 4 6 + log 4 12 + log 4 5 = log 4 (
) = log 4
log a x 2 + log a y + log a 2 = log a (
) = log a
2. The subtraction rule
 x
Rule 2: log a x - log a y = log a  
 y 
Tip: You can only subtract logs with the same base when applying this rule.
log 6 18 - log 6 ( 12 ) = log 6
©The Dublin School of Grinds
= log 6
=
Page 35
Mills & Kelly (Power of Maths)


log k 6( x + 2) - log k 3( x + 2) = log k 




log a x + log a y - log a z = log a 


4

 = log (
k


3
) = log k (
)




3. Multiplication by a number rule
Rule 3: k log a x = log a x k
3 log 2 5 = log 2
= log 2
1 log 16 = log
2
2
2
= log 2
4 log a x - 2 log a y = log a


= log a 


- log a





4. Change of Base
Rule 4: log a x =
Change to base 2: log 3 ( x -1) =
Change to base e: log a x =
Change to base 8: log16 8 =
log b x
log b a
log 2
log 2
log e
log e
log8
log8
=
1
log8
Tip: If you invert a log, you interchange the base with the expression inside the log.
log a b =
Invert: log 2 5 =
Invert:
1
log b a
1
1
=
log 3 6
©The Dublin School of Grinds
Page 36
Mills & Kelly (Power of Maths)
All the rules together
1. Breaking logs down
Technique
Breaking down a log into a string of individual logs
(a) Multiplication → + logs (addition)
(b) Division → − logs (subtraction)
(c) Power → Multiplies logs
Break the following into individual logs:
log 2 ( xy 2 ) = log 2
+ log 2
 1- x 
log 3  2  = log 3 (
 x 
=
log 4
= log 2
)
 x + 1
x +1
= log 4  3 
3
 y 
y
=
log 3
 x + 1
log 4  3 
 y 
=

log 4 (

) - log 4
=
(log
)-
=
log 4 (
4
(
log 2
- log 3
)-
log 3 (
+
)-


log 4
)
log 4
Example 25: The magnitude M of an earthquake on the Richter scale is given by
2
 E 3
M = log10   , where E is the energy, in joules, released in the earthquake and E0 = 104·4 J.
 E0 
Show that M = 23 [log10 E - log10 E0 ].
Evaluate log10 E0.
The 1906 San Francisco earthquake released 6 × 1016 J of energy. What was its
magnitude on the Richter scale, if log10 6 = 0∙778?
Solution
(a)
(b)
(c)
Cont...
©The Dublin School of Grinds
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Mills & Kelly (Power of Maths)
2. Bringing logs together
Technique
Combining a string of individual logs into a single log
(a) + Logs → Multiply together on the top
(b) – Logs → Multiply together on the bottom
(c) A number multiplying a log moves into the log as a power
Example 26: Evaluate log 4 8( x 2 -1) - log 4 ( x + 1) - log 4 ( x -1).
Solution
For you to practice
Exercise 10: page 53
For you to practice (Section 2)
Revision Questions: page 56
©The Dublin School of Grinds
Page 38
Mills & Kelly (Power of Maths)
Exercises
EXERCISE 1
(c) (x − 4)2
1. Multiply out and simplify the following:
(a) (x + 2)(x + 5)
(d) (5x − 4)2
(b) (3x + 7)(2x + 3)
(e) (x 2 − 11)2
(c) ( y + 5)( y + 8)
(f) (4y − 5)2
(d) (3x − 5)(2x − 1)
(g) (ax − b)2
(e) x(x − 3)(2x + 4)
(h) (a + 1)2 − (a − 1)2
(f) (2x 2 + x + 1)(x + 1)
4. Multiply out and simplify the following:
2
(g) (x − x + 5)(x − 3)
(a) (x − 2)(x + 1)(x + 2)
(h) (x − 1)(3x 2 + 5x − 7)
(b) (2x − 1)(x + 3)(2x + 1)
2
(i) (−2x + 5x − 6)(1 − x)
(c) (2x + 3)(3x − 1)(2x − 3)
(d) (x + 1)3
2. Multiply out the following. The answers are
a difference of two squares expression, which
simplifies the process.
[Hint: (x + 1)2(x + 1)]
(e) (2x − 1)3 [Hint: (2x − 1)2(2x − 1)]
(f) (x − 2)(2x + 1)(x − 3)
(a) (x + 2)(x − 2)
5. (a) If a = x – 3 and b = 2x + 5, find the
following, in terms of x:
(b) (2x − 1)(2x + 1)
(c) (4x − 1)(4x + 1)
(i)
(d) (x 2 + 1)(x 2 − 1)
a+b
(ii) a − b
(e) (3x − 2)(3x + 2)
2
(iii) a
(f) (4x – 3y)(4x + 3y)
(iv) b2
(v)
ab
(vi) a2b
(b) If p = x 2 − x + 1 and q = 2x 2 + 3x – 2,
find the following, in terms of x:
(g) (x 2 − 5)(x 2 + 5)
(h) (xn − 3)(xn + 3)
3. Multiply out the following perfect squares:
(i) p + q
(iii) p + 2q
(ii) p − q
(iv) 2q − 3p
(a) (x + 2)2
(b) (3x + 2)2
(c) If p = 2x 2 + 5x − 1 and q = 4x3 − 2x 2 + 5x − 1,
find the following, in terms of x:
(i) p + q
(iii) 3p − 2q
(ii) q − p
(iv) q – xp
6. Find the values of the following expressions:
(a) 3x + 11y if x = −2 and y = 3⋅5
(b) 4x − 3y + 8 if x = −1, y = −6
2
(c) 5x if x = −3
©The Dublin School of Grinds
(d) −5x 2 if x = −3
(e) (5x)2 if x = 3
(f) x 2 − 3y 2 if x = 3 and y = −2
(g) 2x 2 + 5x – 7 if x = 5
(h) −5x 2 + 7x – 3 if x = −2
(i) (2x + 3y)2 if x = −3 and y = 4
(j) 2x 2y − 3xy 2 – 7 if x = 3 and y = −2
(k) 3x 2y 2 − 5x + 7(x − y)2 if x = −2 and y = 0⋅5
Page 39
Mills & Kelly (Power of Maths)
EXERCISE 2
5. (a) Find the fifth term in ( p + q)8.
1. Use your calculator to evaluate:
5
C0 , 10C0 , 18C0 , 6C0 , 8C0. What is nC0?
(b) Find the fourth term in (x + y)7.
(c) Find the third term in (2x + y)6.
2. Use your calculator to evaluate:
5
C5 , 10C10 , 18C18 , 6C6 , 8C8 and hence evaluate nCn
6. Find the term with:
3. Using your calculator show:
(a) 5C2 = 5C3
(b)
10
(a) p3 in the expansion of ( p + q)7
(c) 7C3 = 7C4
C4 = 10C6
(d)
18
(d) Find the sixth term in ( p − q)7.
(b) q5 in the expansion of ( p + q)8
C5 = 18C13
(c) (0⋅6)4 in the expansion of (0⋅4 + 0⋅6)12
Make a conclusion.
(d) (0⋅85)5 in the expansion of (0⋅85 + 0⋅15)9
4. Expand out the following:
(a) (x + 1)4
(e) (2x + 3y)3
(e) p6 in the expansion of ( p + q)10
(b) ( p + q)8
(f) (0⋅6 + 0⋅4)6
(f) q3 in the expansion of ( p + q)8
(c) (q + p)5
(g) (0⋅8 + 0⋅2)4
(d) (x − 2y)4
©The Dublin School of Grinds
Page 40
Mills & Kelly (Power of Maths)
EXERCISE 3
1. Factorise the following by taking out the highest
common factor:
4. Factorise the following difference of two
squares and simplify:
(a) 3x 2 + 9x − 18
(a) 4x 2 − 1
(b) 8a2 − 16a2b2
(b) 25x 2 − y 2
(c) 7x 2y 2 − 14x 2y
(c) x 2 − a2b2
(d) 3(x − 2y) − 5x(x − 2y)
(d) 4m2 − 81n2
(e) m(a + b) − 3n(a + b)
(e) (x + y)2 − z2
(f) (2x − y)2 − (x + y)2
2. Factorise the following by grouping:
(a) ax 2 + 2ax + x + 2
(g) (x + 1)2 − z2
(b) ax − bx + ay − by
(h) ( Yoke)2 − (Thing)2
(c) 2x − 6 − bx + 3b
(i) (3a + 2b)2 − (2a − 3b)2
(d) x 2z − 2x 2 − 2y 2 + y 2z
(j) 5042 − 4962
5. Factorise the following sums and differences
of two cubes:
(e) 3x − 8y − 2 + 12xy
2
2
2 2
(f) 21 − 3ax − 14by + 2abx y
3. Factorise the following trinomials:
2
(a) x + 5xy − 14y
2
2
(b) 10x + 13x − 3
2
(c) 7x − 22xy + 3y
2
2
2
(d) 2a + ab − 3b
(f ) 125x3 − 64a3b3
(b) x3 + 27y3
(g) (x − 2)3 + 8
(c) 8x3 − 27
(h) (x − 2)3 + (x + 2)3
(d) 1000 − 27y3
(i) x3 − (1 − y)3
(e) a3b3 + c3
(j) (Thing)3 − ( Yoke)3
6. Factorise the following fully:
(e) (a − 1)2 + 2(a − 1) − 15
(a) 2x 2 − 8
(f) 30x 2 − 17x + 2
(b) 18a2 − 8b2
(g) b2x 2 + 2bxc + c2
(c) 36x 2 + 15xy − 9y 2
(h) 4p2 − 4p + 1
©The Dublin School of Grinds
(a) x3 + 64
Page 41
Mills & Kelly (Power of Maths)
(d) x 2y + 2x 2 − y − 2
(f) x6 − y6
(e) 28 − 7x 2
(g) a2 − 2ab + b2 − a + b
(f) 4x 2 − 24xy + 36y 2
(h) x3 + x − 2x 2
(g) −2x 2 + 4xy − 2y 2
(i) x4 + 2x 2 + 1
(h) (a + 1)2 − 9
(j) a4 − a + ba3 − b
(i) 16(x − 1)2 − 4
8. (a) Derive x3 − y3 = (x − y)(x 2 + xy + y 2) from
x3 + y3 = (x + y)(x 2 − xy + y 2).
