Rational and
irrational
numbers
1
Manning’s formula is a
formula used to estimate the
flow of water down a river in
a flood event, measured in
metres per second. The
2
---
1
---
R3 S2
formula is v = ----------- , where
n
R is the hydraulic radius, S is
the slope of the river and n is
the roughness coefficient.
What will be the flow of
water in the river if R = 8,
S = 0.0025 and n = 0.625?
To find the answer to this
problem you will need to
have an understanding of
fractional indices, which in
this chapter we will be
relating to your work on
surds from last year.
areyou
2
Maths Quest 8 for Victoria
READY?
Are you ready?
Try the questions below. If you have difficulty with any of them, extra help can be
obtained by completing the matching SkillSHEET. Either click on the SkillSHEET icon
next to the question on the Maths Quest 10 CD-ROM or ask your teacher for a copy.
1.1
Identifying surds
1 Which of the following are surds?
a
1.2
10
b
48
98
c 5 12
d 3 72
3 Simplify each of the following.
b 2 32 – 5 45 – 4 180 + 10 8
Multiplying and dividing surds
4 Simplify each of the following.
7 ×
10
b 2 3 ×4 6
6
c ------2
5 6
d ------------10 3
Rationalising denominators
5 Rationalise the denominator of each of the following.
1
a ------5
1.9
d 4 2
Adding and subtracting surds
a
1.5
49
2 Simplify each of the following.
a 2 6 –4 3 +7 3 –5 6
1.4
c
Simplifying surds
a
1.3
b
7
2
b ------3
3
c ------6
7 5
d – ---------5 7
c (2.5)6
d (0.3)4
c 24a3b ÷ 6ab5
d (2m4)2
Evaluating numbers in index form
6 Evaluate each of the following.
a 72
b 34
1.10
Using the index laws
7 Simplify each of the following.
a x3 × x7
b 4y3 × 5y8
Chapter 1 Rational and irrational numbers
3
Classifying numbers
The number systems used today evolved from a basic and practical need of primitive
people to count and measure magnitudes and quantities such as livestock, people,
possessions, time and so on.
As societies grew and architecture
and engineering developed, number
systems became more sophisticated.
Number use developed from solely
whole numbers to fractions, decimals
and irrational numbers.
We shall explore these different
types of numbers and classify them
into their specific groups.
The Real Number System contains
the set of rational and irrational
numbers. It is denoted by the symbol
R. The set of real numbers contains a
number of subsets which can be classified as shown in the chart below.
Real numbers R
Irrational numbers I
(surds, non-terminating
and non-recurring
decimals, π ,e)
Negative
Z–
Rational numbers Q
Integers
Z
Zero
(neither positive
nor negative)
The relationship which exists between the
subsets of the Real Number System can be
illustrated in a Venn diagram as shown on
the right.
We can say N ⊂ Z, Z ⊂ Q, and so on,
where ⊂ means ‘is a subset of’.
Non-integer rationals
(terminating and
recurring decimals)
Positive
Z+
(Natural
numbers N)
ε =R
Q (Rational numbers)
Z (Integers)
N
(Natural
numbers)
Rational numbers (Q)
A rational number (ratio-nal) is a number
that can be expressed as a ratio of two whole
a
numbers in the form --- , where b ≠ 0.
b
Rational numbers are given the symbol Q. Examples are:
.
1
2
3
9
--- , --- , ------ , --- , 7, –6, 0.35, 1.4
5
7
10
4
I
(Irrational
numbers)
4
M a t h s Q u e s t 1 0 fo r N ew S o u t h Wa l e s 5 . 3 P a t h w ay
Integers (Z)
Rational numbers may be expressed as integers. Examples are:
5
--1
= 5,
= −4,
–4
-----1
27
-----1
------ = −15
= 27, – 15
1
The set of integers consists of positive and negative whole numbers and 0 (which is
neither positive nor negative). They are denoted by the letter Z and can be further
divided into subsets. That is:
Z = {. . ., −3, −2, −1, 0, 1, 2, 3, . . .}
Z + = {1, 2, 3, 4, 5, 6, . . .}
Z − = {−1, −2, −3, −4, −5, −6, . . .}
Positive integers are also known as natural numbers (or counting numbers) and are
denoted by the letter N. That is:
N = {1, 2, 3, 4, 5, 6, . . .}
Integers may be represented on the number line as illustrated below.
–3 –2 –1 0 1 2 3 Z
The set of integers
Z – –6 –5 –4 –3 –2 –1
The set of negative integers
1 2 3 4 5 6 N
The set of positive integers
or natural numbers
Note: Integers on the number line are marked with a solid dot to indicate that they are
the only points in which we are interested.
Non-integer rationals
Rational numbers may be expressed as terminating decimals. Examples are:
7
-----10
= 0.7,
1
--4
= 0.25,
5
--8
= 0.625,
9
--5
= 1.8
These decimal numbers terminate after a specific number of digits.
Rational numbers may be expressed as recurring decimals (non-terminating or
periodic decimals). For example:
.
1
--- = 0.333 333 . . . or 0.3
3
9
-----11
5
--6
3
-----13
..
= 0.818 181 . . . or 0.8 1 (or 0.81 )
.
= 0.833 333 . . . or 0.83
.
.
= 0.230 769 230 769 . . . or 0.2 30769 (or 0.230 769 )
–3.743 3
1–
These decimals do not terminate, and the specific
2 1.63
–2 –4
3.6
digit (or number of digits) is repeated in a pattern.
Recurring decimals are represented by placing a dot
–4 –3 –2 –1 0 1 2 3 4 Q
or line above the repeating digit or pattern.
Rational numbers are defined in set notation as: Q = rational numbers
a
Q = --- , a, b ∈ Z, b ≠ 0 where ∈ means ‘an element of’.
b
Rational numbers may be represented on the number line as shown above.
{
}
Chapter 1 Rational and irrational numbers
5
Irrational numbers (I)
Numbers that cannot be expressed as a ratio between two whole numbers are called
irrational numbers. Irrational numbers are denoted by the letter I. Numbers such as
surds (for example 7 , 10 ), decimals that neither terminate nor recur, and π and e
are examples of irrational numbers. The numbers π and e are examples of transcendental numbers; these will be discussed briefly later in this chapter.
Irrational numbers may also be represented on the number line with the aid of a ruler
and compass.
An irrational number (ir-ratio-nal) is a number that cannot be expressed as a
ratio of two whole numbers in the form --a- , where b ≠ 0.
b
Irrational numbers are given the symbol I. Examples are:
7
13 , 5 21 , ------- , π, e
9
Irrational numbers may be expressed as decimals. For example:
7,
5 = 2.236 067 977 5 . . .
0.03 = 0.173 205 080 757 . . .
18 = 4.242 640 687 12 . . .
2 7 = 5.291 502 622 13 . . .
π = 3.141 592 653 59 . . .
e = 2.718 281 828 46 . . .
These decimals do not terminate, and the digits do not repeat themselves in any particular pattern or order (that is, they are non-terminating and non-recurring).
Rational and irrational numbers belong to the
set of real numbers (denoted by the symbol R).
π
– 1–2
–
π
– 12 – 5
2
They can be positive, negative or 0. The real
4
numbers may be represented on a number line
as shown at right (irrational numbers above the
–4 –3 –2 –1 0 1 2 3 4 R
line; rational numbers below it).
To classify a number as either rational or irrational:
1. Determine whether it can be expressed as a whole number, a fraction or a
terminating or recurring decimal.
2. If the answer is yes, the number is rational; if the answer is no, the number is irrational.
Consider an isosceles right-angled triangle of side length 1 unit.
By Pythagoras’ theorem, (OB)2 = (OA)2 + (AB)2; therefore the length of the
hypotenuse is 2 units.
By using a compass, we can transfer the length of the
B
hypotenuse OB to the number line (labelled C). This
distance can now be measured using a ruler. Although
2 units
1 unit
this distance will be inaccurate due to the equipment used,
A
C
there is an exact point on the number line for each
O
2 2 R
0 1 unit 1
irrational number.
This geometric model can be extended to any irrational number in surd form.
