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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
C:thlbcrtz__i__images__i__e65.png
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
C:thlbcrtzEXPONENTS_F2_filesimage001.gif = C:thlbcrtzEXPONENTS_F2_filesimage002.gif= 3 X 3
= 32
Also C:thlbcrtzEXPONENTS_F2_filesimage001.gif = 34-2 = 32
Generally:

C:thlbcrtz__i__images__i__e75.png
Example 1.
Find i) C:thlbcrtzEXPONENTS_F2_filesimage004.gif = 87-5
= 82
ii) C:thlbcrtzEXPONENTS_F2_filesimage005.gif= 52n-n
= 5n
Example 2.
If C:thlbcrtzEXPONENTS_F2_filesimage006.gif = 81 find n
Solution:
C:thlbcrtzEXPONENTS_F2_filesimage006.gif = 81
(C:thlbcrtzEXPONENTS_F2_filesimage007.gif) = 34
33n – 4 = 34

Equate the exponents
3n – 4 = 4
n= C:thlbcrtzEXPONENTS_F2_filesimage008.gif
NEGATIVE EXPONENTS
Suppose C:thlbcrtzEXPONENTS_F2_filesimage009.gif = 32 – 4 = 3-2
Also C:thlbcrtzEXPONENTS_F2_filesimage009.gif = C:thlbcrtzEXPONENTS_F2_filesimage010.gif
= C:thlbcrtzEXPONENTS_F2_filesimage011.gif
C:thlbcrtz__i__images__i__E37.png
and Inversely xn = C:thlbcrtzEXPONENTS_F2_filesimage013.gif
Example
Find
( i) 2-3 = C:thlbcrtzEXPONENTS_F2_filesimage014.gif= C:thlbcrtzEXPONENTS_F2_filesimage015.gif
(ii) 9-1/2 = C:thlbcrtzEXPONENTS_F2_filesimage016.gif
(iii) C:thlbcrtzEXPONENTS_F2_filesimage017.gif = 33 = 27
EXERCISE 2
1. Given 23n x 16 x 8n = 4096 find n
2. Given C:thlbcrtzEXPONENTS_F2_filesimage018.gif= 56 find y
3. If 32n+1 – 5 = 76 find n
4. Given 2y = 0.0625.Find y
C:thlbcrtz__i__images__i__a771.png
6. Find the value of x
(i). 81-1/2 = x
ii) 2-x = 8
ZERO EXPONENTS
Suppose,
C:thlbcrtzEXPONENTS_F2_filesimage021.gif = C:thlbcrtzEXPONENTS_F2_filesimage022.gif = 1
C:thlbcrtz__i__images__i__ee14.png
30 = 1
C:thlbcrtz__i__images__i__a2412.png
Example
Show that 90 = 1
Consider C:thlbcrtzEXPONENTS_F2_filesimage023.gif = C:thlbcrtzEXPONENTS_F2_filesimage024.gif = C:thlbcrtzEXPONENTS_F2_filesimage025.gif = 1
Also C:thlbcrtzEXPONENTS_F2_filesimage023.gif = 92-2 = 90
90 = 1 hence shown
Also
(i) C:thlbcrtzEXPONENTS_F2_filesimage026.gif m = C:thlbcrtzEXPONENTS_F2_filesimage027.gif
(ii) (x C:thlbcrtzEXPONENTS_F2_filesimage028.gif y)m = xm C:thlbcrtzEXPONENTS_F2_filesimage028.gif 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) (C:thlbcrtzEXPONENTS_F2_filesimage029.gif)3
C:thlbcrtzEXPONENTS_F2_filesimage030.gif= C:thlbcrtzEXPONENTS_F2_filesimage031.gif = C:thlbcrtzEXPONENTS_F2_filesimage032.gif
2. Show that 2-1 = C:thlbcrtzEXPONENTS_F2_filesimage033.gif
Solution:
2-1 = C:thlbcrtzEXPONENTS_F2_filesimage033.gif
C:thlbcrtzEXPONENTS_F2_filesimage033.gif = C:thlbcrtzEXPONENTS_F2_filesimage033.gif
consider LHS
2-1 = C:thlbcrtzEXPONENTS_F2_filesimage033.gif
L H S = R H S
Therefore
2-1 = C:thlbcrtzEXPONENTS_F2_filesimage033.gif 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

C:thlbcrtz__i__images__i__a3331.png
C:thlbcrtz__i__images__i__a2121.png
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
C:thlbcrtz__i__images__i__a1221.png
Solution
Consider the exponents of powers when C:thlbcrtz__i__images__i__F30.png is squared, we get
C:thlbcrtz__i__images__i__F110.png
Let x be positive number and let n be a natural number. Then
C:thlbcrtz__i__images__i__a2123.png
Examples:
(1) Find C:thlbcrtz__i__images__i__F52.png

C:thlbcrtz__i__images__i__F61.png
C:thlbcrtz__i__images__i__F71.png

C:thlbcrtz__i__images__i__f81.png
Thus if x is a negative number, and n is an odd number

C:thlbcrtz__i__images__i__f91.png


Exercise 2.
1. Show that 2-2 = C:thlbcrtzEXPONENTS_F2_filesimage034.gif
Solution:
Consider LHS
2-2 = C:thlbcrtzEXPONENTS_F2_filesimage035.gif = C:thlbcrtzEXPONENTS_F2_filesimage034.gif
2-2 = C:thlbcrtzEXPONENTS_F2_filesimage034.gif
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 C:thlbcrtzEXPONENTS_F2_filesimage036.gif
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