(j) 2a2 − 578
(b) Expand out (a − 4b)(a + b) + 3ab and
hence factorise this expression.
(k) 3x3 − 24
(l) x 2y3 − 27x 2
(c) Expand out (2x + 1)2 + (x − 1)2 − 2(7x − 1)
and hence factorise this expression.
(m) a2b4 − 8a2b
(n) −2x3 − 54
(d) A is the square PQRS of side x. B is the
square shown of side y.
(o) cos3 q + sin3 q
(a) x 2 + 2xy + y 2 − 81
y
(c) a2 + 8ab + 16b2 − c2
2
y
B
P y
2
(d) a − 2ab + b − 16c
C
x
Q
Using Area A – Area B = Area C + Area D, show
that x 2 − y 2 = (x − y)(x + y).
(e) x4 − y4
©The Dublin School of Grinds
R
D
(b) a2 + 6a + 9 − b2
2
x
S
7. Factorise the following fully:
Page 42
Mills & Kelly (Power of Maths)
EXERCISE 4
1. An entrepreneur bought x phones at €30 each and sold y of them at €98 each. By completing the table,
find the entrepreneurʼs net profit, in terms of x and y.
Sold
Bought
Number of phones
Price/unit
Total
F
E
2. An L-shaped flowerbed is shown with |DC| = x m and |CB| = y m.
If [AB] is 1 m longer than [CD] and [ED] is 2 m longer than [CB],
find expressions for:
(a) the perimeter P,
D
x
C
y
(b) the area A of the flowerbed, in terms of x and y.
A
B
3. The product of a number x and the square of another number y is greater than the product of the
number x squared and the number y. Find an expression for D, the difference between the bigger
number and the smaller number, in terms of x and y.
4. A petty cash box contains 10x one cent coins, 10x two cent coins, 5x five cent coins, 20x ten cent coins,
15x twenty cent coins, 7x fifty cent coins, 5x one euro coins and 3x two euro coins. It also contains 2y
five euro notes, 3y ten euro notes and one 20 euro note.
Complete the table below and find an expression for the total amount A of cash in the box in cents.
Cash type
1c
2c
5c
10c
20c
50c
1€
2€
5€
10€
20€
Number
Value in
cents
5. (a) For the rectangle shown, find an expression in terms of x and y:
(i)
for the perimeter P of the rectangle,
Rectangle
(ii) for the area A of the rectangle.
(b) If x = 20 and y = 30, what is the length of the perimeter?
What is the area of the rectangle?
©The Dublin School of Grinds
Page 43
ym
xm
Mills & Kelly (Power of Maths)
6. A farmer constructs a fence around a field in the
shape of a trapezium ABCD with [AD] parallel
to [BC] and [AB] perpendicular to [BC].
Find an expression in terms of r for:
(a) the circumference of the circle,
(b) the perimeter of the square,
(c) the area of the circle,
River
A
D
xm
(d) the area of the square,
(e) the area of the shaded region.
2y m
Fence
Fence
B
C
(a) If |BC| is 4 m longer than |AD| and |DC| is
2 m longer than |AB|, find an expression for
the length L of fencing in terms of x and y.
9. (a) A man can swim at 2 m/s and walk at
1⋅5 m/s. If he takes x seconds to swim
from A to B and y seconds to walk from B
to D, find an expression for the length of
the journey ABCD, in terms of x and y.
(b) Find an expression for the area A enclosed
by the fence in terms of x and y.
(c) Find y.
D
Water
7. (a) If x is a whole number, find an expression
in terms of x and y for:
(i) the next whole number,
(ii) the sum of these two consecutive
whole numbers.
(b) Find the values of the sum of two
consecutive whole numbers if x is the first
number by copying and completing the
following:
x=1
x=2
x=3
x=4
x=5
x=6
x=7
Road
C
B
Sum of two consecutive whole numbers
1+2=3
A
(b) A woman can swim at 1⋅8 m/s and walk
at 1⋅3 m/s. If she takes (x + 30) seconds to
swim from A to C and ( y − 10) seconds to
walk from C to D, find an expression for
the length of the journey ACD, in terms of
x and y.
10. ABCD is a rectangular frame with a picture
inside, as shown.
D
(x + 2) m
C
1m
1m
Picture
1m
xm
1m
A
Find an expression in terms of x for the
area of:
What conclusion can you make?
8. r is the radius of a circle, centre O, inscribed
in a square ABCD.
D
B
(a) rectangle ABCD,
(b) the picture,
C
(c) the border.
r
O
A
©The Dublin School of Grinds
B
Page 44
Mills & Kelly (Power of Maths)
EXERCISE 5
3. (a) If x − 1 is a factor of x 2 − k x + 3, find k.
1. Simplify the following and give your answer
in descending powers of x:
(b) If x + 3 is a factor of k x 2 + 4x – 6, find k.
(a) 2x − 5 + x − 2
(c) If 2x − 1 and 3x + 2 are factors of
ax 2 + bx + c, find a, b and c.
(b) 5x 2 − 7x − 6 – x 2 + 1
(d) If 2x − 3 is a factor of 4x 2 − 4x − 3, find
the other factor.
(c) 3x 2 + 2x 3 − 5x + 7x 3 − 5x 2 + 1
(d) 2 − 3x + 5x 2 − 8x 3 + 7x − 3
(e) If 4x − 1 is a factor of 8x 2 + k x + 7, find
the other factor and k.
(e) −x 3 + 5x 2 − 2x + 3x 3 − 2x 2 + 7x
2. Multiply out the following and give your
answer in order of descending powers of x:
(a) (x + 1)(x + k)
(f) If ax 2 + bx + 8 = (3x − 1)(5x − k), find k,
a and b.
(f) (x 2 − x + 1)(x − k)
(g) Find the quadratic polynomial with
factors:
2
(b) (2x + 1)(k x + 1) (g) (2x − 1)(k x − 5)
(c) (x − 1)(k x + 2)
(h) (3x − 1)(x 2 + k x − 2)
2
(d) (1 − 2x)(x − k)
(i) (2x + 1)(3x + kx − 3)
(e) (x 2 + x)(x + k)
(j) (x − c)(x 2 + k x + d)
(i) 5x − 1 and 2x – 3
(ii) 10x − 2 and 2x – 3
(iii) 5x − 1 and 4x – 6
(c) If x 2 − px + q is a factor of
x 3 + 3px 2 + 3qx + r, show:
(h) Find the quadratic polynomial with
factors:
__
__
(i) x + √2 , x − √2
__
__
(ii) x + √3 − 1, x − √3 + 1
__
__
(iii) 2x + √3 , x − 2√3
(i) q = −2p 2,
(ii) r = −8p 3.
(d) If x2 − a2 is a factor of f (x) = x 3 + x 2 + px + q:
4. (a) If x 2 + x + 2 is a factor of x 3 + k x 2 + lx + 2,
find the other factor and k and l.
(i) show p = q = –a 2,
(ii) write f (x) in form (x 2 − a 2)(x + r).
(b) If 2x2 + bx − 5 is a factor of 2x 3 − x2 + kx + 10,
find the other factor and b and k.
(e) If 3 − x is a factor of P (x) = 6 + x − 4x 2 + x 3,
find the other factors of P (x).
(c) ax 2 + 2 is a factor of 18x 3 + k x 2 + 6x – 2.