π (pi)
The symbol π (pi) is used for a particular number; that is, the circumference of a circle
whose diameter length is 1 unit. It can be approximated as a decimal that is nonterminating and non-recurring. Therefore, π is classified as an irrational number. (It is
also called a transcendental number and cannot be expressed as a surd.)
6
M a t h s Q u e s t 1 0 fo r N ew S o u t h Wa l e s 5 . 3 P a t h w ay
In decimal form, π = 3.141 592 653 589 793 23 . . . It has been calculated to
29 000 000 (29 million) decimal places with the aid of a computer.
WORKED Example 1
Specify whether the following numbers are rational or irrational.
1
--4
a
b
c
16
11
d 2π
e 0.28
THINK
a
b
1
--4
f
3
64
g
3
22
h
3 1--8
WRITE
a
is already a rational number.
1
--4
16 = 4
b
1
Evaluate
16 .
2
The answer is an integer, so classify
is rational.
16 is rational.
16 .
c
Evaluate
11 .
2
The answer is a non-terminating and
non-recurring decimal; classify
d
1
2
Use your calculator to find the value
of 2π.
The answer is a non-terminating and
non-recurring decimal; classify 2π.
Evaluate
2
The answer is a whole number, so
3
64 .
2π is irrational.
e 0.28 is rational.
3
64 = 4
3
64 is rational.
3
22 = 2.802 039 330 66 . . .
3
22 is irrational.
64 .
3
g
1
Evaluate
22 .
2
The result is a non-terminating and
non-recurring decimal; classify
h
d 2π = 6.283 185 307 18 . . .
f
1
classify
g
3
11 is irrational.
11 .
e 0.28 is a terminating decimal; classify it
accordingly.
f
11 = 3.316 624 790 36 . . .
c
1
3
22 .
3 1
--- .
8
1
Evaluate
2
The result is a number in a rational
form.
h
3 1
--8
=
3 1
--8
is rational.
1
--2
Chapter 1 Rational and irrational numbers
7
remember
a
1. Rational numbers (Q) can be expressed in the form --- , where a and b are whole
b
numbers and b ≠ 0. They include whole numbers, fractions and terminating and
recurring decimals.
a
2. Irrational numbers (I) cannot be expressed in the form --- , where a and b are
b
whole numbers and b ≠ 0. They include surds, non-terminating and nonrecurring decimals, and numbers such as π and e.
3. Rational and irrational numbers together constitute the set of Real numbers (R).
1A
WORKED
Example
1
Classifying numbers
1 Specify whether the following numbers are rational (Q) or irrational (I).
4
--5
4
b
f
0.04
g 2 1---
h
5
i
l
m
14.4
n
1.44
o π
q 7.32
r – 21
s
1000
t 7.216 349 157 . . .
v 3π
w
x
1
-----16
y
k −2.4
25
-----9
p
u – 81
c
7
--9
a
2
100
d
3
62
2
9
--4
e
7
j 0.15
3
0.0001
2 Specify whether the following numbers are rational (Q), irrational (I) or neither.
a
1
--8
f
3
k
3
p
64
-----16
q
u
22 π
--------7
v
b
625
c
81
g – 11
h
21
l
π
--7
2
-----25
3
– 1.728
11
-----4
d
e −6 1---
0
--8
7
π
1.44
---------4
i
( –5 )2
3
n − -----
o
3
27
t
1
------4
x 4 6
y
( 2)
m
3
r
6
------2
11
s
w 6 4
8
--0
j
1
--------100
3 multiple choice
Which of the following best represents a rational number?
Aπ
B
4
--9
C
9
-----12
D
3
3
4
8
M a t h s Q u e s t 1 0 fo r N ew S o u t h Wa l e s 5 . 3 P a t h w ay
4 multiple choice
Which of the following best represents an irrational number?
A − 81
6
--5
3
D
22
Which of the following statements regarding the numbers −0.69,
correct?
7,
B
C
343
5 multiple choice
A
B
π
--3
π
--- ,
3
49 is
is the only rational number.
7 and
49 are both irrational numbers.
C −0.69 and
49 are the only rational numbers.
D −0.69 is the only rational number.
6 multiple choice
------ ,
Which of the following statements regarding the numbers 2 1--- , − 11
2
3
correct?
------ and
A − 11
3
624 ,
3
99 is
624 are both irrational numbers.
B
624 is an irrational number and
C
624 and
3
3
99 is a rational number.
99 are both irrational numbers.
------ is an irrational number.
D 2 1--- is a rational number and − 11
2
3
Surds
We have classified a particular group of numbers as irrational and will now further
examine surds — one subset of irrational numbers — and some of their associated
properties.
A surd is an irrational number that is represented by a root sign or a radical sign,
for example:
,
Examples of surds include:
7,
3
,
4
5,
3
4
11 ,
15
Examples that are not surds include:
9,
16 ,
3
125 ,
4
81
Numbers that are not surds can be simplified to rational numbers, that is:
9 = 3,
16 = 4 ,
3
125 = 5 ,
4
81 = 3
Chapter 1 Rational and irrational numbers
9
WORKED Example 2
Which of the following numbers are surds?
a
25
b
10
c
1
--4
d
3
e
11
4
59
f
3
343
THINK
WRITE
a
Evaluate 25 .
The answer is rational (since it is a
whole number), so state your
conclusion.
a
25 = 5
25 is not a surd.
Evaluate 10 .
The answer is irrational (since it is a
non-recurring and non-terminating
decimal), so state your conclusion.
b
10 = 3.162 277 660 17 . . .
10 is a surd.
c
1
--4
=
1
--4
is not a surd.
1
2
b
1
2
c
d
Evaluate
2
The answer is rational (a fraction);
state your conclusion.
1
2
e
1
2
f
1
--- .
4
1
1
2
Evaluate 3 11 .
The answer is irrational (a nonterminating and non-recurring
decimal), so state your conclusion.
d
Evaluate 4 59 .
The answer is irrational, so classify
4
59 accordingly.
e
Evaluate 3 343 .
The answer is rational; state your
conclusion.
f
3
3
4
4
1
--2
11 = 2.223 980 090 57 . ..
11 is a surd.
59 = 2.771 488 002 48 . . .
59 is a surd.
3
343 = 7
343 is not a surd.
So b, d and e are surds.
3
Proof that a number is irrational
In Mathematics you are required to study a variety of types of proofs. One such method
is called proof by contradiction.
This method is so named because the logical argument of the proof is based on an
assumption that leads to contradiction within the proof. Therefore the original assumption must be false.
a
An irrational number is one that cannot be expressed in the form --- (where a and b
b
are integers). The next worked example sets out to prove that
2 is irrational.
10
M a t h s Q u e s t 1 0 fo r N ew S o u t h Wa l e s 5 . 3 P a t h w ay
WORKED Example 3
Prove that
2 is irrational.
THINK
1
Assume that
2 is rational; that is, it
a
can be written as --- in simplest form.
b
We need to show that a and b have no
common factors.
2
Square both sides of the equation.
3
Rearrange the equation to make a2 the
subject of the formula.
If x is an even number, then x = 2n.
WRITE
a
2 = --- , where b ≠ 0
b
2
4
6
Since a is even it can be written as
a = 2r.
Square both sides.
7
Equate [1] and [2].
5
a
2 = ----2
b
2
a = 2b2
[1]
∴ a2 is an even number and a must also
be even; that is, a has a factor of 2.
∴ a = 2r
a2 = 4r2
But
a2 = 2b2 from [1]
∴ 2b2 = 4r2
[2]
2
8
Repeat the steps for b as previously
done for a.
4r
b2 = -------2
= 2r2
2
∴ b is an even number and b must also be even;
that is, b has a factor of 2.
Both a and b have a common factor of 2.
This contradicts the original assumption that
a
2 = --- , where a and b have no common factor.
b
∴ 2 is not rational.
∴ It must be irrational.
The ‘dialogue’ included in the worked example should be present in all proofs and is an
essential part of the communication that is needed in all your solutions.
Note: An irrational number written in surd form gives an exact value of the number;
whereas the same number written in decimal form (for example, to 4 decimal places)
gives an approximate value.
remember
A number is a surd if:
1. it is an irrational number (equals a non-terminating, non-recurring decimal)
2. it can be written with a radical sign (or square root sign) in its exact form.
