Form 2 Mathematics – EXPONENT AND RADICALS

EXPONENTS:

– Is the repeated product of real number by itself

e.g. i) 2 x 2 x 2 x 2 = 2⁴

ii) 6 x 6 x 6 x 6 x 6 = 6⁵

iii) a x a x a x a x a = a⁵

LAWS OF EXPONENTS

MULTIPLICATION RULE

Suppose;

4 x 4 x 4 = 4³

Then, 4³ = power

4 = base

3 = exponent

Suppose, 3² x 3⁴ = 3⁽²⁺⁴⁾ = 3⁶

3² x 3⁴ = 3 x 3 x 3 x 3 x 3 x 3 = 3⁶

Example 1
Simplify the following

i) 6⁴ x 6⁸ x 6⁶ x 6¹

ii) y⁴ x y⁰ x y³

Solution:

i) 6⁴ x 6⁸ x 6⁶ x 6¹ = 6 ^4+8+6+1

= 6¹⁹

ii) y⁴ x y⁰ x y³

Solution:

Y⁴ x y⁰ x y³ = y⁴⁺⁰⁺³

= y⁷

Example 2
Simplify the following

i) 3² x 5⁴ x 3³ x 5²

ii) a³ x b³ x b⁴ x a⁵ x b²

Solution:

i) 3² x 5⁴ x 3³ x 5² = 3²⁺³ x 5⁴⁺²

= 3⁵ x 5⁶

ii) a³ x b³ x b⁴ x a⁵ x b² = a³⁺⁵ x b³

= a⁸ x b⁹

Example 3

If 2^Y x 16 x 8^Y = 256, find y

Solution:

2^y x 2⁴ x 8^y = 256

2^y x 2⁴ x 8^y = 2⁸

2^y x 2⁴ x (2³)^y = 2⁸

y + 4 + 3y = 8

y + 3y = 8 – 4

4y = 4

Y = 1

Exercise 1:

1. Simplify

i) 3⁴ x 4³ x 3⁸ x 3⁴ x 4² = 3⁴⁺⁸⁺⁴ x 4³⁺² = 3¹⁶ x 4⁵

ii) a² x a³ x a⁴ x b² x b³ = a²⁺³⁺⁴ x b²⁺³ = a⁹ x b⁵

2. If 125^m x 25² = 5¹⁰ find m

Solution:

125^m x 25² = 5¹⁰

5^3m x 5⁴ = 5¹⁰

3m + 4 = 10

3m = 10 – 4

3m=6

m = 2

3. If x⁷ = 2187. Find x

Solution:

X⁷ = 2187

X⁷ = 3⁷

X = 3

QUOTIENT LAW

= = 3 X 3

= 3²

Also = 3^4-2 = 3²

Generally:

Example 1.

Find i) = 8^7-5

= 8²

ii) = 5^2n-n

= 5ⁿ

Example 2.

If = 81 find n

Solution:

= 81

() = 3⁴

3^{3n – 4} = 3⁴

Equate the exponents

3n – 4 = 4

n=

NEGATIVE EXPONENTS

Suppose = 3^{2 – 4} = 3^-2

Also =

=

and Inversely xⁿ =

Example

Find

( i) 2^-3 = =

(ii) 9^-1/2 =

(iii) = 3³ = 27

EXERCISE 2

1. Given 2³ⁿ x 16 x 8ⁿ = 4096 find n

2. Given = 5⁶ find y

3. If 3²ⁿ⁺¹ – 5 = 76 find n

4. Given 2^y = 0.0625.Find y

6. Find the value of x

(i). 81^-1/2 = x

ii) 2^-x = 8

ZERO EXPONENTS

Suppose,

= = 1

3⁰ = 1

Example

Show that 9⁰ = 1

Consider = = = 1

Also = 9^2-2 = 9⁰

9⁰ = 1 hence shown

Also

(i) ^m =

(ii) (x y)^m = x^m y^m

Example

(1)Find
i) (5 x 4)²

Solution:

(5 x 4)² = 5² x 4²

5 x 5 x 4 x 4 = 400

ii) ()³

= =

2. Show that 2^-1 =

Solution:

2^-1 =

=

consider LHS

2^-1 =

L H S = R H S

Therefore

2^-1 = hence shown

FRACTIONAL EXPONENTS AND EXPONENTS OF POWERS

EXPONENTS OF POWERS

Consider (5⁴)³=(5x5x5x5)³
=(5x5x5x5)x(5x5x5x5)x(5x5x5x5)
=5x5x5x5x5x5x5x5x5x5x5x5
=5¹²
Similarly (5⁴)³=5^4×3

Examples:
1.Simplify (a (x⁴)⁵

(b) (8⁶)³

Solution
(a) (x⁴)⁵=x^4×5
=x²⁰

(b) (8⁶)³= 8^6x3
=8¹⁸

2.Write ^{23x 42 as a power of single number}
Solution
2³x 4² ,but 4=2²

therefore 42=(22)2
42=22×2
=24
23x 24=23+4
∴23x 24=27
^{FRACTIONAL EXPONENT}

Solution

Consider the exponents of powers when is squared, we get

Let x be positive number and let n be a natural number. Then

Examples:

(1) Find

Thus if x is a negative number, and n is an odd number

Exercise 2.

1. Show that 2^-2 =

Solution:

Consider LHS

2^-2 = =

2^-2 =

LHS = RHS hence shown

2. Evaluate

27^2/3 x 729 ^1/3 ÷ 243

Solution:

27 ^2/3 x 729 ^1/3 ÷ 243

(3³)^2/3 x (3⁶)^1/3 ÷ 3⁵

3² x 3² ÷ 3⁵

3^2+2-5

= 3^-1 or

3. Find the value of m

(1/9)^2m x (1/3)^-m ÷ (1/27)² = (1/3)^-3m

Solution:

(1/3²)^2m x 1/3^-m ÷ (1/3³)² = 1/3^-3m

(1/3)^4m x (1/3)^-m ÷ (1/3)⁶ = (1/3)^3m

3^-4m x 3^-m ÷ 3^-6 = 3^-3m

-4m + -m – 6 = -3m

-5m – 6 = -3m

6 = -2m

m = -3

4. Given 2^x x 3^y = 5184 find x and y

Solution:

2^x = 5184 2^x x 3^y = 2⁶ x 3^y

2^x = 2⁶ By comparison

2^x = 2⁶ 2^x = 2⁶

X = 6

3^y = 5184 3^x = 3⁴

3^y = 3⁴

y = 4

The value of x and y is 6 and 4 respectively

RADICALS

-A number involving roots is called a surd or radical.
-Radical is a symbol used to indicate the square root, cube root or n^th root of a number.
-The symbol of a radical is

Example of Radicals
(i)

(ii)

(iii)

PRIME FACTORS

Example 1

Find (i) by prime factorization

Solution:

=
= 2×7

= 14

ii) by prime factorization

solution:

=

= 2 x 3

= 6

iii) by prime factorization

solution:

=

= 2

Example 2

If = 8^x find x

Solution:

= = 8^x

= (2³)^1/3 = 2^3x

= 2¹ = 2^3x

x=

Exercise 3

1. Find the following

i)

Solution

=

= 2 x 2 x 2 x 2 x 2

= 32

=32

ii)

Solution

=

= 5

2. Simplify

a) Solution

=

= 5

b) =

= 3 x 5

= 15

3. Find = 16^y find y

= = 2^4y

2 ² = 2^4y

2 = 4y

y =

4. Find x if

=49^1/3

Solution

= = 49^1/3

343^1/x = 7^3/x = (7²)^1/3

7^3/x = 7 ^2/3

=

2x = 9

x =

ii) = 81^x

solution

== 81^x

= 3² = 3^4x

= 2 = 4x

x =

OPERATION ON RADICAL

ADDITION

Example1.

