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 2Y x 16 x 8Y = 256, find y
Solution:
2y x 24 x 8y = 256
2y x 24 x 8y = 28
2y 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 125m x 252 = 510 find m
Solution:
125m x 252 = 510
53m 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 = 34-2 = 32
Generally:

Example 1.
Find i) = 87-5
= 82
ii) = 52n-n
= 5n
Example 2.
If = 81 find n
Solution:
= 81
() = 34
33n – 4 = 34

Equate the exponents
3n – 4 = 4
n=
NEGATIVE EXPONENTS
Suppose = 32 – 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 2y = 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 = 92-2 = 90
90 = 1 hence shown
Also
(i) m =
(ii) (x y)m = xm ym
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=54×3

Examples:
1.Simplify (a (x4)5
(b) (86)3

Solution
(a) (x4)5=x4×5
=x20

(b) (86)3= 86×3
=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
272/3 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
32+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 2x x 3y = 5184 find x and y
Solution:
2x = 5184 2x x 3y = 26 x 3y
2x = 26 By comparison
2x = 26 2x = 26
X = 6
3y = 5184 3x = 34
3y = 34
y = 4
The value of x and y is 6 and 4 respectively

-A number involving roots is called a surd or radical.
-Radical is a symbol used to indicate the square root, cube root or nth root of a number.
-The symbol of a radical is
(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 = 8x find x
Solution:
= = 8x
= (23)1/3 = 23x
= 21 = 23x
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 = 16y find y
= = 24y
2 2 = 24y
2 = 4y
y =
4. Find x if
=491/3
Solution
= = 491/3
3431/x = 73/x = (72)1/3
73/x = 7 2/3
=
2x = 9
x =
ii) = 81x
solution
= = 81x
= 32 = 34x
= 2 = 4x
x =

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 rational denominator.
The best choice can follow the following rules:-
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 2 both 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

subscriber

## 1 Comment

• ### David Garjaye, October 26, 2023 @ 6:38 pmReply

I’m very excited. I’m also grateful to you for this latest mathematics app.
I’m David Garjaye, a Liberian.
Thanks.

By

By

By

By