11
Chapter 1 Rational and irrational numbers
1B
WORKED
Example
1 Which of the numbers below are surds?
a
81
b
g
3
--4
h
m
3
48
3 3
-----27
32
n
361
s
3
169
t
7
--8
y
5
32
z
80
1.1
c
16
d
1.6
e
i
1000
j
1.44
k 4 100
100
p
125
q
o
3
u
4
16
3
v ( 7)
2
f
0.16
w
3
11
l 2 + 10 Identifying
surds
6+ 6
r 2π
33
x
0.0001
2 multiple choice
The correct statement regarding the set of numbers
A
B
C
D
WORKED
Example
3
3
27 and
6
--9
6
--9
{
6
--- ,
9
20 ,
54 ,
3
27 ,
}
9 is:
9 are the only rational numbers of the set.
is the only surd of the set.
and
20 are the only surds of the set.
20 and
54 are the only surds of the set.
3 Prove that the following numbers are irrational, using a proof by contradiction:
a 3
b 5
c
7
4 multiple choice
Which of the numbers of the set
A
21 only
B
1
--8
{
1 3 1
--- , ------ ,
4
27
only
1
--- ,
8
21 ,
C
1
--8
3
and
}
8 are surds?
3
8
1
--8
D
and
21 only
3+1
} is not
5 multiple choice
{
1
Which statement regarding the set of numbers π, -----,
49
true?
A 12 is a surd.
B 12 and
C π is irrational but not a surd.
D 12 and
12 ,
16 ,
16 are surds.
3 + 1 are not rational.
6 multiple choice
{
Which statement regarding the set of numbers 6 7 ,
not true?
A
144
--------16
when simplified is an integer.
C 7 6 is smaller than 9 2 .
B
144
--------16
144
--------- ,
16
and
7 6, 9 2,
18 ,
}
25 is
25 are not surds.
D 9 2 is smaller than 6 7 .
7 Complete the following statement by selecting appropriate words, suggested in
brackets:
6
a is definitely not a surd, if a is . . . (any multiple of 4; a perfect square and cube).
8 Find the smallest value of m, where m is a positive integer, so that
3
SkillS
HEET
2
Surds
16m is not a surd.
12
M a t h s Q u e s t 1 0 fo r N ew S o u t h Wa l e s 5 . 3 P a t h w ay
Operations with surds
In Year 9 we looked at surds and various operations with surds. In this section, we will
revise each of these operations.
Simplifying surds
To simplify a surd means to make a number (or an expression) under the radical sign
( ) as small as possible. To simplify a surd (if it is possible), it should be rewritten as
a product of two factors, one of which is a perfect square, that is, 4, 9, 16, 25, 36, 49,
64, 81, 100 and so on.
We must always aim to obtain the largest perfect square when simplifying surds so
that there are fewer steps involved in obtaining the answer. For example,
written as
4 × 8 = 2 8 ; however,
32 = 2 × 2 2 ; that is
8 can be further simplified to 2 2 , so
32 = 4 2 . If, however, the largest perfect square had been
selected and 32 had been written as
would be obtained in fewer steps.
16 × 2 =
16 ×
2 = 4 2 , the same answer
WORKED Example 4
Simplify the following surds. Assume that x and y are positive real numbers.
a
384
b 3 405
c − 1--- 175
8
d 5 180 x 3 y 5
THINK
WRITE
a
a
1
b
Express 384 as a product of two
factors where one factor is the
largest possible perfect square.
384 =
64 × 6
Express 64 × 6 as the product of
two surds.
=
3
Simplify the square root from the
perfect square (that is, 64 = 8).
=8 6
1
Express 405 as a product of two
factors, one of which is the largest
possible perfect square.
2
Express
surds.
3
Simplify
4
Multiply together the whole numbers
outside the square root sign
(3 and 9).
2
32 could be
81 × 5 as a product of two
81 .
64 × 6
b 3 405 = 3 81 × 5
= 3 81 × 5
= 3×9 5
= 27 5
Chapter 1 Rational and irrational numbers
THINK
WRITE
c
c − 1--- 175 = − 1--- 25 × 7
d
1
Express 175 as a product of two factors
in which one factor is the largest
possible perfect square.
2
Express
surds.
3
Simplify
4
Multiply together the numbers outside
the square root sign.
1
Express each of 180, x3 and y5 as a
product of two factors where one factor
is the largest possible perfect square.
2
Separate all perfect squares into one
surd and all other factors into the other
surd.
Simplify 36x 2 y 4 .
= 5 × 36x 2 y 4 × 5xy
Multiply together the numbers and the
pronumerals outside the square root
sign.
= 30xy 2 5xy
3
4
8
13
8
25 × 7 as a product of 2
= − 1--- ×
8
25 × 7
= − 1--- × 5 7
25 .
8
= − 5--- 7
8
d 5 180x 3 y 5 = 5 36 × 5 × x 2 × x × y 4 × y
= 5 × 6 × x × y 2 × 5xy
Addition and subtraction of surds
Surds may be added or subtracted only if they are alike.
Examples of like surds include
7 , 3 7 and – 5 7 . Examples of unlike surds
include 11 , 5 , 2 13 and – 2 3 .
In some cases surds will need to be simplified before you decide whether they are
like or unlike, and then addition and subtraction can take place. The concept of adding
and subtracting surds is similar to adding and subtracting like terms in algebra.
WORKED Example 5
Simplify each of the following expressions containing surds. Assume that a and b are
positive real numbers.
a 3 6 + 17 6 – 2 6
b 5 3 + 2 12 – 5 2 + 3 8
c
1
--2
100a 3 b 2 + ab 36a – 5 4a 2 b
THINK
WRITE
a All 3 terms are alike because they
contain the same surd ( 6 ) .
Simplify.
a 3 6 + 17 6 – 2 6 = ( 3 + 17 – 2 ) 6
= 18 6
Continued over page
14
M a t h s Q u e s t 1 0 fo r N ew S o u t h Wa l e s 5 . 3 P a t h w ay
THINK
WRITE
b
b 5 3 + 2 12 – 5 2 + 3 8
1
Simplify surds where possible.
= 5 3+2 4×3–5 2+3 4×2
= 5 3+2×2 3–5 2+3×2 2
= 5 3+4 3–5 2+6 2
c
2
Add like terms to obtain the
simplified answer.
1
Simplify surds where possible.
= 9 3+ 2
c
1
--2
100a 3 b 2 + ab 36a – 5 4a 2 b
=
1
--2
× 10 a 2 × a × b 2 + ab × 6 a – 5 × 2 × a b
=
1
--2
× 10 × a × b a + ab × 6 a – 5 × 2 × a b
= 5ab a + 6ab a – 10a b
2
Add like terms to obtain the
simplified answer.
= 11ab a – 10a b
Multiplication and division of surds
Multiplying surds
To multiply surds, multiply together the expressions under the radical signs. For
example, a × b = ab , where a and b are positive real numbers.
When multiplying surds it is best to first simplify them (if possible). Once this has
been done and a mixed surd has been obtained, the coefficients are multiplied with each
other and then the surds are multiplied together. For example,
m a × n b = mn ab
WORKED Example
6
Multiply the following surds, expressing answers in the simplest form. Assume that x and
y are positive real numbers.
a
11 ¥ 7
b 5 3¥8 5
c 6 12 ¥ 2 6
THINK
WRITE
a Multiply the surds together, using
a × b = ab (that is, multiply
expressions under the square root sign).
Note: This expression cannot be
simplified any further.
a
11 × 7 =
=
d
11 × 7
77
15 x 5 y 2 ¥ 12 x 2 y
Chapter 1 Rational and irrational numbers
THINK
WRITE
b Multiply the coefficients together and
then multiply the surds together.
b 5 3×8 5 = 5×8× 3× 5
15
= 40 × 3 × 5
= 40 15
c
1
Simplify
c 6 12 × 2 6 = 6 4 × 3 × 2 6
12 .
= 6×2 3×2 6
= 12 3 × 2 6
2
Multiply the coefficients together
and multiply the surds together.
= 24 18
3
Simplify the surd.