Evaluate

i) +3

Solution: + 3=(1 + 3)

=4

ii) +

Solution

=+

(2²)^{1/2 (}3²)^1/2 + (2²)^1/2 (2²)^1/2

= (2 x 3) + (2 x 2)

= 6 + 4

= 10

SUBTRACTION

Example

Evaluate

i) 3 – 2

Solution

= 3 n-2

= (3 x 2 x 3 2 x 2 x 2 )

= 18 8

= 10

ii)

Solution

=

=(2 x 3) (2 x 2)

= 6 4

= 2

MULTIPLICATION

Example

Find i) x

solution

x =

=

=

= 2 x 2 x 2 x 3

= 24

ii) 3 x 3

Solution

3 x 3

(5 x 3) x (3 x 3)

= 15 x 9

= 135

DIVISION

Example 1

Find i)

Solution: =

=

=

=

EXERCISE 4.

1. Find 2 + 3

Solution: 2 +3

= (2 x 2 x 3)+ (3 x 2 x 2)

= 12 +12

= 24

(ii )3

Solution:

3 = 3 + 3

= 3 + 3

=(3 x 2) +(3 x 2 x 3)

= 6 +12

= 18

(iii) 6 2

Solution:

6 2 6 = 2

= (6 x 2) (2 x 3)

= 12 6

= 6

iv) +

Solution:

+

+3

4

(v) + 2250

Solution:

+ = +2250

= 2 + 2250

=2 + 2250

=2 + 2250

2. Simplify

(i) x

=

=

=

= 24

ii)

()

= (2 x 3 – 4 )

= (6 – 4 )

= (2 )

= 4

(iii) 3 x 2

Solution:

= 3 x 2

= 3 x 2 x 3 x (2 x 2)

= 18 x 4

= 72

(iv) (15 )

Solution:

(15 )= 15

= 15 X 3

= 45

RATIONALIZATION OF THE DENOMINATOR

– Rationalizing the denominator involves the multiplication of the denominator by a suitable radical resulting in a rationaldenominator.
The best choice can follow the following rules:-
(i) If a radical is a single term(that is does not involve + or -),the proper choice is the radical itself,that is

(ii)If the radical involves operations(+ or -),choose a radical with the same format but with one term with the opposite operation.

Examples

The same technique can be used to rationalize the denominator.

Example 1

Rationalize i)

Solution = x

=

(ii)

Solution:

= x

=

=

(iii)

Solution:

= x

=

=

=

=

Example 2:

Rationalize (i)

Solution:

= x

=

=

=

=

=

=

(ii) Rationalize

Solution:

= x

=

=

=

=

=

=

=

EXERCISE 5

1. Evaluate

(i) ()()

Solution:
(1) ()() = (() -4()

= – 6 – 12 + 12

(ii) ()()

Solution:
(iii) ()() = () + ()

= a + + + b

= a + b + 2

(iv) ()()

Solution:

()() = () + ()

= m + – – n

= m – n

(v) ()()

Solution:

()() = ( – ()

= p – + – q

= p – q

2. Rationalize

(i)

Solution:

= x

=

=

=

=

=

(ii)

Solution:

=

=

= – ( )

EXERCISE 6

Rationalize the following denominator

(i)

Solution:

=

=

=

=

(ii)

Solution:

=

=

=

=

(iii)

Solution:

=

=

=

=

(iv)

Solution:

=

=

=

SQUARE ROOT OF A NUMBER

Example

Find( i)

Solution

ii)
Solution:

(iii)

Solution:

TRANSPOSITION OF FORMULA

A formula expresses a rule which can be used to calculate one quantity where others are given,when one of the given quantity is expressed in terms of the other quantity the process is called transposition of formula.

Example 1

The following are examples of a formula

a. A =

b. v =

c. I =

d. A = (a +b)h

e. T = 2r

Example 2

The simple interest (I) on the principal (p) for time (T) years. Calculated at the rate of R% per annual is given by formula

I =

Make T the subject of a formula

Solution:

100 x I = x 100

=

=

T =

Example 3.