RADICALS
-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 C:thlbcrtzEXPONENTS_F2_filesimage037.gif
C:thlbcrtzEXPONENTS_F2_filesimage038.gifExample of Radicals
(i) C:thlbcrtzEXPONENTS_F2_filesimage039.gif
(ii) C:thlbcrtzEXPONENTS_F2_filesimage042.gif
(iii)C:thlbcrtzEXPONENTS_F2_filesimage044.gif
PRIME FACTORS
Example 1
Find (i)C:thlbcrtzEXPONENTS_F2_filesimage046.gif by prime factorization
Solution:
C:thlbcrtzEXPONENTS_F2_filesimage047.gif = C:thlbcrtzEXPONENTS_F2_filesimage048.gif
= 2×7
= 14
C:thlbcrtzEXPONENTS_F2_filesimage049.gifii) C:thlbcrtzEXPONENTS_F2_filesimage050.gif by prime factorization
solution:
C:thlbcrtzEXPONENTS_F2_filesimage050.gif= C:thlbcrtzEXPONENTS_F2_filesimage051.gif
= 2 x 3
= 6
iii) C:thlbcrtzEXPONENTS_F2_filesimage052.gif by prime factorization
solution:
C:thlbcrtzEXPONENTS_F2_filesimage052.gif = C:thlbcrtzEXPONENTS_F2_filesimage053.gif
= 2 C:thlbcrtzEXPONENTS_F2_filesimage054.gif
Example 2
If C:thlbcrtzEXPONENTS_F2_filesimage042.gif= 8x find x
Solution:
C:thlbcrtzEXPONENTS_F2_filesimage042.gif= C:thlbcrtzEXPONENTS_F2_filesimage055.gif= 8x
= (23)1/3 = 23x
= 21 = 23x
x= C:thlbcrtzEXPONENTS_F2_filesimage036.gif

C:thlbcrtz__i__images__i__c78.png
Exercise 3
1. Find the following
i) C:thlbcrtzEXPONENTS_F2_filesimage057.gif
Solution
C:thlbcrtzEXPONENTS_F2_filesimage057.gif= C:thlbcrtzEXPONENTS_F2_filesimage058.gif
= 2 x 2 x 2 x 2 x 2
= 32
C:thlbcrtzEXPONENTS_F2_filesimage057.gif =32
ii) C:thlbcrtzEXPONENTS_F2_filesimage059.gif
Solution
C:thlbcrtzEXPONENTS_F2_filesimage059.gif = C:thlbcrtzEXPONENTS_F2_filesimage060.gif
= 5
2. Simplify
a) C:thlbcrtzEXPONENTS_F2_filesimage061.gif Solution
C:thlbcrtzEXPONENTS_F2_filesimage061.gif= C:thlbcrtzEXPONENTS_F2_filesimage062.gif
= 5 C:thlbcrtzEXPONENTS_F2_filesimage063.gif
b) C:thlbcrtzEXPONENTS_F2_filesimage064.gif=C:thlbcrtzEXPONENTS_F2_filesimage065.gif
= 3 x 5 C:thlbcrtzEXPONENTS_F2_filesimage066.gif
= 15 C:thlbcrtzEXPONENTS_F2_filesimage066.gif
3. Find C:thlbcrtzEXPONENTS_F2_filesimage067.gif = 16y find y
C:thlbcrtzEXPONENTS_F2_filesimage067.gif=C:thlbcrtzEXPONENTS_F2_filesimage068.gif = 24y
2 2 = 24y
2 = 4y
y = C:thlbcrtzEXPONENTS_F2_filesimage033.gif
4. Find x if
C:thlbcrtzEXPONENTS_F2_filesimage069.gif=491/3
Solution
C:thlbcrtzEXPONENTS_F2_filesimage069.gif = C:thlbcrtzEXPONENTS_F2_filesimage070.gif= 491/3
3431/x = 73/x = (72)1/3
73/x = 7 2/3
C:thlbcrtzEXPONENTS_F2_filesimage071.gif = C:thlbcrtzEXPONENTS_F2_filesimage029.gif
2x = 9
x =C:thlbcrtzEXPONENTS_F2_filesimage072.gif
ii) C:thlbcrtzEXPONENTS_F2_filesimage073.gif= 81x
solution
C:thlbcrtzEXPONENTS_F2_filesimage073.gif=C:thlbcrtzEXPONENTS_F2_filesimage074.gif = 81x
= 32 = 34x
= 2 = 4x
x = C:thlbcrtzEXPONENTS_F2_filesimage033.gif

OPERATION ON RADICAL
ADDITION
Example1.

Evaluate
i) C:thlbcrtzEXPONENTS_F2_filesimage075.gif+3C:thlbcrtzEXPONENTS_F2_filesimage075.gif
Solution: C:thlbcrtzEXPONENTS_F2_filesimage075.gif+ 3C:thlbcrtzEXPONENTS_F2_filesimage075.gif =(1 + 3) C:thlbcrtzEXPONENTS_F2_filesimage075.gif
=4C:thlbcrtzEXPONENTS_F2_filesimage075.gif
ii) C:thlbcrtzEXPONENTS_F2_filesimage076.gif+ C:thlbcrtzEXPONENTS_F2_filesimage077.gif
Solution
=C:thlbcrtzEXPONENTS_F2_filesimage078.gif+C:thlbcrtzEXPONENTS_F2_filesimage079.gif
(22)1/2 (32)1/2 C:thlbcrtzEXPONENTS_F2_filesimage075.gif+ (22)1/2 (22)1/2 C:thlbcrtzEXPONENTS_F2_filesimage075.gif
= (2 x 3) C:thlbcrtzEXPONENTS_F2_filesimage075.gif+ (2 x 2) C:thlbcrtzEXPONENTS_F2_filesimage075.gif
= 6 C:thlbcrtzEXPONENTS_F2_filesimage075.gif+ 4 C:thlbcrtzEXPONENTS_F2_filesimage075.gif
= 10 C:thlbcrtzEXPONENTS_F2_filesimage075.gif