= 24 9 × 2
= 24 × 3 2
= 72 2
d
1
Simplify each of the surds.
d
15x 5 y 2 × 12x 2 y
4
=
2
15 × x × x × y ×
= x2 × y ×
2
4×3×x ×y
15 × x × 2 × x ×
3×y
= x y 15x × 2x 3y
2
2
3
Multiply the coefficients together
and the surds together.
= x2y × 2x 15x × 3y
Simplify the surd.
= 2x3y 9 × 5xy
= 2x3y 45xy
= 2x3y × 3 5xy
= 6x3y 5xy
When working with surds, we sometimes need to multiply surds by themselves; that is,
square them. Consider the following examples:
( 2 )2
=
2× 2 =
4 = 2
( 5 )2
=
5× 5 =
25 = 5
We observe that squaring a surd produces the number under the radical sign. This is
not surprising, because squaring and taking the square root are inverse operations and,
when applied together, leave the original unchanged.
When a surd is squared, the result is the number (or expression) under the radical
sign; that is,
( a )2
= a , where a is a positive real number.
16
M a t h s Q u e s t 1 0 fo r N ew S o u t h Wa l e s 5 . 3 P a t h w ay
WORKED Example 7
Simplify each of the following.
a ( 6)
2
b (3 5)
2
THINK
WRITE
a Use ( a )2 = a, where a = 6.
a ( 6) = 6
b
1
2
2
2
Square 3 and use ( a )2 = a to
square 5 .
Simplify.
b ( 3 5 ) = 32 × ( 5 )2
=9×5
= 45
Dividing surds
To divide surds, divide the expressions under the radical signs;
a
a
that is, ------- = --- , where a and b are whole numbers.
b
b
When dividing surds it is best to simplify them (if possible) first. Once this has been
done, the coefficients are divided next and then the surds are divided.
WORKED Example
8
Divide the following surds, expressing answers in the simplest form. Assume that x and y
are positive real numbers.
55
a ---------5
48
b ---------3
36 xy
d -----------------------25 x 9 y 11
9 88
c ------------6 99
THINK
a
a
a
Rewrite the fraction, using ------- = --- .
b
b
Divide the numerator by the
denominator (that is, 55 by 5).
Check if the surd can be simplified any
further.
55
a ---------- =
5
=
2
a
Rewrite the fraction, using ------- =
b
Divide 48 by 3.
48
b ---------- =
3
=
3
Evaluate
1
2
3
b
WRITE
1
16 .
a
--- .
b
=4
55
-----5
11
48
-----3
16
Chapter 1 Rational and irrational numbers
THINK
c
d
1
17
WRITE
a
Rewrite surds, using ------- =
b
a
--- .
b
2
Simplify the fraction under the radical
by dividing both numerator and
denominator by 11.
3
Simplify surds.
4
Multiply the whole numbers in the
numerator together and those in the
denominator together.
5
Cancel the common factor of 18.
1
Simplify each surd.
c
9 88
9 88
------------- = --- -----6 99
6 99
9 8
= --- --6 9
9×2 2
= ------------------6×3
18 2
= ------------18
=
2
36xy
6 xy
d ----------------------- = -------------------------------------------25x 9 y 11
5 x 8 × x × y 10 × y
6 xy
= -----------------------5x 4 y 5 xy
2
Cancel any common factors — in this
case xy .
6
= -------------5x 4 y 5
Rationalising denominators
If the denominator of a fraction is a surd, it can be changed into a rational number. In
other words, it can be rationalised.
As we discussed earlier in this chapter, squaring a simple surd (that is, multiplying it
by itself) results in a rational number. This fact can be used to rationalise denominators
as follows.
b
a
b
ab
------- × ------- = ---------- , where ------- = 1
b
b
b
b
If both numerator and denominator of a fraction are multiplied by the surd contained
in the denominator, the denominator becomes a rational number. The fraction takes on
a different appearance, but its numerical value is unchanged, because multiplying the
numerator and denominator by the same number is equivalent to multiplying by 1.
18
M a t h s Q u e s t 1 0 fo r N ew S o u t h Wa l e s 5 . 3 P a t h w ay
WORKED Example
9
Express the following in their simplest form with a rational denominator.
6
a ---------13
2 12
b ------------3 54
17 – 3 14
c ----------------------------7
THINK
WRITE
a
6
a ---------13
1
Write the fraction.
2
Multiply both the numerator and denominator by
the surd contained in the denominator (in this case
13 ). This has the same effect as multiplying the
6
13
= ---------- × ---------13
13
78
= ---------13
13
fraction by 1, because ---------- = 1 .
13
b
1
Write the fraction.
2
Simplify the surds. (This avoids dealing with large
numbers.)
3
Multiply both the numerator and denominator by 6 .
(This has the same effect as multiplying the fraction by
6
1, because ------- = 1.)
6
Note: We need to multiply only by the surd part of the
denominator (that is, by 6 rather than by 9 6 ).
4
5
Simplify
18 .
Divide both the numerator and denominator by 6
(cancel down).
2 12
b ------------3 54
2 12 2 4 × 3
------------- = ------------------3 54 3 9 × 6
2×2 3
= ------------------3×3 6
4 3
= ---------9 6
4 3
6
= ---------- × ------9 6
6
4 18
= ------------9×6
4 9×2
= ------------------9×6
4×3 2
= ------------------54
12 2
= ------------54
2 2
= ---------9
Chapter 1 Rational and irrational numbers
THINK
WRITE
c
17 – 3 14
c ----------------------------7
1
Write the fraction.
2
Multiply both the numerator and denominator by 7 .
Use grouping symbols to make it clear that the whole
numerator must be multiplied by 7 .
3
Apply the Distributive Law in the numerator.
a(b + c) = ab + ac
19
( 17 – 3 14)
7
= ---------------------------------- × ------7
7
17 × 7 – 3 14 × 7
= ---------------------------------------------------------7× 7
119 – 3 98
= -------------------------------7
4
Simplify
119 – 3 49 × 2
= -----------------------------------------7
98 .
119 – 3 × 7 2
= --------------------------------------7
119 – 21 2
= -------------------------------7
Rationalising denominators using conjugate surds
The product of pairs of conjugate surds results in a rational number. (Examples of pairs
of conjugate surds include
6 + 11 and
6 – 11 ,
a + b and
a – b , 2 5 – 7 and
2 5 + 7 .)
This fact is used to rationalise denominators containing a sum or a difference of
surds.
To rationalise the denominator that contains a sum or a difference of surds, we
multiply both numerator and denominator by the conjugate of the denominator.
Two examples are given below:
1
a– b
1. To rationalise the denominator of the fraction -------------------- , multiply it by -------------------- .
a+ b
a– b
1
a+ b
2. To rationalise the denominator of the fraction -------------------- , multiply it by -------------------- .
a– b
a+ b
A quick way to simplify the denominator is to use the difference of two squares
identity:
(
2
a – b )( a + b ) = ( a ) – ( b )
=a−b
2
20
M a t h s Q u e s t 1 0 fo r N ew S o u t h Wa l e s 5 . 3 P a t h w ay
WORKED Example 10
Rationalise the denominator and simplify the following.
1
a ---------------4– 3
THINK
a
1
2
3
4
6+3 2
b -----------------------3+ 3
WRITE
1
a ---------------Write down the fraction.
4– 3
(4 + 3)
1
Multiply the numerator and denominator = --------------------- × --------------------(4 – 3) (4 + 3)
by the conjugate of the denominator.
(4 + 3)
(Note that --------------------- = 1.)
(4 + 3)
Apply the Distributive Law in the
numerator and the difference of two
squares identity in the denominator.
Simplify.
4+ 3
= -----------------------------2
( 4 )2 – ( 3 )
4+ 3
= ---------------16 – 3
4+ 3
= ---------------13
b
6+3 2
b -----------------------3+ 3
1
Write down the fraction.
2
Multiply the numerator and denominator
by the conjugate of the denominator.
( 6 + 3 2) (3 – 3)
= ----------------------------- × --------------------(3 + 3)
(3 – 3)
(3 – 3)
(Note that --------------------- = 1.)