Find k and a and the other factor.
(f) If x − a is a factor of x 3 – c, show c = a 3.
(d) If x − 1 is a factor of x 3 + 2x 2 − k x + 4,
find k ∈ R and the other factors.
(e) If x − 2 is a factor of P (x) = x 3 − 2x 2 − 3x – k,
find k and the quadratic factor of P(x).
(f) If (x + 1) and (x − 3) are both factors of
k x 3 − 6x 2 + bx – 6, find k, b ∈ R and the
other factor.
2
(h) If (x − 1)2 is a factor of ax 3 + bx 2 + 1,
find a and b.
(i) If (x − a)2 is a factor of x 3 + 3rx + q, show:
(i) r = −a2
(ii) q = 2a 3
3
5. (a) If (x − p) is a factor of x + qx + r, show:
(i) r = 2p 3, (ii) q = −3p2.
(b) If (x − 1) is a factor of x 3 + ax2 − (a + 1)2 + 12,
a > 0, find a ∈ R.
©The Dublin School of Grinds
(g) If x 2 − px + 1 is a factor of ax 3 + bx + c,
show c 2 = a(a − b).
Page 45
(j) If x 2 + ax + b is a factor of x 3 + qx 2 + rx + s
show:
(i) r = b + a(q − a)
(ii) s = b(q − a)
Mills & Kelly (Power of Maths)
EXERCISE 6
1. Simplify the following:
x−2
(a) _____
2−x
x 2 − 2x
(b) _______
2−x
5x 2 + 15x
(c) _________
5x
3
x +x
(d) ______
x2 + 1
x3 + x2
(e) _______
x+1
15x 2 + 11x − 14
(f) _____________
3x − 2
3
3x − 24
(g) ________
3x − 6
3 + 3x 3
(h) ___________
3x 2 − 3x + 3
16x 3 − 28x 2 + 6x
(i) _______________
8x − 2
2
−4x + 36
(j) _________
x−3
2. Simplify the following:
x3 + x2 − x − 1
(g) ____________
x+1
6x 3 + x 2 − 2x
(h) ___________
2x − 1
27x 3 − 1
(i) _______
3x − 1
x 3 − 7x + 6
(j) _________
x 2 + 2x − 3
3. The volume of the box shown is given by
V = 2x 3 + 7x 2 + 7x + 2.
x 3 + 7x 2 + 14x + 8
(a) _______________
x+2
h
x 3 − 4x 2 + 5
(b) __________
x+1
2x + 1
x+1
4x 3 − 11x + 3
(c) ___________
2x − 3
Find: (a) h in terms of x, (b) the surface area
A, in terms of x.
10x 3 − 31x 2 + 27x − 30
(d) ___________________
2x − 5
4. If x − 1 is a factor of x 3 + kx 2 − 4x + 1,
show that k = 2.
x 3 − 25x
(e) _______
x−5
5. If x − 2 is a factor of x 3 + ax 2 + bx + 6,
show that b + 2a = −7.
−6x 3 − 7x 2 + x − 3
(f) _______________
3x 2 − x + 1
©The Dublin School of Grinds
6. If x 2 − 5x + 11 is a factor of x 3 − 2x 2 + ax + 33,
show that a = −4.
Page 46
Mills & Kelly (Power of Maths)
1. Simplify the following:
2p
4
(a) _____ + _____
p−2 2−p
x 1
(b) __ − __
2 x
x+1
x
(c) _____ − _____
x−1 x−2
α β
(d) __ − __
β α
EXERCISE 7
3
3
__
(h) ( __
a − 2 )( a + 2 )
( a3 − 2b )
2 3 x
( j) ( + )( − 4y )
x y 2
(i)
(g)
2 + 2n
−
2 − 2n
+
1 − n2
(d)
(g)
y 3 − 9y
b−9
2
4
x
+
x−3
__________
4x 2 − 7x + 3
4
(h)
(i)
4
x −y
______
x2 − y2
6
6
x −y
__________________
(x 3 − y 3)(x 2 − xy + y 2)
2
3x
(b) ___
_2_
x
x
__
2
(c) ___
3x
1
x − _x_
(e)
1
2 + _x_
_____
1
2 − _x_
3 10
1 − __ + ___2
x x
(f) __________
2 15
−1 + __ + ___2
x x
at
– at 2
______
(g) a a
__ __
−
t t2
(d) _____
x−1
_____
x
2
x + 6x − 7
_________
x 2 − 5x + 4
2
3
y + 27
______
3a 2 + a
_3_
x
(a) __
x
__
x+2
3x 2 − 11x + 10 ______
×
(e) ____________
2
x −4
3x − 5
×
2
6a + 2a
________
2
6b − 51b − 27
_____________
4. Simplify the following:
a 2 − 3a
a 2 + 8a + 15 _______
×
(d) ___________
ab + 5b
a2 − 9
2x 2 + 11x − 21
)
−5a 2 + 17a − 6
6x 3 − 7x 2 + 2x
(e) _____________ (j) ____________
3−a
2x 2 − x
2a 2 28b 2
(a) ____2 × ____
4ab
35b
3x
−
1
(b) ______
3 − 9x
4b 2 − 2ab
3ab
_________
(c) _________
×
8ab
3a 2 − 6ab
(f )
)(
6x − 18
(c) _______
15 − 5x
2n
______
2. Simplify the following:
2
__ __
3. Simplify the following:
4x 2
(f)
(a) ___
2x
2x
(b) ___2
(g)
4x
2a
a−1
a+1
(h) ______ − _______ + ________
2
4a + 8 5a − 10 5a − 20
x − 2 _____
3x
4
_____
(i) _____
−
+
x2 − 1 x2 + x x + 1
7
1
( j) ___________
+ _________
2
2
x + 13x + 30 x + 5x + 6
2x − 11x + 12
____________
__
(
1
(f) m − __
m
1+n
______
__ 2
x
x
(k) __ − 1 __ + 1 (4x − 8)
2
2
1 x − 3 4x − 1
(e) __ + _____ − ______
24x
9 18x
1−n
______
__
×
y + 3y
______
y+3
©The Dublin School of Grinds
Page 47
Mills & Kelly (Power of Maths)
EXERCISE 8
1. Evaluate the following without using your
calculator:
(a) 33
(h) (−2)4
(b) 25
(i)
(c) 54
(d) 10
(j)
8
(e) 12013
(f) (−1)
(k)
4
(g) (−1)21
(l)
(−3)5
__
2
(o)
2
__
2
( 3 _12 )
2
( 2 _34 )
3
( −2 _12 )
(n)
( 21 )
( 32 )
( − 12 )
__
2
( )
4
(m) − __
5
(p)
3
(c) 4
−3
(f) (3⋅6)−1
(g)
(4)
3
__
−1
(h) 2−1 × 3
©The Dublin School of Grinds
(l)
(c) 25 2
__4
__1
−3
__1
−2
(h) 2 × 16 2
(i) 6 × 100
(k) 3 × 4
−1
−1
3 ×2
(l)
(m) (−8)−2
2
__3
__5
(o) (3 × 2 )
Page 48
(
(s)
8
( − ___
27 )
(t)
8
__1
3
8
− ___
27
)
__1
3
__1
−3
0
( 27 )
( − 271 )
1 − __13
___
__1
___ − 3
__1
−2
( )
__1
(9)
1
__
(n) (−8) 3
−1 −3
(r)
1
(w) 2 __
4
__1
−2
(m) (100 000 000) 4
(n) (−3)−3
8
( ___
27 )
(v)
__1
(j) 4 × 49 2
1
________
(q)
(u)
__3
−2
__1
−2
( __94 )
(d) 8 3
__1
−2
__1
−2
(p)
__3
__1
3
(k) ___
2
(e) 2−4
__1
(g) 6 × 36
−1
(d) 5−2
(o) 100
(b) 25 2
(f) 64
(i) 2 × 3−1
2
(j) ___
−1
3
(b) 3−2
__3
(a) 9 2
(e) 36
2. Evaluate the following without using your
calculator:
(a) 2−1
3. Evaluate the following exactly without using
your calculator:
2
( ) (2)
25
(x) ___
16
(y)
__1
−2
3
__
( 2) × ( 9 )
1
– __
0
16
___
− __32
Mills & Kelly (Power of Maths)
6. Simplify the following, giving all your
answers with positive powers:
(a + 3b)2(a + 3b)6
x 2 × (xy)2
_______________
(l)
(a) ________
y
(a + 3b)3
4. Write:
__
p
(a) 8√2 in the form 2 , p > 0
__
27
√3
p
(b) _____ in the form 3 , p > 0
3
__
4
√2
1
(c) ____ in the form __p , p > 0
32
2
__
(d)
49√7
_____
p
√7
(d)
__2
(e)
28x 4
(c) ____
7x 3
in the form 7 , p > 0
__
3
5x 3 × 6x 4
(b) ________
15x 2
2
3
125 × __
5
________
25 × √5
p
in the form 5 , p > 0
23
(e)
15
(g)
p
( 2 ) __× 2 × 3
4
3
__
1
___
in the form
−2
3__
q , p > 0, q > 0
2
27 × 3
p
__ in the form 3 , p > 0
(i) _______
9√3
(j)
(3)
5
__
−1
__3
−2
)3
( 25
9
p
in the form 5 , p > 0
× _____
× _____
__2
−2
3
1253
p
5. Write in the form a or
7
(a) a × a
2
(b) (a 7)3
7
−3
(c) a × a
a7
(d) ___3
a
__
3
(e) √ a
__
(f) a√a
3
__
(g) (a√ a )3
a14
(h) ___
a2
a
(i) __3
a
©The Dublin School of Grinds
1
__
a
p,
(j)
where p ∈R, p > 0:
√a
(m)
3
72x 5
42x 5
________
7x 2 × 3x 3
x
2_______
× 2 3x
(j)
(k)
(o)
( )
2
x___
y
2y 2
__3
(q)
4x 2
___
__1
2x 2
−3
(r)
25y
_____
5y− 2
__4 __5
(s)
8x 3 y 2
________
__1
__3
4x 3 × 2y 2
4
3z 3
7y 2 × 25y 2
__________
2 × 22x × 3 × 3x
a−2
___
2a 2
(p) ___
y−2
2 2x
5x
(2 ) × (3 )
_____________
__3
(t)
16(
x 2y) 2
_______
__
8√y x
5xy 3 × 14xy
7. The population P of a certain strain of bacteria
is given by P = 1500 × 3t, where t is the time
in hours.
