MODULE IV
FOR THE FINAL PERIOD
Learning Objectives
At the end of Module III, the students are expected to be able to:
1. Explain and determine the roots of real numbers;
2. Recognize the principal nth root of a real number;
3. Simplify expressions containing rational exponents;
4. Define Radicals and name parts of a radical;
5. Explain the laws of radicals;
6. Operate on radical expressions;
7. Define algebraic equation;
8. Explain the different types of equations;
9. Define linear equation in one variable;
10. Solve for the solution set of a linear equation;
11. Translate worded problems into mathematical statements; and,
12. Solve mathematical problems involving linear equation.
ISHRM 2014
Learning Material
College Algebra
MT 111/MT 211 Module IV
MODULE IV
4.1 ROOTS AND RADICAL EXPRESSION
Definition: If a and b are real numbers and n is a positive integer greater than 1
such that bn = a, then b is called an nth root of a.
Illustrations:
1. Since 22 = 4 and (-2)2 = 4, then 2 and -2 are square roots of 4.
2. (-4)3 = - 64, hence – 4 is the cube root of – 64.
3. 3 and – 3 are fourth roots of 81 since 34 = 81 and (- 3)4 = 81.
In the above illustrations, 1 and 3 have more than one root. To identify which root is
needed in a problem, the concept of principal nth root is defined herewith.
𝑛
Definition: Let the principal nth root of a be denoted by √𝑎 .
𝑛
If a is a positive real number, then √𝑎 is the positive nth root of a. If a is a
𝑛
negative number and n is a positive odd integer, then √𝑎 is the negative nth root of a. If
𝑛
a is equal to zero, then √0 = 0.
Illustrations:
1. The principal square root of 4 is 2, thus √4 = 2. However, - 2 is also a square
root but not the principal square root of 4. This root is written as - √4 = - 2.
2. The principal cube root of – 64 is – 4.
4
3. 3 is the principal fourth root of 81 so √81 = 3. Although – 3 is also a fourth
𝑛
root but this is written as - √81 = - 3.
1
The principal nth root of a real number a may also be denoted by 𝑎𝑛 . That is, if b
1
is the principal nth root of a real number a, then 𝑎𝑛 = b. Hence, from the above
1
1
1
illustration: 42 = 2, (−64)3 = - 4 and 814 = 3.
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College Algebra
MT 111/MT 211 Module IV
𝑚
A more general expression containing a rationalexponent is𝑎 𝑛 . For relatively
𝑚
1
prime integers m and n, that is, they have no common factor except 1, 𝑎 𝑛 = ( 𝑎𝑛 )ᵐ or
𝑚
1
𝑎 𝑛 =(𝑎ᵐ)𝑛 , provided each expression is defined, that is the expression will result to a real
number since the variables satisfy the conditions in the definition.
Examples:
Simplify the following expressions containing rational exponents.
2
1
1. 83 = (83 )² = 22 = 4
2. (
32
243
4
4
5
) =
325
4
2435
1
=
( 325 )⁴
1
(2435 )⁴
2⁴
=
3⁴
=
16
81
3. Assume that x and y are both positive.
1
3
(𝑥 𝑦‾¹)
−
1
3
3
−
2
= (𝑥 )
3
2
(𝑦‾¹)
3
2
−
1
2
−
3
2
= 𝑥 𝑦 =
3
𝑦2
1
𝑥2
4. Assume that x is positive.
1
3
1
1 3
1 1
𝑥 2 (3𝑥 2 + 2𝑥 −2 ) = 3𝑥 2+2 + 2𝑥 2−2
4
= 3𝑥 2 + 2𝑥 0 = 3𝑥 2 + 2
1
1
2
1
1
5. (𝑥 3 − 𝑦 3 ) (𝑥 3 + 𝑥 3 𝑦 3 ) = x – y
4.2 RADICALS
𝑛
1
If a is a real number and n is a positive integer greater than 1, then √𝑎 = 𝑎𝑛
𝑛
provided a is positive whenever n is even.The expression √𝑎 is called a radical, a is
𝑛
called the radicand and n the index. √𝑎 is the radical form of the principal nth root of a
1
while 𝑎𝑛 is the exponential form. The two notations may be used interchangeably.