C:thlbcrtz__i__images__i__aaa22.png

SUBTRACTION
Example
Evaluate
i) 3 C:thlbcrtzEXPONENTS_F2_filesimage080.gif– 2 C:thlbcrtzEXPONENTS_F2_filesimage081.gif
Solution
= 3 C:thlbcrtzEXPONENTS_F2_filesimage082.gif n-2C:thlbcrtzEXPONENTS_F2_filesimage083.gif
= (3 x 2 x 3 C:thlbcrtzEXPONENTS_F2_filesimage084.gifC:thlbcrtzEXPONENTS_F2_filesimage085.gif2 x 2 x 2 C:thlbcrtzEXPONENTS_F2_filesimage084.gif )
= 18 C:thlbcrtzEXPONENTS_F2_filesimage084.gifC:thlbcrtzEXPONENTS_F2_filesimage085.gif8 C:thlbcrtzEXPONENTS_F2_filesimage084.gif
= 10 C:thlbcrtzEXPONENTS_F2_filesimage084.gif
ii) C:thlbcrtzEXPONENTS_F2_filesimage076.gifC:thlbcrtzEXPONENTS_F2_filesimage085.gif C:thlbcrtzEXPONENTS_F2_filesimage077.gif
Solution
C:thlbcrtzEXPONENTS_F2_filesimage076.gifC:thlbcrtzEXPONENTS_F2_filesimage085.gifC:thlbcrtzEXPONENTS_F2_filesimage077.gif= C:thlbcrtzEXPONENTS_F2_filesimage078.gifC:thlbcrtzEXPONENTS_F2_filesimage085.gifC:thlbcrtzEXPONENTS_F2_filesimage079.gif
=(2 x 3) C:thlbcrtzEXPONENTS_F2_filesimage075.gifC:thlbcrtzEXPONENTS_F2_filesimage086.gif(2 x 2) C:thlbcrtzEXPONENTS_F2_filesimage075.gif
= 6 C:thlbcrtzEXPONENTS_F2_filesimage075.gifC:thlbcrtzEXPONENTS_F2_filesimage085.gif4C:thlbcrtzEXPONENTS_F2_filesimage075.gif
= 2 C:thlbcrtzEXPONENTS_F2_filesimage075.gif

C:thlbcrtz__i__images__i__c612.png
MULTIPLICATION
Example
Find i) C:thlbcrtzEXPONENTS_F2_filesimage087.gif x C:thlbcrtzEXPONENTS_F2_filesimage077.gif
solution
C:thlbcrtzEXPONENTS_F2_filesimage087.gif x C:thlbcrtzEXPONENTS_F2_filesimage077.gif = C:thlbcrtzEXPONENTS_F2_filesimage088.gif
= C:thlbcrtzEXPONENTS_F2_filesimage089.gif
= C:thlbcrtzEXPONENTS_F2_filesimage090.gif
= 2 x 2 x 2 x 3
= 24
ii) 3 C:thlbcrtzEXPONENTS_F2_filesimage091.gif x 3C:thlbcrtzEXPONENTS_F2_filesimage092.gif
Solution
3 C:thlbcrtzEXPONENTS_F2_filesimage093.gifx 3 C:thlbcrtzEXPONENTS_F2_filesimage094.gif
(5 x 3) C:thlbcrtzEXPONENTS_F2_filesimage084.gifx (3 x 3) C:thlbcrtzEXPONENTS_F2_filesimage084.gif
= 15 C:thlbcrtzEXPONENTS_F2_filesimage084.gifx 9 C:thlbcrtzEXPONENTS_F2_filesimage084.gif
= 135 C:thlbcrtzEXPONENTS_F2_filesimage084.gif
C:thlbcrtz__i__images__i__c126.png
DIVISION
Example 1
Find i) C:thlbcrtzEXPONENTS_F2_filesimage099.gif
Solution: C:thlbcrtzEXPONENTS_F2_filesimage099.gif = C:thlbcrtzEXPONENTS_F2_filesimage100.gif
= C:thlbcrtzEXPONENTS_F2_filesimage101.gif
= C:thlbcrtzEXPONENTS_F2_filesimage102.gif
= C:thlbcrtzEXPONENTS_F2_filesimage103.gif
C:thlbcrtz__i__images__i__z8.png
EXERCISE 4.
1. Find 2 C:thlbcrtzEXPONENTS_F2_filesimage104.gif+ 3 C:thlbcrtzEXPONENTS_F2_filesimage077.gif
Solution: 2C:thlbcrtzEXPONENTS_F2_filesimage078.gif +3 C:thlbcrtzEXPONENTS_F2_filesimage079.gif
= (2 x 2 x 3)C:thlbcrtzEXPONENTS_F2_filesimage075.gif+ (3 x 2 x 2)C:thlbcrtzEXPONENTS_F2_filesimage075.gif
= 12 C:thlbcrtzEXPONENTS_F2_filesimage075.gif+12 C:thlbcrtzEXPONENTS_F2_filesimage075.gif
= 24 C:thlbcrtzEXPONENTS_F2_filesimage075.gif
(ii )3 C:thlbcrtzEXPONENTS_F2_filesimage105.gif
Solution:
3 C:thlbcrtzEXPONENTS_F2_filesimage105.gif = 3 C:thlbcrtzEXPONENTS_F2_filesimage106.gif + 3 C:thlbcrtzEXPONENTS_F2_filesimage107.gif
= 3 C:thlbcrtzEXPONENTS_F2_filesimage108.gif+ 3 C:thlbcrtzEXPONENTS_F2_filesimage079.gif
=(3 x 2) C:thlbcrtzEXPONENTS_F2_filesimage075.gif+(3 x 2 x 3) C:thlbcrtzEXPONENTS_F2_filesimage075.gif