(3 – 3)
Multiply the expressions in grouping
symbols in the numerator, and apply
the difference of two squares identity
in the denominator.
Simplify.
6×3+ 6×– 3+3 2×3+3 2×– 3
= ----------------------------------------------------------------------------------------------------------2
( 3 )2 – ( 3 )
3
4
3 6 – 18 + 9 2 – 3 6
= -----------------------------------------------------------9–3
– 18 + 9 2
= ------------------------------6
– 9×2+9 2
= ------------------------------------6
–3 2+9 2
= ------------------------------6
6 2
= ---------6
= 2
Chapter 1 Rational and irrational numbers
21
remember
1. To simplify a surd means to make a number (or an expression) under the
radical sign as small as possible.
2. To simplify a surd, write it as a product of two factors, one of which is the
largest possible perfect square.
3. Only like surds may be added and subtracted.
4. Surds may need to be simplified before adding and subtracting.
5. When multiplying surds, simplify the surd if possible and then apply the
following rules:
(a) a × b = ab
(b) m a × n b = mn ab , where a and b are positive real numbers.
6. When a surd is squared, the result is the number (or the expression) under the
radical sign: ( a )2 = a, where a is a positive real number.
7. When dividing surds, simplify the surd if possible and then apply the following
rule:
a
a
b = ------- = --b
b
where a and b are whole numbers, and b ≠ 0.
8. To rationalise a surd denominator, multiply the numerator and denominator by
the surd contained in the denominator. This has the effect of multiplying the
fraction by 1, and thus the numerical value of the fraction remains unchanged,
while the denominator becomes rational:
a ÷
a
a
b
ab
------- = ------- × ------- = ---------b
b
b
b
where a and b are whole numbers and b ≠ 0.
9. To rationalise the denominator containing a sum or a difference of surds,
multiply both the numerator and denominator of the fraction by the conjugate
of the denominator. This eliminates the middle terms and leaves a rational
number.
1C
WORKED
Example
WORKED
4b, c
1.2
a
12
b
24
c
27
d
125
e
54
f
112
g
68
h
180
i
88
j
162
k
245
l
448
Simplifying
surds
2 Simplify the following surds.
a 2 8
e – 6 75
i
1
--9
162
b 8 90
f – 7 80
j
1
--4
192
SkillS
Math
c 9 80
g 16 48
k
1
--9
135
d 7 54
h
l
1
--7
3
-----10
392
175
cad
Example
1 Simplify the following surds.
HEET
4a
Operations with surds
Simplifying
surds
22
M a t h s Q u e s t 1 0 fo r N ew S o u t h Wa l e s 5 . 3 P a t h w ay
WORKED
Example
4d
EXCE
et
reads
L Sp he
Simplifying
surds
am –
rogr Ca
GC p
sio
WORKED
Example
5a
GC p
am –
rogr TI
Surds
WORKED
Example
5b
SkillS
1.3
a
16a 2
b
72a 2
e
338a 3 b 3
f
i 6 162c 7 d 5
Surds
HEET
3 Simplify the following surds. Assume that a, b, c, d, e, f, x and y are positive real
numbers.
Adding and
subtracting
surds
Mat
d
hca
Adding and
subtracting
surds
Example
5c
90a 2 b
d
68a b
g
125x 6 y 4
h 5 80x 3 y 2
j 2 405c 7 d 9
k
3 5
SkillS
1.4
Multiplying
and
dividing WORKED
Example
surds
6
d
Multiplying
and dividing
surds
392e 11 f 11
b 2 3+5 3+ 3
c 8 5+3 3+7 5+2 3
d 6 11 – 2 11
e 7 2+9 2–3 2
f 9 6 + 12 6 – 17 6 – 7 6
g 12 3 – 8 7 + 5 3 – 10 7
h 2 x+5 y+6 x–2 y
5 Simplify the following expressions containing surds. Assume that a and b are positive
real numbers.
125 – 150 + 600
a
200 – 300
b
c
27 – 3 + 75
d 2 20 – 3 5 + 45
e 6 12 + 3 27 – 7 3 + 18
f
g 3 90 – 5 60 + 3 40 + 100
h 5 11 + 7 44 – 9 99 + 2 121
i 2 30 + 5 120 + 60 – 6 135
j 6 ab – 12ab + 2 9ab + 3 27ab
1
--2
98 + 1--- 48 + 1--- 12
3
l
3
150 + 24 – 96 + 108
1
--8
32 – 7--- 18 + 3 72
6
6 Simplify the following expressions containing surds. Assume that a and b are positive
real numbers.
b 10 a – 15 27a + 8 12a + 14 9a
c
150ab + 96ab – 54ab
d 16 4a 2 – 24a + 4 8a 2 + 96a
e
8a 3 + 72a 3 – 98a 3
f
g
9a 3 + 3a 5
h 6 a5b + a3b – 5 a5b
i ab ab + 3ab a 2 b + 9a 3 b 3
k
32a 3 b 2 – 5ab 8a + 48a 5 b 6
1
--2
36a + 1--- 128a – 1--- 144a
4
6
j
a 3 b + 5 ab – 2 ab + 5 a 3 b
l
4a 2 b + 5 a 2 b – 3 9a 2 b
7 Multiply the following surds, expressing answers in the simplest form. Assume that a,
b, x and y are positive real numbers.
a
2× 7
b
6× 7
c
8× 6
d
10 × 10
e
21 × 3
f
27 × 3 3
Mat
hca
1
--2
l
88ef
a 3 5+4 5
a 7 a – 8a + 9 9a – 32a
HEET
1
--2
338a 4
4 Simplify the following expressions containing surds. Assume that x and y are positive
real numbers.
k
WORKED
c
g 5 3 × 2 11
h 10 15 × 6 3
i 4 20 × 3 5
Chapter 1 Rational and irrational numbers
j 10 6 × 3 8
m
p
WORKED
Example
7
WORKED
Example
8
1
-----10
1
--4
48 × 2 2
l
n
xy × x 3 y 2
o
q
15x 3 y 2 × 6x 2 y 3
r
k
60 × 1--- 40
5
12a 7 b × 6a 3 b 4
c ( 12 )2
d ( 15 )2
e (3 2 )2
f (4 5 )2
g (2 7 )2
h (5 8 )2
9 Simplify the following surds, expressing answers in the simplest form. Assume that a,
b, x and y are positive real numbers.
15
a ---------3
8
b ------2
60
c ---------10
128
d ------------8
18
e ---------4 6
65
f ------------2 13
96
g ---------8
7 44
h ---------------14 11
2040
j ---------------30
x4 y3
k --------------x2 y5
l
9 63
------------15 7
9c
16xy
-----------------8x 7 y 9
2 2a 2 b 4
10a 9 b 3
n ---------------------- × --------------------5a 3 b 6
3 a7b
10 Express the following in their simplest form with a rational denominator.
5
a ------2
7
b ------3
4
c ---------11
8
d ------6
15
f ---------6
2 3
g ---------5
3 7
h ---------5
i
8 3
m ---------7 7
8 60
n ------------28
l
16 3
------------6 5
12
e ---------7
5 2
---------2 3
2 35
o ------------3 14
6 + 12
a ----------------------3
15 – 22
b -------------------------6
6 2 – 15
c -------------------------10
2 18 + 3 2
d -----------------------------5
3 5+6 7
e --------------------------8
4 2+3 8
f --------------------------2 3
3 11 – 4 5
g -----------------------------18
2 7–2 5
h --------------------------12
6 2– 5
j ----------------------4 8
6 3–5 5
k --------------------------7 20
l
7 12 – 5 6
-----------------------------6 3
3 5+7 3
--------------------------5 24
SkillS
Rationalising
denominators
4 3
j ---------3 5
11 Express the following in their simplest form with a rational denominator.
i
1.5
Math
cad
Example
15a 3 b 3 × 3 3a 2 b 6
b ( 5 )2
5 14
k ------------7 8
WORKED
1
--2
HEET
9a, b
3a 4 b 2 × 6a 5 b 3
a ( 2 )2
xy
12x 8 y 12
m --------------- × ----------------------x5 y7
x2 y3
Example
48 × 2 3
8 Simplify each of the following.
i
WORKED
1
--9
23
Rationalising
denominators
24
1.6
WORKED
Example
10
SkillS
HEET
M a t h s Q u e s t 1 0 fo r N ew S o u t h Wa l e s 5 . 3 P a t h w ay
Conjugate
pairs
1.7
SkillS
HEET
Applying the difference
of two squares rule to
surds
Work
T
SHEE
1.1
12 Rationalise the denominator and simplify.
1
1
a ---------------b -------------------5+2
8– 5
4
c ----------------------------2 11 – 13
5 3
d --------------------------3 5+4 2
8–3
e ---------------8+3
12 – 7
f ----------------------12 + 7
3–1
g ---------------5+1
3 6 – 15
h -------------------------6+2 3
i
5– 3
----------------------4 2– 3
1
1 Explain whether
3 is rational or irrational.