ISHRM 2014
Learning Material
College Algebra
MT 111/MT 211 Module IV
Illustrations:
1
1. 42
= √4 =
2
1
2. − 42 = − √4 = - 2
1
3
3. (−27)3 = √−27= - 3
1
4
4. 6254 = √625 = 5
3
1
5. 492 = (492 )³ = (√49) ³ = 7³ = 343
Simplify the following expressions.
1. √. 0081
1
2. 362
1
3. 643
2
1
4
4. 𝑥 2 (𝑥 3 − 𝑥 3 )
7
5
5. 𝑥 2 𝑥 2 𝑥
3
2
−
2
6. 325
7.
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MT 111/MT 211 Module IV
1
2
1
2
1
2
1
2
( 𝑥 + 𝑦 )(𝑥 − 𝑦 )
Laws of Radicals
1
𝑛
Where: √𝑎 = 𝑎𝑛 , if a and are real numbers and n is a positive integer.
Theorem 1. Raising to the Index
𝑛
𝑛
( √𝑎)n = a
or 𝑎𝑛 = a
This theorem states that when a radical is being raised to a power
equal to its index, and assuming the radicand is positive and the index, the
index and the exponent cancel off.
5
Example: ( √−6 )5 = - 6
Theorem 2.Root of a Product
𝑛
𝑛
1
𝑛
1
1
𝑛
√𝑎 • √𝑏 = √𝑎𝑏 or 𝑎𝑛 • 𝑏 𝑛 = (𝑎𝑏)𝑛 = √𝑎𝑏
This theorem states that when radicals with the same index are
being multiplied, only the radicands are multiplied while the index
remains the same for the product.
3
3
3
3
3
Examples: 1. √8 • √2 = √16= √8 • 2= 2 √2
2. √9 • √𝑥 = √9𝑥 = 3√𝑥
3
3
3
3. √−6𝑢²𝑣⁴ • √9𝑢⁵𝑣² = √−54𝑢⁷𝑣⁶
3
3
3
3
= √−27 • √2𝑢 • √𝑢⁶ • √𝑣⁶
6
6
3
3
= - 3(𝑢3 )(𝑣 3 ) √2𝑢= - 3u2v2 √2𝑢
Theorem 3. Root of a Quotient
𝒏
√𝒂
𝒏
√𝒃
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College Algebra
MT 111/MT 211 Module IV
𝒏
𝒂
= √ from:
𝒃
1
𝑛
√𝑎
𝑛
√𝑏
=
𝑎𝑛
1
𝑏𝑛
𝑎
𝑛
𝑎
= ( )𝑛1 = √ b ≠ 0
𝑏
𝑏
3
Examples:
1.
√𝑥⁵
3
√𝑥²
3
= √
𝑥⁵
𝑥²
3
= √𝑥³ = x
Theorem 4. Root of a root
𝒏 𝒎
𝒏𝒎
√ √𝒂 =
√𝑎
3
6
6
Examples: 1. √ √64𝑧 = √64𝑧 = 2 √𝑧
3 4
2. √ √𝑥𝑦²
=
12
√𝑥𝑦²
Simplify the following radical expressions and equations:
5
5
5
5
5
1. √64 = √32 • 2 = √32 • √2 = 2√2
2.
√𝑥² + 2𝑥𝑦 + 𝑦² = √(𝑥 + 𝑦)² = x + y
3
3
3
3
3. √81𝑥⁵𝑦⁷ = √(27𝑥 3 𝑦 6 )(3𝑥 2 𝑦) = √27𝑥³𝑦⁶ • √3𝑥²𝑦
3
= 3xy²√3𝑥²𝑦
4. √54
4
1
2
4
2
= ( 5⁴) = 5 = 52 = 25
1
1
5. √(32)² = (32²)4 = 322 = √32 = √16 • 2
= √16 • √2 = 4√2
ISHRM 2014
Learning Material
College Algebra
MT 111/MT 211 Module IV
Addition and subtraction of Radicals
Similar radicals are radicals of the same radicand and index. For
3
example, 4√3 and 3√3 are similar radicals while √2 and √2 are not
similar radicals since they have different indices although their radicands
are the same.