2
a__
___
(a) Find the population P1 after 4 hours.
a2
(k) ___
a−3
(l)
36
x
____
x ×2
(h) _______
2x × x3
(3xy)2 × 2xy
(i) __________
9(x 2y)3
2
3
(n)
3
4
16√2
__ in the form 2 p, p > 0
(h) _____
√8
__2
(g)
x 3
x 2
70x 3(xy)4
(f) _________2
5xy × 7xy
10 × 2 × 10
p
(f) _______________4 in the form 10 , p > 0
19
−2
2 × 10 × ( 10 )
xyab 2
(m) ______
abx 2 y 2
(b) Find the population P2 after 10 hours.
__
(c) Compare these populations by dividing
P2 by P1.
√ a2
__
√a × (a 2)3
__________
8. The mass M of a radioactive material in
grams (g) left after t hours is given by
20
M = ___t .
16
(a) Show that M = 20 × 2−4t.
a−2 × (a 3)−1
__1
−
__1
(____________
a−2) 2 × (a 3) 2
__
(n)
√a
__
__3
3
−2
a × (a )2
√__________
(o)
−2 2
−1
(a ) × a
(b) Compare the mass M2 left after (x + 7 )
hours with the mass M1 left after (x + 6)
hours by dividing M2 by M1.
Page 49
Mills & Kelly (Power of Maths)
9. The mass M of a drug in milligrams (mg)
in a person’s body after t hours is given by
10. The value V in € of an investment after t years
is given by V = 5000(1⋅08)t.
_____
M=
√8
3 _____
8000 .
3t
(a) Compare the amount V2 after (t + 3) years
with the amount V1 after t years, giving
your answer to four decimal places.
(a) Show that M = 20 × 2−3t.
(b) Find M after 1 hour.
(c) Find the percentage change in M between
t and (t + 1).
(b) Find the value of the investment after
five years, correct to the nearest euro.
(c) Find the value of the investment after eight
years, correct to the nearest euro.
EXERCISE 9
1. Write the following surds in their simplest form:
___
____
(a) √12
(d) √512
__
___
8
(b) √27
(e) __
9
___
_____
18
(c) √1210
(f) ___
25
√
√
©The Dublin School of Grinds
Page 50
____
____
(g)
√___
100
(h)
√4x 3
147
____
_____
(i) √ 8x 2 y 3
(j)
√
√
12
x3
____
y2
_____
(k)
16
z 61
_____
9x 2y 2
Mills & Kelly (Power of Maths)
2. Simplify the following surds:
___
___
___
(a) √11 − 2√11 + 4√11
__
__
(b) 5 + 7√3 − 9 + 8√3
__
__
__
(c) x + √y − 3√y + 2x − 5√y
___
___
____
(d) √28 + √63 − √175
____
____
____
(e) √125 − 2√180 + √245
__
___
___
___
(f) √8 + √12 + √18 + √27
___
___
____
(g) √75 − 3√48 +__√147
__
__
(h) √x___
+ x√x −___
√x 3 ____
3
3
(r)
(s)
3 3
(i) a√ay + y√a y + √a y
_____
_______
_____
________
(j) √x − 3 − √4x − 12 + 2√x − 3 − √25x − 75
3. Multiply
__ out and simplify the following surds:
3
(a) (√2 )
__
__
(b) √2 × √3
__
___
(c) 2√6 × 5√18
__ __
__
__
(d) √6 (3√2 + 2√3 + √6 )
__
2
(e) (√3 + 1)
__
__
(f) (√3 − 1)(√3 + 1)
___ __
__
(g) √ab (a√b + b√a )
__
___
(h) (√7 + √13 )2
__
___ __
___
(i) (√7 + √13 )(√7 − √13 )
__
__
(j) (2√x + 3√y )2
_____
_____ _____
_____
(k) (√x + 2 + √x + 1 )(√x + 2 − √x + 1 )
__ __
__
(l) √x (2√x − 3) + 5(√x − x)
_____
_____
1
(m) __ (√a + x − √a − x )2
2
__
__
(n) (x + √2 )(x − 1)(x − √2 )
1
__
1
__
(o) (x 2 + 1)(x 2 − 1)
(p)
(q)
( __ )
(
y__
√x − ___
___
y
√x
2
)
2
4. Rationalise each denominator:
__
6
√
1__
__
(f) ____
(a) ___
(k)
2√7
√5
x__
1__
(g) ___
(b) − ___
(l)
√__y
√2
2__
√__x
(c) ___
(h) ___
(m)
3
√
√y
4__
(d) ____
__1
(i) ______
5√2
(n)
√2 + 1
−3__
____
1 __
(e)
(j) _______
2√3
(o)
3 − 2√2
__________
__ 6 __
3√
__2 − 2√3
√__5 + 1
______
√5 −__1
5√x
________
__
3 − 2√x
__
__
√__3 + √__2
________
√3 − √2
__________
__ x _____
√a − √a – x
__
5. If y = 2 − √3 , evaluate y 2 − 4y + 1.
6. BCD is an isosceles triangle.
D
4 + x2
4 + x2
h
B
8 – 2x 2
C
Find:
(a) the height h in terms of x,
(b) the area of the triangle in terms of x.
__
__
__
__
7. If x = √3 + √2 and y = √3 − √2 ,
find x2 + xy + y2. _______
______
y2 − x2
x2 √
________
__
.
8. Show 1 − 2 =
y
y
√
( ___√1__x__− 1)( ___√1__x + 1__)
( 4√y + ___√3__y )( 4√y − ___√3__y )
©The Dublin School of Grinds
__
___
√x + 2__
√x
___________
√
1
9. If V = 20h2 ____2 − 0·05 , show that V can be
25h
_______
written as V = 2h √ 4 − 5h2 .
Page 51
Mills & Kelly (Power of Maths)
10. Kepler showed that the periodic time P for
a satellite to make a complete orbit around
a planet of mass M at a distance r from the
centre of____
the planet is given by
P = 2π
√
12. The mass M of an object moving at a speed
M0
______ , where M is the
v is given by M = _______
0
v2
1 − __2
√
GM
, where G is a constant.
(a) Show that M can be written as
_______
2
2
M
0 c √c − v
____________
M=
.
2
2
c − v
5M0
4c
(b) If v = ___, show that M = ____ .
5
3
r
M
c
mass at rest and c is the speed of light.
3
r
____
13. A simple pendulum consists of a mass M on
the end of a string of length L moving back
and forth as shown.
(a) Write down the periodic times:
L
L
(i) P1 for a satellite at r = R,
(ii) P2 for a satellite at r = 4R.
P1 1
(b) Show that ___ = __.
P2 8
M
The time to swing from right to left and back
11. If x, y and z are consecutive terms in a
y z
geometric sequence, then __x = __y . Show that
the following numbers are consecutively in a
geometric sequence:
___
___
(a) 5, √35 , 7
(b) a, √ab , b
©The Dublin School of Grinds
M
__
√
L
again is given by T = 2π __
g , where g is a
constant.
(a) Write down the time T1 if L = d.
d
(b) Write down the time T2 if L = __ .
4
T
__1
(c) Show that = 2.