= 6C:thlbcrtzEXPONENTS_F2_filesimage075.gif +12 C:thlbcrtzEXPONENTS_F2_filesimage075.gif
= 18 C:thlbcrtzEXPONENTS_F2_filesimage075.gif
(iii) 6 C:thlbcrtzEXPONENTS_F2_filesimage109.gif C:thlbcrtzEXPONENTS_F2_filesimage085.gif 2 C:thlbcrtzEXPONENTS_F2_filesimage110.gif
Solution:
6 C:thlbcrtzEXPONENTS_F2_filesimage109.gif C:thlbcrtzEXPONENTS_F2_filesimage085.gif 2 C:thlbcrtzEXPONENTS_F2_filesimage110.gif6 = C:thlbcrtzEXPONENTS_F2_filesimage111.gif C:thlbcrtzEXPONENTS_F2_filesimage085.gif2 C:thlbcrtzEXPONENTS_F2_filesimage112.gif
= (6 x 2) C:thlbcrtzEXPONENTS_F2_filesimage113.gifC:thlbcrtzEXPONENTS_F2_filesimage085.gif(2 x 3) C:thlbcrtzEXPONENTS_F2_filesimage113.gif
= 12 C:thlbcrtzEXPONENTS_F2_filesimage113.gifC:thlbcrtzEXPONENTS_F2_filesimage114.gif6C:thlbcrtzEXPONENTS_F2_filesimage113.gif
= 6 C:thlbcrtzEXPONENTS_F2_filesimage113.gif
iv) C:thlbcrtzEXPONENTS_F2_filesimage115.gif+ C:thlbcrtzEXPONENTS_F2_filesimage116.gif
Solution:
C:thlbcrtzEXPONENTS_F2_filesimage115.gif + C:thlbcrtzEXPONENTS_F2_filesimage117.gif
C:thlbcrtzEXPONENTS_F2_filesimage115.gif+3 C:thlbcrtzEXPONENTS_F2_filesimage115.gif
4 C:thlbcrtzEXPONENTS_F2_filesimage115.gif
(v) C:thlbcrtzEXPONENTS_F2_filesimage118.gif+ 2250
Solution:
C:thlbcrtzEXPONENTS_F2_filesimage118.gif+ C:thlbcrtzEXPONENTS_F2_filesimage119.gif = C:thlbcrtzEXPONENTS_F2_filesimage120.gif +2250
= 2 C:thlbcrtzEXPONENTS_F2_filesimage121.gif + 2250
=2 C:thlbcrtzEXPONENTS_F2_filesimage122.gif + 2250
=2 C:thlbcrtzEXPONENTS_F2_filesimage122.gif+ 2250
2. Simplify
(i) C:thlbcrtzEXPONENTS_F2_filesimage081.gif x C:thlbcrtzEXPONENTS_F2_filesimage092.gif
= C:thlbcrtzEXPONENTS_F2_filesimage089.gif
= C:thlbcrtzEXPONENTS_F2_filesimage090.gif
= C:thlbcrtzEXPONENTS_F2_filesimage123.gif
= 24
ii) C:thlbcrtzEXPONENTS_F2_filesimage084.gif C:thlbcrtzEXPONENTS_F2_filesimage124.gif
C:thlbcrtzEXPONENTS_F2_filesimage084.gif (C:thlbcrtzEXPONENTS_F2_filesimage125.gif C:thlbcrtzEXPONENTS_F2_filesimage083.gif )
= C:thlbcrtzEXPONENTS_F2_filesimage084.gif(2 x 3 C:thlbcrtzEXPONENTS_F2_filesimage084.gif– 4 C:thlbcrtzEXPONENTS_F2_filesimage084.gif )
= C:thlbcrtzEXPONENTS_F2_filesimage084.gif(6 C:thlbcrtzEXPONENTS_F2_filesimage084.gif– 4 C:thlbcrtzEXPONENTS_F2_filesimage084.gif )
= C:thlbcrtzEXPONENTS_F2_filesimage084.gif(2 C:thlbcrtzEXPONENTS_F2_filesimage084.gif )
= 4
(iii) 3 C:thlbcrtzEXPONENTS_F2_filesimage126.gif x 2 C:thlbcrtzEXPONENTS_F2_filesimage127.gif
Solution:
= 3 C:thlbcrtzEXPONENTS_F2_filesimage128.gifx 2 C:thlbcrtzEXPONENTS_F2_filesimage129.gif
= 3 x 2 x 3 x (2 x 2) C:thlbcrtzEXPONENTS_F2_filesimage130.gif
= 18 x 4C:thlbcrtzEXPONENTS_F2_filesimage131.gif
= 72 C:thlbcrtzEXPONENTS_F2_filesimage131.gif
(iv) C:thlbcrtzEXPONENTS_F2_filesimage132.gif (15 C:thlbcrtzEXPONENTS_F2_filesimage075.gif )
Solution:
C:thlbcrtzEXPONENTS_F2_filesimage132.gif (15 C:thlbcrtzEXPONENTS_F2_filesimage075.gif )= 15 C:thlbcrtzEXPONENTS_F2_filesimage040.gif
= 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:-
(i) If a radical is
a single term(that is does not involve + or -),the proper choice is the radical itself,that is
C:thlbcrtz__i__images__i__c99.png
(ii)If the radical involves operations(+ or -),choose a radical with the same format but with one term with the opposite operation.