In adding or subtracting similar radicals, we only add or subtract
the integral factors.
Examples:
1. √3 + 2√3 = (1 + 2)√3 = 3√3
2. 10√6 + 5√6 - 7√6 = (10 +5 - 7)√6 = 8√6
3. √9 + √45 - √320 = 3 + √9 • 5 - √64 • 5
= 3 + 3 – 8 √5 = 3 - 5√5
1
1
1
4. √2 + √2 = 2√2
5. √𝑎 + √𝑏 = √𝑎 + √𝑏
3
3
3
3
6. √𝑥³𝑦 + √𝑥𝑦³= x √𝑦 + 𝑦 √𝑥
ISHRM 2014
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College Algebra
MT 111/MT 211 Module IV
Solve the following:
1.
3
3
3
√4 + 2 √4 - 5√4
3
5
2. (√2)⁶ + √25 - √64 + √−32
3.
√8𝑥⁴𝑦⁷ + √32𝑥²𝑦
4.
8√7 - 7𝑥 √7 + √875
5.
√𝑎²𝑥 − 𝑎² + √𝑎4 𝑥 − 𝑎⁴
6.
√8𝑥² − 8 + √2𝑥² − 2
7.
√16𝑥 − 16 - √𝑥³ − 𝑥²
8.
3√50 + 4√18 - 36√32
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College Algebra
MT 111/MT 211 Module IV
3
3
3
Multiplication and division of Radicals
The second theorem of radicals is very useful in this operation.
Multiplying and dividing radicals follow from the properties of radicals,
that is,
𝑛
𝑛
𝑛
√𝑎
𝑛
√𝑎 • √𝑏 = √𝑎𝑏 and
𝑛
√𝑏
𝑛
= √
𝑎
𝑏
(b ≠ 0)
If the radicals to be multiplied or divided have different indices,
derive the least common multiple (LCM) of the indices. Then multiply or
divide the radicals having the same index.
Examples:
3
1. √18𝑥 • √4𝑥 =
6
6
√18𝑥 = √9 • 2𝑥 = √3² • 2𝑥 = √3⁶ • 2³𝑥³ = 3 √2³𝑥³
3
3
6
√4𝑥 = √2²𝑥 = √2⁴𝑥²
6
6
6
3
therefore: √18𝑥 • √4𝑥 = 3 √2³𝑥³ • √2⁴𝑥² = 3 √2⁷𝑥⁵
6
6
3 • 2 √2𝑥⁵ = 6 √2𝑥⁵
2.
3
4
12
12
√𝑥𝑦² ÷ √𝑥²𝑦 = √𝑥⁴𝑦⁸ ÷ √𝑥⁶𝑦³ =
12
√𝑥⁴𝑦⁸
12
√𝑥⁶𝑦³
12
√𝑥⁶𝑦³
• 12
√𝑥⁶𝑦³
12
=
√𝑥¹⁰𝑦¹¹
𝑥𝑦
3.
√4𝑥 • √12𝑥³ = √(4𝑥)(12𝑥 3 ) = √48𝑥⁴ = √16𝑥⁴3
= 4x√3
4. (√𝑦 − 1)(√𝑦 − 1) = √(𝑦 − 1)(𝑦 − 1) = √(𝑦 − 1)²
= y–1
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MT 111/MT 211 Module IV
5. (√𝑥 + √𝑦)(√𝑥 − √𝑦) = (√𝑥 )2 – (√𝑦 )2 = x - y
3
6.
√64𝑥⁵
3
= √
3
√4𝑥²
64𝑥⁵
4𝑥²
3
3
3
= √16𝑥³ = 𝑥 √8 • 2 = 2x√2
Solve the following:
3
4
1. √𝑚 • √𝑚 • √𝑚
2.
3.