T2
Page 52
Mills & Kelly (Power of Maths)
EXERCISE 10
1. Evaluate the following exactly:
( )
( )
1
(f) logp __2
p
(a) log10 100
−5
(b) log10 (10 )
(c) log10
(g) log3 e x + log3 e 2x − 1
y
x4
__
(h) log x __
+
log
x
y
x3
3
___
( )
(g) log5 √ 25
( )
1
___
1
2
(i) logk (x − 1) + logk _____
x+1
(j) logx 2 x + logx 2 −x
(h) log__1 16
10
__
(d) loge√e
2
(i) log√__2 4
()
1
(e) logk __
k
( )
4. Write the following as a single log (in the
form log a x):
(j) log__1 27
(a) log2 6 − log2 3
9
(b) log3 3a − log3 a
2. Write the following logs in exponential
(power) form:
(a) log17 289 = 2
(f) log3 3 = 1
(b) log4 16 = 2
(g) loga 1 = 0
1
(c) log4 ___ = −2
16
(c) loge (x 2 − 1) − loge (x + 1)
1
(d) log5 7 − log5 __
7
(e) log4 (x 3 − 1) − log4 (x 2 + x + 1)
(h) logb a = c
(f) log3 e x + 1 − log3 e x − 1
(d) log16 4 = _12
(i) log8 16 = _43
( )
()
1
1
(e) log16 __ = − __
4
2
()
(g) log9 (3x + 6) − log9 (x + 2)
(h) logb (ba + b2) − logb (a + b)
__
__
(i) log√__x √xy − log√__x √y
(j) log16 8 = _4
3
()
1
(j) loga x − loga __
x
3. Write the following as a single log (in the
form loga x) and simplify, where possible:
5. Write the following as a single log (in the
form loga x):
(a) log2 3 + log2 5
(a) 3 log2 5
(b) log3 a + log3 4a
(b) −2 log3 7
(c) loge (1 − x) + loge (1 + x)
1
_
(c) _12 log4 25
(d) log5 7 + log5 7
__
__
(e) log5√y + log5√y
(d) − _2 log2 36
(e) 2 logk y
3
(f) log4 (x − 1) + log4 (x 2 + x + 1)
©The Dublin School of Grinds
Page 53
Mills & Kelly (Power of Maths)
_____
(f) 2 log5 √ 1 + x 2
1
_
2 2
(g) − 2 log3 z y
__3
(h) _23 log7 8 y 2
(v)
log10 30
(vi)
log10 81
( )
( )
()
1
(j) _12 log5 __2
y
6. (a) Write log3 5 in base 2.
(x)
(b) Write log7 4 in base 5.
(c) Write loga y in base b.
log3 12
8. (a) Write the following as a single log and
hence, evaluate exactly, where possible:
(d) Write log5 (x − 2) in base 7.
(e) Write log10 a in base e.
(f) Write log5 3 in base 3.
(g) Write log4 x in base x.
(i)
log10 25 + log10 4
(ii)
log3 21 − log3 7
(iii) _12 log10 4 + log10 35 − log10 7
(h) Write logx 5in base 5.
(iv)
1
(i) Write _____ in base x.
log2 x
1
(j) Write _____ in base x.
loge x
log3 2 + 2 log3 3 − log318
(b) Write the following as a single log:
7. (a) Write the following as a string of logs:
__
2
(i)
loga x √y
(i)
2 log5 x + log5 y
(ii)
3 logk x + 2 logk y − _14 logk z
3
(iii) _12 log3 (x − 1) − _2 log3 ( x + 1 )
log2 (u 2v 3)
__
x
3
√
(iii) log5 ____
p 2q 3
(iv)
( )
(v)
7{log3 (x + 5) + log3 2 − 3 log3(4x − 1)}
__
__
1
log4 a 2 + log4 a + log4√a
x+1
1
_
_____
1
_
2
4 loga (x + 3x + 2) + 4 loga x + 2
( )
( )
(vi)
( )
(viii) x loge x − _12 loge x + e loge x
3
27x
____
(iv)
log3
(v)
1
loga __2
a
logk (x + 3)2(2x − 1)3
(vi)
log10 60
16
(vii) log10 ___
27
400
(viii) log10 ____
27
(ix) log10 48
(i) x loge 2
(ii)
(iv)
(vii) _12{loge 9 − 3 loge(x2 + 1) + 2 loge(3x + 2)}
3y 6
(
(x + 7)7
______
(vii) log2 ________
4√2x − 3
( )
x−1
(viii) log5 _____
x+1
__1
9. If log2 y = log2 x + 2, show that y = 4x.
)
10. If loge A + 2 loge y = 3 loge y + D,
show that y = Ae−D.
11. If f (x) = loga x, show that
2
f (x + h) − f (x)
____________
(b) If a = log10 2 and b = log10 3, express the
following in terms of a and b:
(i)
log10 6
(ii)
log10 24
(x + h) − x
(
h
__
)
__1
h
= loga 1 + x .
12. If log5 y = 2 log5 x − log5 (x + 1) + c,
x2
show that y = _____ 5c.
x+1
()
9
(iii) log10 __
8
©The Dublin School of Grinds
Page 54
Mills & Kelly (Power of Maths)
16. Show that:
13. Use the rules of logs to show that:
_____
_____
(a) loga (x + √ x 2 − 1 ) + loga (x − √ x 2 − 1 ) = 0
__
_____
__
_____
(b) logb (√ x + √ x − 1 ) + logb (√ x − √ x − 1 ) = 0
x
(a) loga a = x
(b) a loga x = x
14. Simplify the following:
7
(a) log3 3
(b) 3
log3 4
__
(c) log5√5
(d) 10
17. The loudness L of a sound in decibels is given
I
by L = 10 log10 __ , where I is the intensity of
I0
()
log10 3
(e) logx (x 8)
1
__
(f) e 2
the sound in W m−2 and I0 = 10−12 W m−2.
loge x
(a) Find:
(i) the loudness in decibels of normal
conversation which has an intensity
of I = 10−7 W m−1,
Check the answers to (a), (b), (c) and (d) on
your calculator.
logc a
15. (a) Using logb a = _____, solve for logc a.
logc b
Hence, simplify:
(ii) the loudness of amplified rock music
which has an intensity of 10−1 W m−2.
log2 3 × log3 4 × log4 5 × log5 6 × log6 7 × log7 8
(b) Simplify:
log2 2 × log2 4 × log2 8 × … × log2 2n
(b) If a sound has a loudness L1 for intensity I1
and loudness L2 for intensity 1000I1, show
that L2 − L1 = 30.
1
1
1
1
1
1
18. Show that _____ + _____ + _____ + _____ + _____ = _______ .
log2 x log3 x log4 x log5 x log6 x log720 x
19. Evaluate the following exactly:
(a) log3 354
(b) 3
log3 5 − log3 2
(c) 10
2 − log10 2
(d) log3 8 × log8 9
20. If log10 x = (1 + p) and log10 y = (1 – p), show that x y = 100.
©The Dublin School of Grinds
Page 55
Mills & Kelly (Power of Maths)
REVISION QUESTIONS
__
__
a
−
b
5
3
−
√__
√5
in the form _______ , a, b ∈N.
1. (a) Express ______
3 + √5
2
(b) If 2x − 1 is a factor of 2x 3 + x 2 + k x + 6, k ∈Z,
find k and the other factors.
(c) If ax + b is a factor of 2ax 2 + (2b − a)x + c,
find the second factor and show that b = −c.
__
__
y−x
_____
2. (a) If x = 1 − √3 and
__ y = 1 + √3 , express xy
in the form a√b , a ∈Z, b ∈ N.
x2 − 4
7. (a) If x = 999 999 999 999, evaluate ______ exactly.
x−2
_____
2009
√ 10 _____ exactly.
_______________
(b) (i) Evaluate _____
2011
√ 10 − √ 102007
___
___
2 2
(ii) Simplify 1 + √2 x − √8x 2 .
(
(c) Show: (i) 2n + 1 + 2n = 3 × 2n
(iii)
2
2
− b 2 c______
− d2 .
(c) If _a_ = _c_ , show that a______
=
b d
a2 + b 2 c 2 + d 2
_7_
=
32009 + 32010 3
5 × 2x − 8 × 2x − 2
_______________
2x − 2x−1
=6
__1 __3
per week is given by N1 = 50x 2 y 2 , where x is the
average number of workers that attend per week
and y is the average number of hours worked by
each worker per week.
(a) Find the number of phones produced in a week
in which the average attendance is 256 workers
and the average number of hours worked is 36.
)
(b) For another factory producing the same
__3 __1
phone N2 = 20 x2y2 . Find the number of phones
produced by this factory in a week in which
the average attendance is 256 workers and the
average number of hours worked is 36.
N1 5y
(c) Show that ___ = ___ .
N2 2x
(d) Show that for a 40-hour week in each factory,
an attendance of 100 workers in each will
produce the same number of phones per week.