Examples
C:thlbcrtz__i__images__i__a716.png
The same technique can be used to rationalize the denominator.

Example 1
Rationalize i) C:thlbcrtzEXPONENTS_F2_filesimage133.gif
Solution C:thlbcrtzEXPONENTS_F2_filesimage133.gif= C:thlbcrtzEXPONENTS_F2_filesimage133.gif x C:thlbcrtzEXPONENTS_F2_filesimage134.gif
= C:thlbcrtzEXPONENTS_F2_filesimage135.gif
(ii) C:thlbcrtzEXPONENTS_F2_filesimage136.gif
Solution:
C:thlbcrtzEXPONENTS_F2_filesimage136.gif = C:thlbcrtzEXPONENTS_F2_filesimage136.gif x C:thlbcrtzEXPONENTS_F2_filesimage137.gif
= C:thlbcrtzEXPONENTS_F2_filesimage138.gif
= C:thlbcrtzEXPONENTS_F2_filesimage139.gif
(iii) C:thlbcrtzEXPONENTS_F2_filesimage140.gif

Solution:
C:thlbcrtzEXPONENTS_F2_filesimage140.gif = C:thlbcrtzEXPONENTS_F2_filesimage140.gifx C:thlbcrtzEXPONENTS_F2_filesimage141.gif
= C:thlbcrtzEXPONENTS_F2_filesimage142.gif
= C:thlbcrtzEXPONENTS_F2_filesimage143.gif
= C:thlbcrtzEXPONENTS_F2_filesimage144.gif
= C:thlbcrtzEXPONENTS_F2_filesimage145.gif
Example 2:
Rationalize (i) C:thlbcrtzEXPONENTS_F2_filesimage146.gif
Solution:
C:thlbcrtzEXPONENTS_F2_filesimage146.gif = C:thlbcrtzEXPONENTS_F2_filesimage146.gif x C:thlbcrtzEXPONENTS_F2_filesimage147.gif
= C:thlbcrtzEXPONENTS_F2_filesimage148.gif
= C:thlbcrtzEXPONENTS_F2_filesimage149.gif
= C:thlbcrtzEXPONENTS_F2_filesimage150.gif
= C:thlbcrtzEXPONENTS_F2_filesimage151.gif
= C:thlbcrtzEXPONENTS_F2_filesimage152.gif
= C:thlbcrtzEXPONENTS_F2_filesimage153.gif
(ii) Rationalize C:thlbcrtzEXPONENTS_F2_filesimage154.gif
Solution:
C:thlbcrtzEXPONENTS_F2_filesimage154.gif = C:thlbcrtzEXPONENTS_F2_filesimage154.gif x C:thlbcrtzEXPONENTS_F2_filesimage155.gif
= C:thlbcrtzEXPONENTS_F2_filesimage156.gif
= C:thlbcrtzEXPONENTS_F2_filesimage157.gif
= C:thlbcrtzEXPONENTS_F2_filesimage158.gif
= C:thlbcrtzEXPONENTS_F2_filesimage159.gif
= C:thlbcrtzEXPONENTS_F2_filesimage160.gif
= C:thlbcrtzEXPONENTS_F2_filesimage161.gif
= C:thlbcrtzEXPONENTS_F2_filesimage162.gif
EXERCISE 5
1. Evaluate
(i) (C:thlbcrtzEXPONENTS_F2_filesimage163.gif)(C:thlbcrtzEXPONENTS_F2_filesimage164.gif)
Solution:
(1) (C:thlbcrtzEXPONENTS_F2_filesimage163.gif)(C:thlbcrtzEXPONENTS_F2_filesimage164.gif) = (C:thlbcrtzEXPONENTS_F2_filesimage165.gif(C:thlbcrtzEXPONENTS_F2_filesimage166.gif) -4(C:thlbcrtzEXPONENTS_F2_filesimage164.gif)
= C:thlbcrtzEXPONENTS_F2_filesimage167.gif – 6C:thlbcrtzEXPONENTS_F2_filesimage131.gif – 12C:thlbcrtzEXPONENTS_F2_filesimage168.gif + 12C:thlbcrtzEXPONENTS_F2_filesimage084.gif
(ii) (C:thlbcrtzEXPONENTS_F2_filesimage169.gif)(C:thlbcrtzEXPONENTS_F2_filesimage170.gif)
Solution:
(iii) (C:thlbcrtzEXPONENTS_F2_filesimage169.gif)(C:thlbcrtzEXPONENTS_F2_filesimage170.gif) = C:thlbcrtzEXPONENTS_F2_filesimage171.gif(C:thlbcrtzEXPONENTS_F2_filesimage170.gif) + C:thlbcrtzEXPONENTS_F2_filesimage172.gif(C:thlbcrtzEXPONENTS_F2_filesimage170.gif)
= a + C:thlbcrtzEXPONENTS_F2_filesimage098.gif + C:thlbcrtzEXPONENTS_F2_filesimage173.gif + b
= a + b + 2C:thlbcrtzEXPONENTS_F2_filesimage098.gif
(iv) (C:thlbcrtzEXPONENTS_F2_filesimage174.gif)(C:thlbcrtzEXPONENTS_F2_filesimage175.gif)
Solution:
(C:thlbcrtzEXPONENTS_F2_filesimage174.gif)(C:thlbcrtzEXPONENTS_F2_filesimage175.gif) = C:thlbcrtzEXPONENTS_F2_filesimage176.gif(C:thlbcrtzEXPONENTS_F2_filesimage177.gif) + C:thlbcrtzEXPONENTS_F2_filesimage178.gif(C:thlbcrtzEXPONENTS_F2_filesimage175.gif)
= m + C:thlbcrtzEXPONENTS_F2_filesimage179.gifC:thlbcrtzEXPONENTS_F2_filesimage179.gif – n
= m – n
(v) (C:thlbcrtzEXPONENTS_F2_filesimage174.gif)(C:thlbcrtzEXPONENTS_F2_filesimage175.gif)
Solution:
(C:thlbcrtzEXPONENTS_F2_filesimage180.gif)(C:thlbcrtzEXPONENTS_F2_filesimage181.gif) = C:thlbcrtzEXPONENTS_F2_filesimage182.gif(C:thlbcrtzEXPONENTS_F2_filesimage181.gifC:thlbcrtzEXPONENTS_F2_filesimage182.gif(C:thlbcrtzEXPONENTS_F2_filesimage181.gif)
= p – C:thlbcrtzEXPONENTS_F2_filesimage183.gif + C:thlbcrtzEXPONENTS_F2_filesimage183.gif – q
= p – q
2. Rationalize
(i) C:thlbcrtzEXPONENTS_F2_filesimage184.gif
Solution:
C:thlbcrtzEXPONENTS_F2_filesimage184.gif = C:thlbcrtzEXPONENTS_F2_filesimage184.gif x C:thlbcrtzEXPONENTS_F2_filesimage185.gif
= C:thlbcrtzEXPONENTS_F2_filesimage186.gif
= C:thlbcrtzEXPONENTS_F2_filesimage187.gif
= C:thlbcrtzEXPONENTS_F2_filesimage188.gif
= C:thlbcrtzEXPONENTS_F2_filesimage189.gif
= C:thlbcrtzEXPONENTS_F2_filesimage152.gif
(ii) C:thlbcrtzEXPONENTS_F2_filesimage190.gif
Solution:
C:thlbcrtzEXPONENTS_F2_filesimage191.gif
= C:thlbcrtzEXPONENTS_F2_filesimage192.gif
= C:thlbcrtzEXPONENTS_F2_filesimage193.gif
= – ( C:thlbcrtzEXPONENTS_F2_filesimage194.gif)
EXERCISE 6
Rationalize the following denominator
(i)C:thlbcrtzEXPONENTS_F2_filesimage195.gif
Solution:
C:thlbcrtzEXPONENTS_F2_filesimage196.gif
= C:thlbcrtzEXPONENTS_F2_filesimage197.gif
= C:thlbcrtzEXPONENTS_F2_filesimage198.gif
= C:thlbcrtzEXPONENTS_F2_filesimage199.gif
= C:thlbcrtzEXPONENTS_F2_filesimage200.gif
(ii)C:thlbcrtzEXPONENTS_F2_filesimage201.gif
Solution:
C:thlbcrtzEXPONENTS_F2_filesimage202.gif
= C:thlbcrtzEXPONENTS_F2_filesimage203.gif
= C:thlbcrtzEXPONENTS_F2_filesimage204.gif
= C:thlbcrtzEXPONENTS_F2_filesimage205.gif
= C:thlbcrtzEXPONENTS_F2_filesimage206.gif
(iii) C:thlbcrtzEXPONENTS_F2_filesimage207.gif
Solution:
C:thlbcrtzEXPONENTS_F2_filesimage208.gif
= C:thlbcrtzEXPONENTS_F2_filesimage209.gif
= C:thlbcrtzEXPONENTS_F2_filesimage210.gif
= C:thlbcrtzEXPONENTS_F2_filesimage211.gif
= C:thlbcrtzEXPONENTS_F2_filesimage212.gif
(iv) C:thlbcrtzEXPONENTS_F2_filesimage213.gif
Solution:
C:thlbcrtzEXPONENTS_F2_filesimage214.gif
= C:thlbcrtzEXPONENTS_F2_filesimage215.gif
= C:thlbcrtzEXPONENTS_F2_filesimage216.gif
= C:thlbcrtzEXPONENTS_F2_filesimage217.gif