√128𝑥⁵𝑦⁶
√2𝑥⁴𝑦²
2
√3 + √4
4. (√3 + √4)²
3
3
5. ( √𝑥 + 1)(√𝑥 2 + 2𝑥 + 1)
3
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College Algebra
MT 111/MT 211 Module IV
3
6. √3
• √3
7. √5
• √9
3
WORK PROJECT / LEARNING ACTIVITY NO.1, MODULE IV
Name: ___________________________________ Yr/Sec: ____________ Date: ____________
A. Simplify the following:
3
3
1. √27 • √2
2. √16 • √𝑥
2
3. 643
27 2
4.( 64 )3
1
4
5. (𝑎3 𝑏‾¹)− 3 =
2
4
2
6. 𝑥 3 (3𝑥 3 + 2𝑥 −3 )
1
7. 252
1
8.− 362
1
9. (−64)3
1
10. 814
3
11. 362
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College Algebra
MT 111/MT 211 Module IV
WORK PROJECT / LEARNING ACTIVITY NO. 2, MODULE IV
Name: ___________________________________ Yr/Sec: ____________ Date: ____________
B. Addition and Subtraction of Radicals
1. 5√3𝑥³ + √12𝑥
2. 4√6 − 8√6 + 9√6 − 7√6
3
4
3
4
3. 2√3 − 4√5 + 10 √3 − 8√5
4. 16√45 + 9√175
3
3
5. √27 + 5√8
6. 10√5 + 11√5
7. 5√4 − 10√36𝑥 − 5√121 − 4√625𝑥
3
3
8. √40 + √320
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College Algebra
MT 111/MT 211 Module IV
WORK PROJECT / LEARNING ACTIVITY NO.3, MODULE IV
Name: ___________________________________ Yr/Sec: ____________ Date: ____________
C. Multiplication and Division of Radicals
1. √5 (√80 + √20 − √45 )
2. (√10 − √6 )(√10 + √6 )
3. (√𝑎2 − 𝑏 2 )(√𝑎 + 𝑏 )(√𝑎 − 𝑏 )
4. (√2 + √8 )
5.
6.
7.
√2 + √3 − √5
√6
2
√2 + √3 + √5
√2 − √3
√5 + √6 − √7
3
8.
2
√64𝑥⁵𝑦
3
√8𝑥𝑦
ISHRM 2014
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College Algebra
MT 111/MT 211 Module IV
4.3
ALGEBRAIC EQUATIONS
An equation is a mathematical statement that shows two expressions are
equal. The equality of such expressions is denoted by the symbol of equality ( = ).
There are different types of equations: Identity, Conditional, and Equivalent
equations.
1. The equation (x – 2)2 = x2 - 4x + 4 is an identity equation because the
equation is satisfied for any real number of x that will be substituted in (x –
2)2.
The equation (x + 3) (x – 3) = x2 – 9 is also an identity equation.
2.
The equation 3x + 2 = 2x - 3 is a conditional equation because it is
satisfied only when x = - 5.
3.
The equations 5x + 4 = 4x -3 and 7x + 39 = 4x + 18 are equivalent
equations because they have the same solution set which is {- 7}.
The equations 7x – 21 = 0 and 7x = 21 are equivalent equations because
the two equations are satisfied only when x = 3.
Identify if the following equations are identity or conditional equations:
1. (xn - 1)2 = x2n - 2xn + 1
2. x + 10 = 5x - 16
3. x2 - 9 = (x - 3) (x + 3)
4. 5 - 3x = 7x + 9
5. x + 4 = 4 - x
6. x2 + 9 = (x - 3) (x + 2)
ISHRM 2014
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College Algebra
MT 111/MT 211 Module IV
4.4 LINEAR EQUATIONS
A linear equation in a single variable is a mathematical equation that
involves a single variable with 1 as the highest exponent.
Some examples of linear equation in the variable xare:
x - 1 = 3, 3x + 2 = 4, and
𝑥
2
= 5𝑥 −
1
3
The value of the variable x that will satisfy the equation is called the
solution set of the linear equation. In the equation x - 1 = 3, when the value of x
is 4, the equation is true. Hence, x = 4 is the solution of the given equation.