3x − 5
1
4. (a) Show that ______ + _____ , x ≠ 2, simplifies to
x−2 2−x
a constant. __
__
√x
√x
__
__
+ ______
as a single fraction.
(b) Express ______
√x − 1 √x + 1
(c) (i) If a − b(c − x)2 = 6x − 7 − x 2 for all x ∈R.
Find a, b, c ∈N.
(ii) If x 2 + x + 1 is a factor of
2x 3 + ax 2 − x + b, find a, b ∈Z.
9. The stopping distances in metres (m) of a car
__5
5. (a) Expand ( p + q) .
3
(b) What is the term with q 3 in the expansion of
( p + q)7?
(c) Show that p3 + q 3 − ( p + q)3 = −3pq( p + q).
3
3
simplifies to a
6. (a) Show that ______p + ______
1+x
1 + x −p
constant.
6y
3
(b) (i) Express ________ − ___ as a single fraction.
x(x
__ + 4y) 2 x
__
2
(ii) If 2x − √3 is a factor of 4x − kx + √3 ,
find the other factor and k ∈R.
2
(c) If x − ax − 3 is a factor of x 3 − 5x 2 + bx + 9,
find a, b ∈N.
©The Dublin School of Grinds
2011
8. The number of phones produced by a factory
3. (a) If 5x 2 − 20x + 8 = a(x + b)2 + c for all x, find
a, b, c ∈Z.
by
cz
ax
(b) If _____ = _______ = _______ , show that
b − c 2(c − a) 3(a − b)
6ax + 3by + 2cz = 0.
__
__
1
1
(c) Express x 2 + √2 + __2 x 2 − √2 + __2 in the
x
x
1
__
form x n + n , where n ∈N.
x
)(
2008
3
+3
___________
(ii)
3
4
(b) Simplify ___________
.
− ___________
2
2
x + 12x + 20 x + 14x + 40
(
)
Page 56
___
v3
travelling at v km/h is given by S1 = 20 for a wet
__4
__
v3
road and S2 = 6 for a dry road.
(a) Copy and complete the table below. Give the
values in the last two columns, correct to two
decimal places, and make a conclusion.
5
__
v km/h
4
__
v3
27
64
125
S__1
3
v3
S1
S2
__
3√ v
(b) Show that = ____
.
S2 10
Mills & Kelly (Power of Maths)
(c) If the speed of a car is 74⋅088 km/h, compare
its stopping distance on a wet road to that on
a dry road.
1000
(d) At v = _____ km/h, show that S1 = S2.
27
10. (a) Show that
6
3
log
10 x − log10 x
_______________
log10 x 2 − log10 x
12. (a) A boy rows downstream for 2 km at a speed of
(x − 2) km/h from a point A to point B.
Write down an expression for the time of the
journey in terms of x, x > 0.
2 km
= 3.
A
log2 a log2 b log2 c
(b) If _____ = _____ = _____ = log2 x,
p
2p
3p
(b) A man rows the same journey at a speed of
(x + 2) km/h. Write down an expression for
the time he takes.
2
b
___
show that ac = 1.
(c) If x 2 + y 2 = 14x y, show that (x + y)2 = 16x y.
loga x + loga y
x+y
Hence, show that _____________ = loga _____ .
2
4
1
11. The power P of a lens is given by P = __ where
f
f is its focal length in metres (m).
(c) Which one takes the shorter time? Why?
( )
(a) The focal length of lens A
is x m.Write down an
expression for its power,
in terms of x.
(c) When lens A and B are
combined, the power
of the combination is
obtained by adding their
powers together. Find an
expression for the power
of the combination as a
single fraction, in terms
of x.
(d) Write down an expression for the difference
in times. Give your answer as a single
a
fraction in the form _______ .
2
x − b2
(e) If x = 12, find this difference to the nearest
second.
13. (a) The number of bacteria in a sample A after
t minutes is given by 50 × 2t + 6.
Lens A
(b) Lens B has a focal length
of − (x + 2) m. Write down
an expression for the
power of lens B, in terms
of x.
Express this number in the form a × 2t.
Find the number of bacteria initially in the
sample.
(b) The number of bacteria in a sample B
after t minutes is given by the expression
100 × 4t.
Lens B
Write this expression in the form b × 22t.
Find the number of bacteria initially in this
sample.
Lens A
Lens B
(c) Find the ratio of the number of bacteria in
sample B to that in sample A after t minutes
in the form 2 p. What is this ratio after:
(i) 2 minutes,
(ii) 5 minutes,
1
_
(d) If x = 2 , find the power of this combination
as a fraction. If the focal length of the
combination is one divided by the power,
find this focal length as a fraction.
©The Dublin School of Grinds
B
(iii) 10 minutes?
(d) Make a conclusion.
Page 57
Mills & Kelly (Power of Maths)
Answers
Section 2
Chapter 3
Exercise 1
1. (a) x 2 + 7x + 10 (b) 6x 2 + 23x + 21 (c) y 2 + 13y + 40
(d) 6x2 – 13x + 5 (e) 2x3 – 2x2 – 12x (f ) 2x3 + 3x2 + 2x + 1
(g) x 3 – 4x 2 + 8x – 15 (h) 3x 3 + 2x 2 – 12x + 7
(i) 2x 3 – 7x 2 + 11x – 6 2. (a) x 2 – 4 (b) 4x 2 – 1
(c) 16x 2– 1 (d) x 4 – 1 (e) 9x 2 – 4 (f ) 16x 2 – 9y 2
(g) x 4 – 25 (h) x 2n – 9 3. (a) x 2 + 4x + 4
(b) 9x 2 + 12x + 4 (c) x 2 – 8x + 16 (d) 25x 2 – 40x + 16
(e) x 4 – 22x 2 + 121 (f ) 16y 2 – 40y + 25
(g) a 2x 2 – 2abx + b 2 (h) 4a 4. (a) x 3 + x 2 – 4x – 4
(b) 4x 3 + 12x 2 – x – 3 (c) 12x 3 – 4x 2 – 27x + 9
(d) x 3 + 3x 2 + 3x + 1 (e) 8x 3 – 12x 2 + 6x – 1
(f ) 2x 3 – 9x 2 + 7x + 6 5. (a) (i) 3x + 2 (ii) – x – 8
(iii) x 2 – 6x + 9 (iv) 4x 2 + 20x + 25 (v) 2x 2 – x – 15
(vi) 2x 3 – 7x 2 – 12x + 45 (b) (i) 3x 2 + 2x – 1
(ii) – x 2 – 4x + 3 (iii) 5x 2 + 5x – 3 (iv) x 2 + 9x – 7
(c) (i) 4x3 + 10x – 2 (ii) 4x3 – 4x2 (iii) – 8x3 + 10x2 + 5x – 1
(iv) 2x 3 – 7x 2 + 6x – 1 6. (a) 32·5 (b) 22 (c) 45
(d) – 45 (e) 225 (f ) –3 (g) 68 (h) –37 (i) 36 (j) –79
(k) 56·75
©The Dublin School of Grinds
Exercise 2
1. 1, 1, 1, 1, 1; 1 2. 1, 1, 1, 1, 1; 1
3. (a)–(d) nCr = nCn − r 4. (a) x 4 + 4x 3 + 6x 2 + 4x + 1
(b) p 8 + 8p 7q + 28p 6q 2 + 56p 5q 3 + 70p 4q 4 +
56p 3q 5 + 28p 2q 6 + 8pq 7 + q 8
(c) q 5 + 5q 4p + 10q 3p 2 + 10q 2p 3 + 5qp 4 + p 5