SQUARE ROOT OF A NUMBER
Example
Find( i) C:thlbcrtzEXPONENTS_F2_filesimage218.gif
Solution

C:thlbcrtz__i__images__i__imgexp11.jpg
ii) C:thlbcrtzEXPONENTS_F2_filesimage220.gif
Solution:
C:thlbcrtz__i__images__i__imgexp2.jpg
(iii)C:thlbcrtzEXPONENTS_F2_filesimage222.gif
Solution:
C:thlbcrtz__i__images__i__imgexp3.jpg
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 = C:thlbcrtzEXPONENTS_F2_filesimage224.gif
b. v = C:thlbcrtzEXPONENTS_F2_filesimage225.gif
c. I = C:thlbcrtzEXPONENTS_F2_filesimage226.gif
d. A = C:thlbcrtzEXPONENTS_F2_filesimage033.gif (a +b)h
e. T = 2C:thlbcrtzEXPONENTS_F2_filesimage227.gifrC:thlbcrtzEXPONENTS_F2_filesimage228.gif

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 = C:thlbcrtzEXPONENTS_F2_filesimage229.gif
Make T the subject of a formula
Solution:
100 x I = C:thlbcrtzEXPONENTS_F2_filesimage229.gif x 100
C:thlbcrtzEXPONENTS_F2_filesimage230.gif = C:thlbcrtzEXPONENTS_F2_filesimage231.gif
C:thlbcrtzEXPONENTS_F2_filesimage230.gif = C:thlbcrtzEXPONENTS_F2_filesimage232.gif
T = C:thlbcrtzEXPONENTS_F2_filesimage230.gif
Example 3.
Given that
Y = mx + c, make m the subject
Solution:
Y = mx +c
C:thlbcrtzEXPONENTS_F2_filesimage233.gif = C:thlbcrtzEXPONENTS_F2_filesimage234.gif
m = C:thlbcrtzEXPONENTS_F2_filesimage233.gif
Example 4
Given that p = w C:thlbcrtzEXPONENTS_F2_filesimage235.gif
Make a the subject.
Solution:
P = w C:thlbcrtzEXPONENTS_F2_filesimage235.gif
Divide by w both sides
C:thlbcrtzEXPONENTS_F2_filesimage236.gif= C:thlbcrtzEXPONENTS_F2_filesimage237.gif C:thlbcrtzEXPONENTS_F2_filesimage235.gif
C:thlbcrtzEXPONENTS_F2_filesimage236.gif = C:thlbcrtzEXPONENTS_F2_filesimage235.gif
Multiply by (1 – a) both sides
C:thlbcrtzEXPONENTS_F2_filesimage236.gif (1 – a) = (1 C:thlbcrtzEXPONENTS_F2_filesimage085.gif a)C:thlbcrtzEXPONENTS_F2_filesimage235.gif
C:thlbcrtzEXPONENTS_F2_filesimage236.gif (1 – a) = 1 + a
C:thlbcrtzEXPONENTS_F2_filesimage236.gifC:thlbcrtzEXPONENTS_F2_filesimage238.gif = 1 + a
C:thlbcrtzEXPONENTS_F2_filesimage236.gif – 1 = a + C:thlbcrtzEXPONENTS_F2_filesimage239.gif
C:thlbcrtzEXPONENTS_F2_filesimage236.gif – 1 = a(1 + C:thlbcrtzEXPONENTS_F2_filesimage236.gif)
Divide by 1 + C:thlbcrtzEXPONENTS_F2_filesimage236.gif both sides
C:thlbcrtzEXPONENTS_F2_filesimage240.gif= C:thlbcrtzEXPONENTS_F2_filesimage241.gif
a = C:thlbcrtzEXPONENTS_F2_filesimage240.gif