To solve a linear equation means to find the solution of the equation. The
following are the steps in solving linear equation in one variable:
1. Transfer all terms containing variables to the left side of the equation,
or transfer all constants to the right side of the equation.
Example:
3x + 2 = 4 transfer 2 to the right side of the equal
sign, becomes 3x = 4 – 2.
2. Simplifying the equation becomes 3x = 2.
3. Divide both sides of the equation by the coefficient {3},
3𝑥
3
=
2
2
, therefore x =
3
3
Our solution shows that the value of x =
2
.
3
4. Check if the value of x is correct which will make the equation true.
2
Substitute x with the value in the equation, re-write the equation and
3
2
simplify 3x + 2 = 4 becomes 3(3) + 2 = 4,
= 4.
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MT 111/MT 211 Module IV
2 + 2 = 4, then 4
Examples:
1. 2x + 3 = 4x - 5
Check:
2x - 4x = - 5 - 3
2(4) + 3 = 4(4) - 5
- 2x = - 88 + 3 = 16 - 5
-2
-2
11 = 11
x = 4
2. (n + 2)(n + 4) = n(n + 8)
n2 + 6n + 8 = n2 + 8n
n2 – n2 + 6n - 8n = - 8
- 2n = - 8
- 2
-2
n = 4
3𝑎
3.
(
+
4
3𝑎
4
+
𝑎
3
𝑎
3
=
13
=
13
3
3
)¹²
9a + 4a = 52
13a = 52
13
13
a = 4
ISHRM 2014
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College Algebra
MT 111/MT 211 Module IV
Solve the following linear equations and check solution set:
1. x + 5 = 8
2. 3y + 3 = y - 4
3. (3z - 8) – 2(z + 5) = 0
4.
5.
10
4𝑥 + 3
𝑦
2𝑦−3
4
=
𝑥
1
−
2
=
1
2𝑦 + 3
6. 2x + 3 = 4x - 1
7. (x - 2) (x + 3) = (x - 3) (x + 5)
8.
9.
5𝑥 + 3
2
8𝑥 + 11
4
=
7𝑥 − 8
3
= 3
10. 3x + 4 = 5x - 10
ISHRM 2014
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College Algebra
MT 111/MT 211 Module IV
4.5 MATHEMATICAL STATEMENTS
Mathematical statements are statements written in algebraic forms which
are equivalent to worded statements.
For examples, translate the following worded phrases into mathematical
statements:
1. “The square of the sum of x and y”
Written as:
(x - y)2
2. “The sum of the squares of x and y”
Written as:
x2 + y2
3. “The sum of x and the product of 5 and y”
Written as:
x + 5y
4. “Five times the difference of the cubes of x and y”
Written as:
5(x3 - y3)
5. “The sum of the three consecutive even integers”
Written as:
x + (x + 2) + (x + 4)
Write a mathematical statement of the following worded problems:
6. The length of the rectangle is twice its width. The perimeter of the
rectangle is 50 inches.
Let x – the width of a rectangle
2x – the length of the rectangle which is twice its width.
The formula for the perimeter of the rectangle:
P = 2(L + W)
Therefore: 50 = 2(2x + x)
ISHRM 2014
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College Algebra
MT 111/MT 211 Module IV
7. A woman is nine times as old as her son and her husband is five years
older than she. The sum of their ages is 81.
Let x - the age of the sun
9x - the age of the woman whose age is 9 times of the age of her
son
9x + 5 - the age of the husband
Therefore:
x + 9x + (9x + 5) = 81
8. In a group of 200 students there are 40 more boys than girls.
Let x - the number of girls
40 + x - the number of boys who are 40 more than the girls.
Therefore:
x + (40 + x) = 200
9. The second number is two greater than the first number while the third
number is three times the first number. The sum of the three numbers
is 17.
Let x - the first number
2 + x - the second number greater than the first number by 2
3x
- the third number three times the first
Therefore:
x + (2 + x) + 3x = 17
10. Business Problem
A man invested part of his ₱5, 000 at a bank that offers 4 % interest
rate and the rest is invested at 5 %. The total annual income from the
two investments is ₱209.00. How much was invested at each of the
rates?