(d) x 4 – 8x 3y + 24x 2y 2 – 32xy 3 + 16y 4
(e) 8x 3 + 36x 2y + 54xy 2 + 27y 3
(f ) (0·6) 6 ‒ 6(0·6) 5 (0·4) + 15(0·6) 4 (0·4) 2 + 20(0·6) 3
(0·4) 3 + 15(0·6) 2 (0·4) 4 + 6(0·6) 1(0·4) 5+ (0·4) 6 = 1
(g) (0·8) 4 + 4(0·8) 3 (0·2) + 6(0·8) 2(0·2) 2 + 4(0·8)
(0·2) 3 + (0·2) 4 = 1 5. (a) 70 p 4q 4 (b) 35x 4 y 3
(c) 240x 4y 2 (d) ‒21p 2q 5 6. (a) 35 p 3q 4 (b) 56 p 3q 5
(c) 495(0·4) 8 (0·6) 4 = 0·042
(d) 126(0·85) 5 (0·15) 4 = 0·028 (e) 210p 6q 4
(f ) 56 p 5q 3
Exercise 3
1. (a) 3(x 2 + 3x – 6) (b) 8a 2(1 – 2b 2) (c) 7x 2y( y – 2)
(d) (x – 2y)(3 – 5x) (e) (a + b)(m – 3n)
2. (a) (x + 2)(ax + 1) (b) (a – b)(x + y) (c) (x – 3)(2 – b)
(d) (z – 2)(x 2 + y 2) (e) (3x – 2)(1 + 4y)
(f ) (3 – 2by 2)(7 – ax 2) 3. (a) (x – 2y)(x + 7y)
(b) (5x –1)(2x + 3) (c) (7x – y)(x – 3y) (d) (2a + 3b)(a – b)
(e) (a – 4)(a + 4) (f ) (5x – 2)(6x – 1) (g) (bx + c)(bx + c)
(h) (2p – 1)(2p – 1) 4. (a) (2x –1)(2x + 1)
(b) (5x – y)(5x + y) (c) (x – ab)(x + ab)
(d) (2m – 9n)(2m + 9n) (e) (x + y – z)(x + y + z)
(f ) 3x(x – 2y) (g) (x – z + 1)(x + z + 1)
(h) (Yoke – Thing)(Yoke + Thing)
(i) (a + 5b)(5a – b) ( j) 8000 5. (a) (x + 4)(x 2 – 4x + 16)
(b) (x + 3y)(x 2 – 3xy + 9y 2) (c) (2x – 3)(4x 2 + 6x + 9)
(d) (10 – 3y)(100 + 30y + 9y2) (e) (ab + c)(a2b2 – abc + c 2)
(f ) (5x – 4ab)(25x 2 + 20abx + 16a 2b 2)
(g) x(x 2 – 6x + 12) (h) 2x(x 2 + 12)
(i) (x + y – 1)(x 2 + y 2 – xy + x – 2y + 1)
( j) (Thing – Yoke)((Thing) 2 + (Thing)(Yoke) + (Yoke) 2)
6. (a) 2(x – 2)(x + 2) (b) 2(3a – 2b)(3a + 2b)
(c) 3(4x + 3y)(3x – y) (d) (y + 2)(x – 1)(x + 1)
(e) 7(2 – x)(2 + x) (f ) 4(x – 3y) 2 (g) –2(x – y) 2
(h) (a – 2)(a + 4) (i) 4(2x – 3)(2x – 1)
( j) 2(a – 17)(a + 17) (k) 3(x – 2)(x 2 + 2x + 4)
(l) x 2(y – 3)(y 2 + 3y + 9) (m) a 2b(b – 2)(b 2 + 2b + 4)
Page 58
Mills & Kelly (Power of Maths)
(n) –2(x + 3)(x 2 – 3x + 9)
(o) (cosq + sinq )(1 – sinq cosq )
7. (a) (x + y – 9)(x + y + 9) (b) (a – b + 3)(a + b + 3)
(c) (a + 4b – c)(a + 4b + c) (d) (a – b – 4c)(a – b + 4c)
(e) (x 2 + y 2)(x – y)(x + y)
(f ) (x – y)(x + y)(x 2 + xy + y 2)(x 2– xy + y 2)
(g) (a – b)(a – b – 1) (h) x(x – 1) 2
(i) (x2 + 1)2 ( j) (a + b)(a – 1)(a2 + a + 1) 8. (b) a2 – 4b 2,
(a – 2b)(a + 2b) (c) 5x 2 – 12x + 4, (5x – 2)(x – 2)
Exercise 4
1. €(98y – 30x) 2. (a) P = (2x + 4y + 6) m
(b) A = (xy + 2y + 2) m 2 3. P = xy 2 – x 2y = xy( y – x)
4. (2005x + 4000y + 2000) c 5. (a) (i) P = (2x + 2y) m
(ii) A = (xy) m 2 (b) 100 m, 600 m 2
6. (a) L = (2x + 4y + 6) m (b) A = (2xy + 4y) m 2
(c) y = 1·5 m 7. (a) (i) x + 1 (ii) 2x + 1
8. (a) 2πr (b) 8r (c) πr 2 (d) 4r 2 (e) 4r 2 – πr 2
9. (a) (2x + 1·5y) m (b) (1·8x + 1·3y + 41) m
10. (a) x(x + 2) m 2 (b) x(x – 2) m 2 (c) 4x m 2
Chapter 4
Exercise 5
1. (a) 3x – 7 (b) 4x 2 – 7x – 5 (c) 9x 3 – 2x 2 – 5x + 1
(d) – 8x 3 + 5x 2 + 4x – 1 (e) 2x 3 + 3x 2 + 5x
2. (a) x 2 + (k + 1) x + k (b) 2kx 2 + (k + 2)x + 1
(c) kx 2 + (2 – k)x – 2 (d) –2x 2 + (1 + 2k)x – k
(e) x 3 + (k + 1)x 2 + kx (f ) x 3 + (–k – 1)x 2 + (k + 1) x – k
(g) 2kx 3 – 10x 2 – kx + 5
(h) 3x 3 + (3k – 1)x 2 + (–k – 6) x + 2
(i) 6x 3 + (2k + 3)x 2 + (k – 6)x – 3
( j) x 3 + (k – c)x 2 + (d – kc)x – cd
3. (a) k = 4 (b) k = 2 (c) a = 6, b = 1, c = –2 (d) 2x + 1
(e) (2x –7), k = –30 (f ) k = 8, a = 15, b = –29
(g) (i) 10x 2 –17x + 3 (ii) 20x 2 – 34x + 6
__
34x + 6 (h) (i) x 2 – 2 (ii) x2 + 2√3 − 4
(iii) 20x 2 – __
(iii) 2x2–3√3 x − 6 4. (a) (x + 1), k = 2, l = 3
(b) (x – 2), b = 3, k = –11 (c) k = –6, a = 6, (3x – 1)
(d) k = 7, (x – 1), (x + 4) (e) k = –6, x 2 – 3 (f ) k = 4,
b = –16, (4x + 2) 5. (b) 3 (d) (ii) f (x) = (x 2 – a 2)(x + 1)
(e) (3 – x), (x + 1), (2 – x) (h) a = 2, b = –3
Exercise 6
1. (a) –1 (b) –x (c) x + 3 (d) x (e) x 2 (f ) 5x + 7
(g) x 2 + 2x + 4 (h) x + 1 (i) 2x 2 – 3x ( j) –4x – 12
2. (a) x 2 + 5x + 4 (b) x 2 – 5x + 5 (c) 2x 2 + 3x – 1
(d) 5x 2 – 3x + 6 (e) x 2 + 5x (f ) –2x – 3 (g) x 2 – 1
©The Dublin School of Grinds
(h) 3x 2 + 2x (i) 9x 2 + 3x + 1 ( j) (x – 2) 3. (a) x + 2
(b) 10x 2 + 22x + 10
Exercise 7
(α − β) (α + β)
x2 − 2
1 − 2x
___________
____________
1. (a)
( –2 (b) _____
(c)
(d)
(
x
αβ
(x − 1)(x − 2)
(m − 1) (m + 1)
a+1
1
(g) 0 (h) ________
(e) − ___ (f ) _____________
m
8x
20(a + 2)
x+2
8
2a
(i ____________ ( j) ____________ 2. (a) ___
(i)
(x + 10)(x + 2)
5b
x(x − 1)(x + 1)
2
y – 3y + 9
b
a
(b) − _13 (c) − ___ (d) __ (e) 1 (f ) 1 (g) _________
4a
y–3
b
(3 + 2a) (3 − 2a)
(2a − 3b) 2
_________
(h) ______________
(i)
(i
36
a2
(2y + 3x) (x − 8y)
( j) ______________ (k) (x – 2) 2 (x + 2) 3. (a) 2x
2xy
+1
6
1
_
_x____
(b) __
2x (c) − 5 (d) 2 (e) 5a – 2 (f ) 3(2b + 1) (g) x − 1
6
3x 2
(h) x 2 + y 2 (i) x + y ( j) 3x – 2 4. (a) __2 (b)
( ___ (c) _16
2
x
+
1
+
2
2x
x
(d) x + 1 (e) ______ (f ) − _____ (g) – t 3
x+3
2x − 1
Chapter 5
Exercise 8
1. (a) 27 (b) 32 (c) 625 (d) 100 000 000 (e) 1 (f ) 1
__
(g) –1 (h) 16 (i) –243 ( j) _14 (k) _49 (l) − _18 (m) __
25 (n) 4
16
121
___
(o) ___
16 (p) − 8
125
49
1
1
1
__
__
2. (a) _12 (b) _19 (c) __
64 (d) 25 (e) 16
4
2
1
1
1
_
_
_
_
__
__
(f ) __
18 (g) 3 (h) 2 (i) 3 (j) 6 (k) 6 (l) 6 (m) 64 (n) − 27
5
(o) __
27
8
3
3. (a) 27 (b) 5 (c) 125 (d) 16 (e) _16 (f ) _14 (g) 1
1
_
_
(h) 8 (i) ___
500 ( j) 28 (k) 2 (l) 3 (m) 1 000 000 (n) –32
3
3
1
2
2
2
_
_
_
_
_
(o) __
10 (p) 2 (q) 3 (r) − 3 (s) − 2 (t) 1 (u) 3 (v) –3 (w) 3
3
3
_7_
_5_
__
_3_
13
1
9
27
__
2
2
6
2
(x) _5 ( y) __
4.