2 Explain whether
1
--- is a surd.
4
3 Simplify
80 .
4 Simplify 6 2 + 11 2 .
5 Simplify
18 +
50 –
72 .
6 Simplify 6 18 × 2 8 .
90
7 Simplify ---------- .
6
2 24
8 Simplify ------------- .
3 3
6 3
9 Express ---------- in simplest form with a rational denominator.
5 2
1+ 2
10 Express ---------------- in simplest form with a rational denominator.
1– 2
Chapter 1 Rational and irrational numbers
25
Fractional indices
1
--2
Consider the expression a . Now consider what happens if we square that expression.
1
--2
( a )2 = a (using the Fourth Index Law, (am)n = am × n)
Now, from our work on surds we know that ( a )2 = a.
1
--2
1
--2
From this we can conclude that ( a )2 = ( a )2 and further conclude that a =
1
--3
We can similarly show that a =
3
a.
a.
WORKED Example 11
Evaluate each of the following without using a calculator.
1
--2
a 9
THINK
a
WRITE
1
--2
Write 9 as
Evaluate.
1
2
b
b 64
1
--3
2
1
--2
a 9 = 9
=3
9.
Write 64 as
Evaluate.
1
1
--3
3
1
--3
b 64 = 3 64
=4
64 .
This pattern can be continued and generalised to produce a
calculator, the
x
1
--n
=
n
a . On a Casio
function can be used to find the value of expressions containing
1
--y
fractional indices. Other models of calculator may have the function x . Check with
your teacher if you are unsure.
WORKED Example 12
Use a calculator to find the value of the following, correct to 1 decimal place.
1
--4
a 10
THINK
b 200
1
--5
WRITE/DISPLAY
Continued over page
26
M a t h s Q u e s t 1 0 fo r N ew S o u t h Wa l e s 5 . 3 P a t h w ay
THINK
a Press
WRITE/DISPLAY
x
4
1
1
--4
a 10 ≈ 1.8
= on a scientific
0
x
0
EXE on a
calculator or 4 SHIFT [ ] 1
Casio graphics calculator. (On a TI graphics calculator,
4
press
b Press
5
MATH
x
5
(5:
2
x
)
0
1
ENTER .)
0
= on a scientific
0
1
--5
b 200 ≈ 2.9
x
0
0
EXE
calculator or 5 SHIFT [ ] 2
on a Casio graphics calculator. (On a TI graphics calculator,
5
press
5
MATH
(5:
x
)
2
0
ENTER .)
0
m
1
--n
Now consider1 the expression ( a ) . Using our work so far on fractional indices, we
m
--n
can say ( a ) =
n
m
a .
m
1
--n
m
---n
We can also say ( a ) = a using the index laws.
m
---n
m
n
We can therefore conclude that a = a .
Such expressions can be evaluated on a calculator either by using the index function,
which is usually either ^ or xy and entering the fractional index, or by separating the
two functions for power and root.
WORKED Example 13
2
--7
Evaluate 3 , correct to 1 decimal place.
THINK
Method 1
Press
3
Press
7
WRITE
xy
ab/c
2
2
--7
=
7
on a scientific
b
(
)
^
2
7
EXE
a /c
calculator or 3
on a Casio graphics calculator. (On a TI graphics calculator,
(
) ENTER .)
^
÷
2
7
press 3
Method 2
x
xy
3
3 ≈ 1.4
= on a scientific
2
x
^
2
EXE on a
calculator or 7 SHIFT [ ] 3
Casio graphics calculator. (On a TI graphics calculator, press
7
MATH
5
(5:
x
)
3
^
2
ENTER .)
1
--2
We can also use the index law a =
fractional indices and surds.
a to convert between expressions that involve
Chapter 1 Rational and irrational numbers
27
WORKED Example 14
Write each of the following expressions in surd form.
1
--2
a 10
THINK
b 5
3
--2
WRITE
a Since an index of 1--- is equivalent to
2
taking the square root, this term can be
written as the square root of 10.
b
3
--2
2
A power of means square root of
the number cubed.
Evaluate 53.
3
Simplify
1
1
--2
a 10 =
3
--2
b 5 =
=
125 .
10
5
3
125
=5 5
In Year 9 you would have studied the index laws and all of these laws are valid for
fractional indices.
WORKED Example 15
Simplify each of the following.
a
1
--m5
×
2
--m5
b
1
--( a2 b3 )6
THINK
a
1
 2--- --2 x 3
c  -----
 3---
 y 4
WRITE
1
---
1
Write the expression.
2
Multiply numbers with the same base by
adding the indices.
1
Write the expression.
2
---
a m5 × m5
3
---
b
= m5
1
---
2
3
Multiply each index inside the grouping
symbols by the index on the outside.
Simplify the fractions.
b ( a2b3 )6
=
2 3
--- --a6b6
=
1 1
--- --a3b2
1
c
1
2
Write the expression.
Multiply the index in both the numerator
and denominator by the index outside the
grouping symbols.
 2--- --2 x 3
c  -----
 3---
 y 4
1
--x3
= ----3
---
y8
28
M a t h s Q u e s t 1 0 fo r N ew S o u t h Wa l e s 5 . 3 P a t h w ay
remember
1. Fractional indices are those that are expressed as fractions.
2. Numbers with fractional indices can be written as surds, using the following
identities:
1
---
an =
n
m
----
a
n
an =
am = ( n a )
m
3. All index laws are applicable to fractional indices.
1D
1.8
SkillS
HEET
WORKED
Example
11
Finding
square roots, cube
roots and other roots
1.9
WORKED
Example
12
SkillS
HEET
Evaluating
numbers
in index
form
d
13
Mat
1
---
b 25 2
1
---
Example
14
1
---
e 27 3
f 125 3
2 Use a calculator to evaluate each of the following, correct to one decimal place.
1
---
1
---
a 81 4
1
---
b 16 4
c 33
1
---
1
---
e 75
f 89
3 Use a calculator to find the value of each of the following, correct to one decimal
place.
3
--8
b 100
d ( 0.6 )