ISHRM 2014
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College Algebra
MT 111/MT 211 Module IV
Let x - the amount invested at 4 % = 0.04x annual income
(5, 000 - x) - the amount invested at 5 % = 0.05(5,000 – x) annual
Therefore: 0.04x + 0.05(5,000 - x) = ₱209.00
0.04x + 250 - 0.05x = ₱ 209. 00
250 - 0.01x = ₱ 209. 00
250 - 209 = 0.01x
41 = 0.01x
0.01 0.01
x = ₱4, 100 amount invested at 4 %
5, 000 – 4, 100 = ₱ 900 amount invested at 5 %
11.
The sum of three numbers is 27. The second number is 2 less than
twice the first number, and the third number is 5 greater than 3 times
the first number. What are the numbers?
Let x - the first number
(2x – 2) - the second number
(3x + 5) - the third number
Therefore:
x + (2x - 2) + (3x + 5) = 27
x + 2x - 2 + 3x + 5 = 27
6x + 3 = 27
6x = 27 - 3
6x = 24
6
6
x = 4
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MT 111/MT 211 Module IV
12.
Berto is 40 years old and his daughter, Cuartita, is 14 years old. In
how many years will he be twice as old as his daughter?
Let x - the number of years required
40 + x - age of Berting after x years
14 + x - age of Cuartita after x years
Therefore:
40 + x = 2(14 + x)
Translate the following verbal expressions into mathematical statements.
1.
“Three subtracted from a number n”
2.
“Five times a number t diminished by two”
3.
“The price of a toy after 10 % discount if the original price is q”
4.
“The average of three numbers a, b, and c”
5.
“Thrice the number x increased by 5”
6.
“Nine more than five times a number”
7.
“The sum of three consecutive odd numbers”
The second number is (x + 2) and the third is (x + 4).
8.
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MT 111/MT 211 Module IV
“Three consecutive even numbers whose sum is 12”
Translate the following worded problem into mathematical statement and solve:
1.
In a group of 300 students there are 30 more boys than girls.
2.
The second number is three greater than the first number while the
third number is five times the first number. The sum of the three
numbers is 24. What are the three numbers/
3.
The length of the rectangle is thrice its width. The perimeter of the
rectangle is 60 inches. How long are the length and the width of
the rectangle?
4.
A girl is two times as old as her younger sister and her eldest sister
is five years older than she. If the sum of their ages is 50, how old
are each of these sisters?
5.
The sum of three numbers is 35. The second number is 3 less than
thrice the first number, and the third number is 6 greater than 4
times the first number. What are the numbers?
ISHRM 2014
Learning Material
College Algebra
MT 111/MT 211 Module IV
WORK PROJECT / LEARNING ACTIVITY NO. 4, MODULE IV
Name: ___________________________________ Yr/Sec: ____________ Date: ____________
A. Translate the following worded phrases into mathematical statements:
1. “The cube of the sum of x and y”
2. “The product of the squares of x and y”
3. “The sum of x and the product of 3 and y”
4. “Eight times the difference of the squares of x and y”
5. “The sum of the three consecutive odd integers”
6.
7.
“Three consecutive even numbers whose sum is 42”
“Twelve times a number m diminished by four”
8.
“The price of a dress after 7 % discount if the original price is q”
9.
“Thrice the number x increased by 5”
10.
“The sum of three consecutive odd numbers whose sum is 45”
ISHRM 2014
Learning Material
College Algebra
MT 111/MT 211 Module IV
WORK PROJECT / LEARNING ACTIVITY NO. 5, MODULE IV
Name: ___________________________________ Yr/Sec: ____________ Date: ____________
A. Translate the following worded phrases into mathematical statements:
1. Business Problem
Miss Rivera invested a certain amount of money at a bank that offers 4 %
interest per year and the rest is invested at 5 %. If her total investment is
₱100,000. How much did Miss Rivera invested at each rates?