Given that

Y = mx + c, make m the subject

Solution:

Y = mx +c

=

m =

Example 4
Given that p = w
Make a the subject.

Solution:

P = w

Divide by w both sides

=

=

Multiply by (1 – a) both sides

(1 – a) = (1 a)

(1 – a) = 1 + a

– = 1 + a

– 1 = a +

– 1 = a(1 + )

Divide by 1 + both sides

=

a =

Alternatively

Example 5

Given that T = 2 write g in terms of other letters

Solution:

T = 2

Divide by 2both side

=

Remove the radical by squares both sides

² = ²

=

Multiply by g both sides

=g

=

Multiply by 4² both sides

4² x = x 4²

T²g = 4²

Divide by T² both sides

∴ g =

Example 6

If A = p +

(i) Make R as the subject formula

(ii) Make P as the subject formula

Solution:

(i) A = p +

= A – P =

Multiply by 100 both sides

= = R

R =

(ii) A = P +

Solution:

Multiply by 100 both sides

100A = 100P + PRT

100A = P(100 + RT)

Divide by 100 + RT both sides

= P

P =

Exercise 7

1. If S = at². Make t the subject of the formula

2. If c = (F – 32) make F the subject of the formula

Solution:

S = at²

Multiply by 2 both sides

s x 2 = at² x 2

2s = at²

Divide by a both sides

=

t² =

Square root both sides

=

t =

2. C = (F – 32)

C = F –

C + =

Multiply by 9 both sides

9C + =

Divide by 5 both sides

F =

More Examples

1. If A = (a + b)

(i) Make h the subject formula

(ii) Make b the subject formula

2. If = –

(i) Make f the subject formula

(ii) Make u the subject formula

Solution:

1. A =

2A = (a + b)x 2

2A = (a + b)

Divide by a + b both sides

=

h =

(ii) Make b the subject formula.

Solution:

A =

2A = (a + b)x 2

2A = (a + b)

2A = ah + bh

2A ah = bh

Divide by h both sides

= b

b =

2. = –

Solution:

= –

=

Divide by u – v both sides

f =

ii) Make u the subject formula

= –

Solution:

=

Multiply by uv both sides

= f(u – v)

uv = fu – fv

fv = fu – uv

fv =u (f – v)

Divide by f – v both sides

u =

Exercise 8

1. If T =

(i) Make t the subject formula

(ii) Make g the subject

2. If P = w

(i) Make w as the ^subject formula

(ii) Make a the subject formula

Solution:

1. (i)T =

Square both sides

T² =

Multiply by 4 both sides

4T² =

4T²g = 9t

Divide by 9 both sides

t =

(ii) Make g the subject formula

T =

Solution:

Square both sides

T² =

Multiply by 4 both sides

4T² =

4T²g = 9t

Divide by 4T² both sides

g =

2)( i) Make w was the subject

Make a the subject

Solution:

P = w

P= w()

Divide by () both sides

w =P

ii) Make a the subject formula

Solution:

P = w

Divide by w both sides

=

=

Multiply by (1 – a) both sides

(1 – a) = (1 a)

(1 – a) = 1 + a

– = 1 + a

– 1 = a +

– 1 = a(1 + )

Divide by 1 + both sides

=

a =

Exercise 9

I. If v = Make R the subject formula

Solution:

v =

Multiply by r + R both sides

v (r + R) = 24R

vr + Rv = 24 R

vr = 24R – Rv

vr = R (24 – v)

Divide by 24 – v both sides

2. If m = n

(i) Make x the subject formula

Solution:

m = n

Multiply by x + y both sides

mx + my = nx – ny

my + ny = nx – mx

my + ny = x(n – m)

divide by n – m both sides

x =

(ii)If T = 2

Make t the subject formula

Solution:

T = 2

Square both sides

T² = 4²

Multiply by a both sides

T²a = 4²kt

Divide by 4²k both sides

t = ²