WORKED
c 81 2
1
---
d 83
a 12
Fractional
indices
1
−
1
---
a 16 2
1
---
Example
hca
1 Evaluate each of the following without using a calculator.
d 52
WORKED
Fractional indices
4
--5
e
5
--9
c 50
3
--4
 3
f
-- 4
 4---
 5
2
--3
2
--3
4 Write each of the following expressions in simplest surd form.
a 7
d 2
1
--2
b 12
5
--2
e 3
1
--2
c 72
3
--2
f 10
1
--2
5
--2
5 Write each of the following expressions with a fractional index.
1.10
SkillS
HEET
Example
Using the
index
laws
15a
5
d
m
b
3
10
c
e 2 t
f
Index
laws
x
3
6
6 Simplify each of the following.
a
3
--45
d
3
--x4
×
1
--45
×
2
--x5
b
1
--28
e
1
--5m 3
h
3
2 --8--- a
×
3
--28
c
1
--a2
×
1
--2m 5
f
3
1 --7--- b
×
3
--0.05a 4
i 5x 3 × x 2
et
reads
L Sp he
EXCE
WORKED
a
g – 4y 2 ×
2
--y9
5
2
×
1
--a3
2
---
× 4b 7
1
---
Chapter 1 Rational and irrational numbers
7 Simplify each of the following.
2 3
--- ---
1 3
--- ---
a a3b4 × a3b4
3
3 2
--- ---
1 2
--1 --- --d 6m 7 × --- m 4 n 5
3
1 1
--- ---
b x5 y9 × x5 y3
1 1
--- ---
1 1 1
--- --- ---
e x3 y2 z3 × x6 y3 z2
1
---
3 4
--- ---
c 2ab 3 × 3a 5 b 5
2 3 1
--- --- ---
3 3
--- ---
f 2a 5 b 8 c 4 × 4b 4 c 4
8 Simplify each of the following.
1
---
1
---
6
--a7
3
--a7
a 32 ÷ 33
d
÷
2
---
1
---
3
--x2
1
--x4
3
---
b 53 ÷ 54
e
÷
c 12 2 ÷ 12 2
4
---
m5
f -----5
---
m9
3
---
2x 4
g --------
3
---
7n 2
h -----------
3
--4x 5
4
--21n 3
i
25b 5
-----------
c
3 4
--- --m8n7
1
--20b 4
9 Simplify each of the following.
a
÷
x3 y2
4 3
--- --x3 y5
4
---
2 1
--- ---
d 10x 5 y ÷ 5x 3 y 4
b
5 2
--- --a9b3
÷
2 2
--- --a5b5
3 3
--- --5a 4 b 5
÷
7 1
--- --p8q4
e -----------------
f ---------------
1 1
--- --20a 5 b 4
2 1
--- ---
7 p3q6
10 Simplify each of the following.
3
--4
a (2 )
d
3
--5
1
-----( a 3 ) 10
3
--7
g 4( p )
WORKED
Example
15b, c
2
--3
1
--4
4
--9
3
--8
m
---n
n
--p
b (5 )
e (m )
h (x )
14
-----15
6
1
--5
i
(7 )
( 2b )
( 3m )
c
(x y )
c
f
1
--3
1
--2
a
--b
b
--c
7
--8
2
11 Simplify each of the following.
1
--2
1
--3
a (a b )
d
1
--3
1
--2
3 3
--- --5 4
( 3a b c )
 4---
 m 5
g  ------
 7--- 
 n8 
2
3
---
b ( a4b )4
1
--3
e
1
--2
2 2
--- --3 5
(x y z )
2
 3--- --3 b 5
h  -----
 4--- 
 c 9
1
--2
3
--5
3 2
 --4- --3a
f  -----
 b
 

 1-- 4x 7 2
i  --------
 3---
 2y 4
3
--3n 8
29
30
M a t h s Q u e s t 1 0 fo r N ew S o u t h Wa l e s 5 . 3 P a t h w ay
12 multiple choice
Note: There may be more than one correct answer.
3
--4
If ( a )
GAM
Rational and
irrational numbers
— 001
Work
1
--a4 ,
is equal to
then m and n could not be:
A 1 and 3
B 2 and 6
C 3 and 8
me
E ti
T
SHEE
m
---n
1.2
D 4 and 9
13 Simplify each of the following.
a
a8
b
3
b9
c
4
m 16
d
16x 4
e
3
8y 9
f
4
16x 8 y 12
27m 9 n 15
h
5
32 p 5 q 10
i
3
216a 6 b 18
g
3
Manning’s formula
At the start of this chapter we looked at Manning’s formula, which is used to
calculate the flow of water in a river during a flood situation. Manning’s formula is
2
--3
1
--2
R S
v = ----------- , where R is the hydraulic radius, S is the slope of the river and n is the
n
roughness coefficient. This formula is used by meteorologists and civil engineers to
analyse potential flood situations.
We were asked to find the flow of water in metres per second in the river if
R = 8, S = 0.0025 and n = 0.625.
1 Use Manning’s formula to find the flow
of water in the river.
2 To find the volume of water flowing
through the river, we multiply the flow
rate by the average cross-sectional area
of the river. If the average crosssectional area is 52 m2, find the volume
of water flowing through the river each
second. (Remember 1 m3 = 1000 L.)
3 If water continues to flow at this rate,
what will be the total amount of water to
flow through in one hour?
4 Use the Bureau of Meteorology’s
website, www.bom.gov.au, to find the
meaning of the terms hydraulic radius
and roughness coefficient.
31
Chapter 1 Rational and irrational numbers
How was Albert Einstein honoured
in 1921?
Simplify each of the following
to solve the code below.
B(x )
2
1 3
2
L
3
R
2
3
2 2
5
Cx y ÷x y
3
4
3
2
Sx ÷ x
1
2
2
5
H(x y )
1
2 5
2
xy 5
x
3
2
3
3x 5
3
3x 5
W12x y ÷ 4x y
1
3
Z
1
3
2
Y
2
2
xy 5
1
1
2
3
1
5
2x 8 y 8
x
2
3
1
1
5
2x 2
2 x3 y
x3
4
2x3 y
x6
5
2
9x 5
13
13
x4 y5
x3
1
x6
3
2 xy ÷ x 8 y 8
4
4
x2
y2
2
3x 5
3x 5
5
3
x
1
5
3




I
3x 15 y 35
9 x 3 y 3 2x 2
x
2
y3
3
4x 5
2 x3 y
xy 5
1
6
O( x)
2
2 x3 y




3
7
N 3(x )
2 2
5
1
x 5 × 6 x5
5
y4
F(3x )
3
1
2
12
4
T ( x)
P
1
E 16x
1
x4 y 3 × x4 y3
4
2
3x 3 × 3 y 3
3
2
xy
2
1
9x 3 y 3
2
5
32
M a t h s Q u e s t 1 0 fo r N ew S o u t h Wa l e s 5 . 3 P a t h w ay
2
1 Simplify 4 63 .
2 Simplify 5 3 + 8 27 – 4 3 + 2 147 .
3 In simplest surd form, what is the product of 3 30 and 5 6 ?
4 Evaluate (3 5 )2.
7 5–6 7
5 Express --------------------------- in simplest form with a rational denominator.
2 3
1
--2
6 State the value of 64 .
1
--3
7 Find 90 correct to two decimal places.
4
--5
8 Find 10 correct to one decimal place.
1
--2
9 Write 6 in surd form.
10 Write
3
5
x using a fractional index.
Negative indices
In Year 9 you would have looked at negative indices and discovered the rules
1
1
a–1 = --- and a–n = ----- .
n
a
a
When using a calculator to evaluate expressions that involve negative indices, we need
1
to familiarise ourselves with the keys needed. Many calculators will have a key for ----- .
y
x
1
To evaluate 5−2 on a calculator using this function, press 5 ----- 2. Other calculators,
y
x
including the Casio and TI graphics calculators, have an x−1 function. To perform the
same calculation, first calculate 52 and then press x−1.
The following worked examples demonstrate how these calculations are performed
on a Casio calculator. Check with your teacher if you are unsure how to use your calculator for these questions.
Chapter 1 Rational and irrational numbers
33
WORKED Example 16
Evaluate each of the following.
a 4−1
b 2−4
THINK
WRITE/DISPLAY
a 4−1 = 0.25
a Press the following keys.
Scientific calculator: 4 x−1
Graphics calculator: 4 SHIFT [x–1]
b Press the following keys.
Scientific calculator: (2 xy 4) x−1
Graphics calculator: 2 ^ 4 SHIFT [x–1]
b 2−4 = 0.0625
1
Consider the index law a–1 = --- . Now let us look at the case in which a is fractional.
a
a –1
Consider the expression  --- .
 b
 a---
 b
–1
1
= -----a
--b
b
= 1 × --a
b
= --a
We can therefore consider an index of –1 to be a reciprocal function.
WORKED Example 17
Write down the value of each of the following.
a
( 2--3- )–1
b
( 1--5- )–1
( 4 )–1
c 1 1---
THINK
WRITE
( 2--3- )–1 take the reciprocal of 2--3- .
To evaluate ( 1--- )–1 take the reciprocal of 1--- .
5
5
a To evaluate
a
b
b
c
1
5
--1
2
Write
1
Write 1 1--- as an improper fraction.
2
Take the reciprocal of 5--- .
( 2--3- )–1 = 3--2( 1--5- )–1 = 5--1-
=5
as a whole number.