Alternatively

C:thlbcrtz__i__images__i__umeme.png
Example 5
Given that T = 2C:thlbcrtzEXPONENTS_F2_filesimage227.gif C:thlbcrtzEXPONENTS_F2_filesimage242.gif write g in terms of other letters
Solution:
T = 2C:thlbcrtzEXPONENTS_F2_filesimage227.gif C:thlbcrtzEXPONENTS_F2_filesimage242.gif
Divide by 2C:thlbcrtzEXPONENTS_F2_filesimage227.gif both side
C:thlbcrtzEXPONENTS_F2_filesimage243.gif = C:thlbcrtzEXPONENTS_F2_filesimage244.gif C:thlbcrtzEXPONENTS_F2_filesimage242.gif
Remove the radical by squares both sides
C:thlbcrtzEXPONENTS_F2_filesimage245.gif2 = C:thlbcrtzEXPONENTS_F2_filesimage246.gif2
C:thlbcrtzEXPONENTS_F2_filesimage247.gif = C:thlbcrtzEXPONENTS_F2_filesimage248.gif
Multiply by g both sides
C:thlbcrtzEXPONENTS_F2_filesimage249.gif =C:thlbcrtzEXPONENTS_F2_filesimage250.gifg
C:thlbcrtzEXPONENTS_F2_filesimage251.gif = C:thlbcrtzEXPONENTS_F2_filesimage249.gif
Multiply by 4C:thlbcrtzEXPONENTS_F2_filesimage227.gif2 both sides
4C:thlbcrtzEXPONENTS_F2_filesimage227.gif2 x C:thlbcrtzEXPONENTS_F2_filesimage249.gif = C:thlbcrtzEXPONENTS_F2_filesimage251.gif x 4C:thlbcrtzEXPONENTS_F2_filesimage227.gif2
T2g = 4C:thlbcrtzEXPONENTS_F2_filesimage227.gif2C:thlbcrtzEXPONENTS_F2_filesimage252.gif
Divide by T2 both sides
g = C:thlbcrtzEXPONENTS_F2_filesimage253.gif
Example 6
If A = p + C:thlbcrtzEXPONENTS_F2_filesimage229.gif
(i) Make R as the subject formula
(ii) Make P as the subject formula
Solution:
(i) A = p + C:thlbcrtzEXPONENTS_F2_filesimage229.gif
= A – P = C:thlbcrtzEXPONENTS_F2_filesimage229.gif
Multiply by 100 both sides
= C:thlbcrtzEXPONENTS_F2_filesimage254.gif= R
R = C:thlbcrtzEXPONENTS_F2_filesimage254.gif
(ii) A = P + C:thlbcrtzEXPONENTS_F2_filesimage229.gif
Solution:
Multiply by 100 both sides
100A = 100P + PRT
100A = P(100 + RT)
Divide by 100 + RT both sides
C:thlbcrtzEXPONENTS_F2_filesimage255.gif = P
P = C:thlbcrtzEXPONENTS_F2_filesimage255.gif
Exercise 7
1. If S = C:thlbcrtzEXPONENTS_F2_filesimage033.gif at2. Make t the subject of the formula
2. If c = C:thlbcrtzEXPONENTS_F2_filesimage256.gif (F – 32) make F the subject of the formula
Solution:
S = C:thlbcrtzEXPONENTS_F2_filesimage033.gifat2
Multiply by 2 both sides
s x 2 = C:thlbcrtzEXPONENTS_F2_filesimage033.gifat2 x 2
2s = at2
Divide by a both sides
C:thlbcrtzEXPONENTS_F2_filesimage257.gif = C:thlbcrtzEXPONENTS_F2_filesimage258.gif
t2 = C:thlbcrtzEXPONENTS_F2_filesimage257.gif
Square root both sides
C:thlbcrtzEXPONENTS_F2_filesimage259.gif = C:thlbcrtzEXPONENTS_F2_filesimage260.gif
t = C:thlbcrtzEXPONENTS_F2_filesimage260.gif
2. C = C:thlbcrtzEXPONENTS_F2_filesimage256.gif(F – 32)
C = C:thlbcrtzEXPONENTS_F2_filesimage256.gifF – C:thlbcrtzEXPONENTS_F2_filesimage261.gif
C + C:thlbcrtzEXPONENTS_F2_filesimage261.gif = C:thlbcrtzEXPONENTS_F2_filesimage262.gif
Multiply by 9 both sides
9C + C:thlbcrtzEXPONENTS_F2_filesimage263.gif = C:thlbcrtzEXPONENTS_F2_filesimage264.gif
Divide by 5 both sides
F = C:thlbcrtzEXPONENTS_F2_filesimage265.gif
More Examples
1. If A = C:thlbcrtzEXPONENTS_F2_filesimage266.gif(a + b)
(i) Make h the subject formula
(ii) Make b the subject formula
2. If C:thlbcrtzEXPONENTS_F2_filesimage267.gif = C:thlbcrtzEXPONENTS_F2_filesimage268.gifC:thlbcrtzEXPONENTS_F2_filesimage269.gif
(i) Make f the subject formula
(ii) Make u the subject formula
Solution:
1. A = C:thlbcrtzEXPONENTS_F2_filesimage270.gif
2A = C:thlbcrtzEXPONENTS_F2_filesimage271.gif(a + b)x 2
2A = C:thlbcrtzEXPONENTS_F2_filesimage272.gif(a + b)
Divide by a + b both sides
C:thlbcrtzEXPONENTS_F2_filesimage273.gif = C:thlbcrtzEXPONENTS_F2_filesimage274.gif
h = C:thlbcrtzEXPONENTS_F2_filesimage273.gif
(ii) Make b the subject formula.
Solution:
A = C:thlbcrtzEXPONENTS_F2_filesimage270.gif
2A = C:thlbcrtzEXPONENTS_F2_filesimage271.gif(a + b)x 2
2A = C:thlbcrtzEXPONENTS_F2_filesimage272.gif(a + b)
2A = ah + bh
2A C:thlbcrtzEXPONENTS_F2_filesimage085.gif ah = bh
Divide by h both sides
C:thlbcrtzEXPONENTS_F2_filesimage275.gif = b
b = C:thlbcrtzEXPONENTS_F2_filesimage275.gif
2. C:thlbcrtzEXPONENTS_F2_filesimage267.gif = C:thlbcrtzEXPONENTS_F2_filesimage268.gifC:thlbcrtzEXPONENTS_F2_filesimage269.gif
Solution:
C:thlbcrtzEXPONENTS_F2_filesimage267.gif = C:thlbcrtzEXPONENTS_F2_filesimage268.gifC:thlbcrtzEXPONENTS_F2_filesimage269.gif
C:thlbcrtzEXPONENTS_F2_filesimage267.gif = C:thlbcrtzEXPONENTS_F2_filesimage276.gif
C:thlbcrtzEXPONENTS_F2_filesimage277.gif
Divide by u – v both sides
f = C:thlbcrtzEXPONENTS_F2_filesimage278.gif
ii) Make u the subject formula
C:thlbcrtzEXPONENTS_F2_filesimage267.gif = C:thlbcrtzEXPONENTS_F2_filesimage268.gifC:thlbcrtzEXPONENTS_F2_filesimage269.gif
Solution:
C:thlbcrtzEXPONENTS_F2_filesimage267.gif = C:thlbcrtzEXPONENTS_F2_filesimage276.gif
Multiply by uv both sides
C:thlbcrtzEXPONENTS_F2_filesimage279.gif = f(u – v)
uv = fu – fv
fv = fu – uv
fv =u (f – v)
Divide by f – v both sides
u = C:thlbcrtzEXPONENTS_F2_filesimage280.gif
Exercise 8
1. If T = C:thlbcrtzEXPONENTS_F2_filesimage281.gif
(i) Make t the subject formula
(ii) Make g the subject
2. If P = w C:thlbcrtzEXPONENTS_F2_filesimage282.gif
(i) Make w as the subject formula
(ii) Make a the subject formula
Solution:
1. (i)T = C:thlbcrtzEXPONENTS_F2_filesimage283.gif
Square both sides
T2 = C:thlbcrtzEXPONENTS_F2_filesimage284.gif
Multiply by 4 both sides
4T2 = C:thlbcrtzEXPONENTS_F2_filesimage285.gif
4T2g = 9t
Divide by 9 both sides
t = C:thlbcrtzEXPONENTS_F2_filesimage286.gif
(ii) Make g the subject formula
T = C:thlbcrtzEXPONENTS_F2_filesimage283.gif
Solution:
Square both sides
T2 = C:thlbcrtzEXPONENTS_F2_filesimage284.gif
Multiply by 4 both sides
4T2 = C:thlbcrtzEXPONENTS_F2_filesimage285.gif
4T2g = 9t
Divide by 4T2 both sides
g = C:thlbcrtzEXPONENTS_F2_filesimage287.gif
2)( i) Make w was the subject
Make a the subject
Solution:
P = w C:thlbcrtzEXPONENTS_F2_filesimage288.gif
PC:thlbcrtzEXPONENTS_F2_filesimage289.gif = w(C:thlbcrtzEXPONENTS_F2_filesimage290.gif)
Divide by (C:thlbcrtzEXPONENTS_F2_filesimage290.gif) both sides
w =P C:thlbcrtzEXPONENTS_F2_filesimage291.gif
ii) Make a the subject formula
Solution:
P = w C:thlbcrtzEXPONENTS_F2_filesimage235.gif
Divide by w both sides
C:thlbcrtzEXPONENTS_F2_filesimage236.gif= C:thlbcrtzEXPONENTS_F2_filesimage237.gif C:thlbcrtzEXPONENTS_F2_filesimage235.gif
C:thlbcrtzEXPONENTS_F2_filesimage236.gif = C:thlbcrtzEXPONENTS_F2_filesimage235.gif
Multiply by (1 – a) both sides
C:thlbcrtzEXPONENTS_F2_filesimage236.gif (1 – a) = (1 C:thlbcrtzEXPONENTS_F2_filesimage085.gif a)C:thlbcrtzEXPONENTS_F2_filesimage235.gif
C:thlbcrtzEXPONENTS_F2_filesimage236.gif (1 – a) = 1 + a
C:thlbcrtzEXPONENTS_F2_filesimage236.gifC:thlbcrtzEXPONENTS_F2_filesimage238.gif = 1 + a
C:thlbcrtzEXPONENTS_F2_filesimage236.gif – 1 = a + C:thlbcrtzEXPONENTS_F2_filesimage239.gif
C:thlbcrtzEXPONENTS_F2_filesimage236.gif – 1 = a(1 + C:thlbcrtzEXPONENTS_F2_filesimage236.gif)
Divide by 1 + C:thlbcrtzEXPONENTS_F2_filesimage236.gif both sides
C:thlbcrtzEXPONENTS_F2_filesimage240.gif= C:thlbcrtzEXPONENTS_F2_filesimage241.gif
a = C:thlbcrtzEXPONENTS_F2_filesimage240.gif
Exercise 9
I. If v = C:thlbcrtzEXPONENTS_F2_filesimage292.gif Make R the subject formula
Solution:
v = C:thlbcrtzEXPONENTS_F2_filesimage292.gif
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 C:thlbcrtzEXPONENTS_F2_filesimage293.gif
(i) Make x the subject formula
Solution:
m = n C:thlbcrtzEXPONENTS_F2_filesimage293.gif
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 = C:thlbcrtzEXPONENTS_F2_filesimage294.gif
(ii)If T = 2C:thlbcrtzEXPONENTS_F2_filesimage295.gif
Make t the subject formula
Solution:
T = 2C:thlbcrtzEXPONENTS_F2_filesimage295.gif
Square both sides
T2 = 4C:thlbcrtzEXPONENTS_F2_filesimage296.gif2C:thlbcrtzEXPONENTS_F2_filesimage297.gif
Multiply by a both sides
T2a = 4C:thlbcrtzEXPONENTS_F2_filesimage296.gif2kt
Divide by 4C:thlbcrtzEXPONENTS_F2_filesimage296.gif2k both sides

t = C:thlbcrtzEXPONENTS_F2_filesimage298.gif2C:thlbcrtzEXPONENTS_F2_filesimage299.gif




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EcoleBooks | MATHEMATICS O LEVEL(FORM TWO) NOTES - EXPONENT AND RADICALS

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1 Comment

  • EcoleBooks | MATHEMATICS O LEVEL(FORM TWO) NOTES - EXPONENT AND RADICALS

    David Garjaye, October 26, 2023 @ 6:38 pm Reply

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

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