(a)
2
(b)
3
(c)
(d)
7
(e)
5
64
_5_
22
_3_
6
3
(f) 1027 (g)
( __3 (h) 23 (i) 3 2 ( j) 53 5. (a) a9 (b) a21
2
_1_
_3_
_3_
1
4
4
(c) a (d) a (e) a 3 (f ) a 2 (g) a4 (h) a12 (i) __2 ( j) a 2
a
_2_
__
_7_
23
1
__
3
2
2
5
(k) a (l) a (m) a (n) 3 (o) a 6. (a) x 4y (b) 2x 5
a
2
1
(c) 4x (d) ___2 (e) 2 (f ) 2x 5 y (g) 2 2x (h) x × 2 4x (i) __3
x
2x
2x − 1
8 4
y
x
5
3
__
b
( j) _____
(k) ___2 (l) (a + 3b) 5 (m) xy (n) _____
12
2
81 z
2x
Page 59
Mills & Kelly (Power of Maths)
_1_
( j) log5 y
9. (b) 2·5 mg (c) 87·5% 10. (a) 1∙2597 (b) €7347 (c) €9255
Exercise 9
__
__
__
___
__
2√ 2
____
1. (a) 2√ 3 (b) 3√ 3 (c) 11√ 10 (d) 16√ 2 (e) 3
___
__
__
__
___
2x √ 3x
7√ 3
3√ 2
____
____
(f ) 5 (g) 10 (h) 2x√ x (i) 2xy √ 2y ( j) ______
y
_
30
___
__
__
4z √z
((k)
k ______ 2. (a) 3√ 11 (b) − 4 + 15√ 3 (c) 3x − 7√ y
3xy
__
__
__
___
(d) 0 (e) 0 (f ) 5√ 2 + 5√ 3 (g) 0 (h) √ x (i) 3ay √ ay
_____
__
__
__
( j) – 4√ x − 3 3. (a) 2√ 2 (b) √ 6 (c) 60√ 3
__ __
__
__ __
(d) 6(1 + √ 2 + √ 3 ) (e) 4 + 2√ 3 (f ) 2 (g) ab(√ a + √ b )
___
__
(h) 20 + 2√ 91 (i) – 6 ( j) 4x + 9y + 12√ xy (k) 1
______
__
(l) –3x + 2√ x (m) a − √a2 − x 2 (n) x3 − x2 − 2x + 2
__
__
__
9
1
4
(o) x – 1 ( p) x − 1 (q ) 16y__− y (r) x__+ x + 4 __
__
y2
x
2√ 2
5
2
2√ 3
_√__
_√__
____
____
__
__
(s) 2 + x − 2 4. (a) 5 (b) − 2 (c) 3 (d) 5
y __
__
__
___
__
__
x√ y_
3
42
√ xy
√
_√__
____
___
____
(e) − 2 (f ) 14 (g) y __(h) y (i) √ 2 −__1 ( j) 3 + 2√ 2
__
__
3 + √5
10x + 15√x
______
__________
(k) 3√2 + 2√3 (l) 2 (m) 9 − 4x
__
__
_____
4x
(n) 5 + 2√6 (o) √a + √a − x 5. 0 6. (a) ____
√
R3
10. (a) (i) P1 = 2π ____
__
GM
_d_
13. (a) T1 = 2π g
(b) 4x(4 – x 2) ____
7. 11
(ii) P2 = 16π
√
__
√
3
R
____
GM
√
_d_
(b) T2 = π g
Exercise 10
__
1. (a) 2 (b) –5 (c) –1 (d) 12 (e) –1 (f ) –2 (g) _23 (h) – 4
3
_
(i) 4 ( j) − 2
_1_
2
1
__
–2
2. (a) 289 = 17 (b) 16 = 4 (c) 16 = 4
(d) 4 = 16 2 (e) _14 = 16
_4_
2
_1_
−2
_3_
(i) 16 = 8 3 ( j) 8 = 16 4
(f ) 3 = 3 1 (g) 1 = a0 (h) a = bc
3. (a) log215 (b) log3 4a 2
(c) loge (1 – x 2) (d) 0 (e) log5 y (f ) log4 (x 3 – 1)
(g) log3 e 3x – 1 (h) 1 (i) logk (x – 1) ( j) 0 4. (a) 1
(b) 1 (c) loge (x – 1) (d) log5 49 (e) log4 (x – 1) (f ) log3 e2
(g) log9 3 = _12 (h) 1 (i) 1 ( j) loga x 2
( )
( )
5. (a) log2 125
___
____
1
1
(b) log3 49 (c) log4 5 (d) log2 216 (e) logk y 2
__
1
(f ) log5 (1 + x 2) (g) log3 zy (h) log7 4y (i) loge 2x
( )
©The Dublin School of Grinds
log5 4
logb y
log2__
5
___
_____
_____
(
(b)
(c)
log5 7
log2 3
log b a
log7 (x − 2)
log___
_________
___
_____
_____
ea
1
1
(d)
(d
(e)
(f )
(g)
loge 10
log3 5
logx 4
log7 5
_____
1
(h)
(i) logx 2 ( j) log xe
log5 x
( )
1
_5_
(o) _____
( p)) a2y2 (q ) 2x (r) y (s) x y (t) 2x2y
2 2
2a y
___
1
7. (a) 121 500 (b) 88 573 500 (c) 729 8. (b) 16
6. (a)
7. (a) (i) 2 loga x + _12 loga y (ii) 2 log2 u + 3 log2 v
(iii) log5 3 + _12 log5 x – 2 log5 p – 3 log5 q
(iv) 2 + 3 log3 x – 6 log3 y (v) –2
(vi) 2 logk (x + 3) + 3log k (2x – 1)
(vii) 7 log2 (x + 7) − 2 − _12 log2 (2x – 3)
(viii) _12 log5 (x − 1) − _12 log5 (x + 1) (b) (i) a + b
(ii) 3a + b (iii) 2b – 3a (iv) a + b + 1 (v) b + 1
(vi) 4b (vii) 4a – 3b (viii) 2a + 2 – 3b (ix) 4a + b
_______
2a + b
8. (a) (i) 2 (ii) 1 (iii) 1 (iv) 0
(x)
_____
b
3 2
x
y
x−1
√
_______
(iii) log3 ________
(b) (i) log5 x 2 y (ii) logk ____
_1_
4
√(x + 1)3
z
7
_____
_3_
2(x + 5)
________
(v) log4 a 2 (vi) loga √x + 1
(iv) log3
(4x − 1)3
_1_
3(3x + 2)
x + e −2
(viii)
lo
g
x
14. (a) 7
(vii) loge ________
e
_3_
2
2
(x + 1)
__
1
_
(b) 4 (c) 2 (d) 3 (e) 8 (f ) √x
{
{
}
( )
(
)
}
15. (a) 3 (b) n(n – 1)(n – 2)…1 = n! 17. (a) (i) 50 dB
5
(ii) 110 dB 19. (a) 54 (b) _2 (c) 50 (d) 2
Revision Questions
__
3
7
−
√5 (b) k = –13, (x – 2), (x + 3)
_______
1. (a)
2
__
___________
1
(c) (2x – 1) 2. (a) −√3 (b)
(x + 2)(x + 4)
1
3. (a) a = 5, b = –2, c = –12 (c) x4 + __4 4. (a) 3
x
_____
2x
(b) x − 1 (c) (i) a = 2, b = 1, c = 3 (ii) a = –1,
5. (a) p 3 + 3p 2q + 3pq 2 + q 3 (b) 35 p 4q 3
__
________
3
6. (a) 3 (b) (i) −
(ii) 2x – 1, k = 2 + 2√3
2(x + 4y)
(c) a = 2, b = 3 7. (a) 1 000 000 000 001
b = –3
2
(b) (i) __
8. (a) 172 800 (b) 491 520
99 (ii) 1 + 2x
__
___1__
_______
2
1
9. (c) 1·26 11. (a) x (b) − x + 2 (c)
x (x + 2)
___2__
__2___
8
5
(d) _5 m−1, _8 m 12. (a) x − 2 hours (b) x + 2 hours
8
(e) 3 min 26 s
(c) Th
The man, bigger speed (d) ______
2
x − 22
13. (a) 3200 × 2t; 3200 (b) 100 × 22t; 100 (c) 2t – 5
(i) _18 (ii) 1 (iii) 32 (d) The number in sample B is less
than the number in sample A for the first 5 minutes
but after that it is always bigger
10
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Mills & Kelly (Power of Maths)