4
4
( 4 )–1 = ( 5--4- )–1
c 1 1---
=
4
--5
34
M a t h s Q u e s t 1 0 fo r N ew S o u t h Wa l e s 5 . 3 P a t h w ay
remember
1
1. To evaluate an expression that involves negative indices, use the ----- or the x–1
y
x
function.
2. An index of –1 can be considered as a reciprocal function and applying this to
a –1 b
fractions gives us the rule  --- = --- .
 b
a
1E
1 Evaluate each of the following.
a 5−1
b 3−1
16
−3
e 2
f 3−2
WORKED
d
Example
Mat
hca
Negative
indices
c 8−1
g 5−2
d 10−1
h 10−4
2 Find the value of each of the following, correct to 3 significant figures.
a 6−1
b 7−1
c 6−2
d 9−3
−3
−2
−2
e 6
f 15
g 16
h 5−4
et
reads
L Sp he
EXCE
Negative indices
Negative
indices
3 Find the value of each of the following, correct to 2 significant figures.
a (2.5)−1
b (0.4)−1
c (1.5)−2
d (0.5)−2
−3
−4
−3
e (2.1)
f (10.6)
g (0.45)
h (0.125)−4
4 Find the value of each of the following, correct to 2 significant figures.
a (−3)−1
b (−5)−1
c (−2)−2
d (−4)−2
−1
−1
−1
e (−1.5)
f (−2.2)
g (−0.6)
h (−0.85)−2
WORKED
Example
17
5 Write down the value of each of the following.
a
e
i
( --45- )–1
( 1--2- )–1
(1 1--2- )–1
b
f
j
( -----103- )–1
( 1--4- )–1
(2 1--4- )–1
c
g
k
( --78- )–1
( 1--8- )–1
(1 -----101- )–1
d
h
l
( -----1320- )–1
( -----101- )–1
(5 1--2- )–1
6 Find the value of each of the following, leaving your answer in fraction form.
a
e
GAM
me
E ti
Rational and
irrational numbers
— 002
Work
T
SHEE
1.3
( --12- )–2
(1 1--2- )–2
b
f
( --25- )–2
(2 1--4- )–2
c
g
( --23- )–3
(1 1--3- )–3
d
(– 1--4- )–1
(–1 --12- )–1
1
d – -----
h
( --14- )–2
(2 1--5- )–3
7 Find the value of each of the following.
( 3 )–1
(– --23- )–2
( 5 )–1
(– --15- )–2
a – 2---
b – 3---
c
e
f
g
h
( 10 )–1
(–2 --34- )–2
8 Consider the expression 2–n. Explain what happens to the value of this expression as n
increases.
Chapter 1 Rational and irrational numbers
35
summary
Copy the sentences below. Fill in the gaps by choosing the correct word or
expression from the word list that follows.
1
a
A number that can be expressed in the form --- , where b ≠ 0, is called a
b
number.
2
An
3
Examples of irrational numbers are
and e.
4
A surd is
as possible.
5
Only
6
Unlike surds can be
a
cannot be expressed in the form --- , where b ≠ 0, and
b
takes the form of a decimal that neither recurs nor terminates.
and numbers such as π
when the number under the radical sign is as small
surds can be added or subtracted.
using the rules
a ×
b =
ab and
m a × n b = mn ab .
7
When a surd is squared the result is a
8
A surd in fractional form can be simplified by
denominator.
9
Raising a number to an index of
1
--2
number.
the
is equivalent to finding the
of the number.
10
1
--n
The rules connecting fractional indices to surds are a =
=
11
m
a .
An index of –1 is equivalent to a
WORD
n
n
a
reciprocal
multiplied
function.
LIST
rational
simplified
irrational
square root
like
whole
surds
m
---n
a
rationalising
and
36
M a t h s Q u e s t 1 0 fo r N ew S o u t h Wa l e s 5 . 3 P a t h w ay
CHAPTER
review
1A
1 multiple choice
6
------ ,
12
Which of the given numbers,
1A
A
.
0.81 , 5, −3.26, 0.5 and
C
6
------ ,
12
0.81 and
. π
0.81 , 5, −3.26, 0.5 , --- ,
5
B
3
-----12
6
-----12
are rational?
π
and --5
D 5, −3.26 and
3
-----12
3
-----12
6
-----12
2 For each of the following, state whether the number is rational or irrational and give the
reason for your answer:
.
a 12
b 121
c 2--d 0.6
e 3 0.08
9
1B
3 multiple choice
Which of the numbers of the given set, { 3 2 , 5 7 , 9 4 , 6 10 , 7 12 , 12 64 }, are
surds?
A 9 4 , 12 64
B 3 2 and 7 12 only
C 3 2 , 5 7 and 6 10 only
D 3 2 , 5 7 , 6 10 and 7 12
m
20
------ , ------ ,
16
m
b if m = 8?
1B
4 Which of
1C
5 Simplify each of the following.
1C
6 multiple choice
2m ,
a if m = 4?
a
50
The expression
A 196x 4 y 3 2y
1C
25m ,
b
3
180
m,
3
8m are surds
c 2 32
d 5 80
392x 8 y 7 may be simplified to:
B 2x 4 y 3 14y
C 14x 4 y 3 2y
D 14x 4 y 3 2
7 Simplify the following surds. Give the answers in the simplest form.
a 4 648x 7 y 9
2 25
b – --- ------ x 5 y 11
5 64
Chapter 1 Rational and irrational numbers
8 Simplify the following, giving answers in the simplest form.
1C
a 7 12 + 8 147 – 15 27
b
1
64a 3 b 3 – 3--- ab 16ab + --------- 100a 5 b 5
4
5ab
1
--2
9 Simplify each of the following.
3 ×
a
b 2 6 ×3 7
5
c 3 10 × 5 6
d ( 5 )2
10 Simplify the following, giving answers in the simplest form.
a
1
--5
675 × 27
2
30
a ---------10
6 45
b ------------3 5
( 7)
d -------------14
3 20
c ------------12 6
12 Rationalise the denominator of each of the following.
2
a ------6
3
b ---------2 6
2
c ---------------5–2
3–1
d ---------------3+1
13 Evaluate each of the following, correct to 1 decimal place.
a 64
1
--3
b 20
1
--2
c 10
1
--3
d 50
1
--4
14 Evaluate each of the following, correct to 1 decimal place.
a 20
2
--3
b 2
3
--4
c ( 0.7 )
3
--5
d
 2---
 3
2
--3
15 Write each of the following in simplest surd form.
1
--2
b 18
1
--2
c 5
3
−
2
d 8
4
--3
16 Evaluate each of the following, without using a calculator. Show all working.
3
--4
1
--4
2
--3
16 × 81
a --------------------1
6 × 16
1C
1C
b 10 24 × 6 12
11 Simplify the following.
a 2
37
1
2 ----- 2
3
1C
1C
1D
1D
1D
1D
b  125 – 27 


--2
17 Evaluate each of the following.
a 4−1
b 9−1
c 4−2
d 10−3
1E
18 Find the value of each of the following, correct to 3 significant figures.
a 12−1
b 7−2
c (1.25)−1
d (0.2)−4
1E
19 Write down the value of each of the following.
1E
a
( 2--3- )–1
b
( -----107- )–1
c
( 1--5- )–1
( 4 )–1
d 3 1---
38
M a t h s Q u e s t 1 0 fo r N ew S o u t h Wa l e s 5 . 3 P a t h w ay
Non-calculator questions
20 multiple choice
The expression
A 25 10
250 may be simplified to:
B 5 10
C 10 5
D 5 50
21 multiple choice
When expressed in its simplest form, 2 98 – 3 72 is equal to:
A –4 2
B –4
C –2 4
D 4 2
22 multiple choice
When expressed in its simplest form,
x x
A ---------2
x3
B --------4
8x 3
-------- is equal to:
32
x3
C --------2
x x
D ---------4
23 Find the value of the following, giving your answer in fraction form.
CHAPTER
test
yourself
1
a
 2---
 5
–1
b
 2---
 3
–2
24 Find the value of each of the following, leaving your answer in fraction form.
a 2−1
b 3−2
c 4−3
d
 1---
 2
–1