Index/Power
The small, raised number next to a normal letter or number, to be multiplied by itself
Positive Indices
Example
What is the value of 22 x 24 ?
= 2 x 2 x 2 x 2 x 2 x 2
= 26
= 2 2 + 3
==> am x an = a m+n
Example
What is the value of 25 ?
22
= 2 x 2 x 2 x 2 x 2
2 x 2
= 2 x 2 x 2
= 23 => 2 5 -2
==> am ÷ an = a m-n
Example
What is the value of (23)2 ?
= (2 x 2 x 2)2 = 82
= 64 = 2 x 2 x 2 x 2 x 2 x 2
= 26
==> (am)n = a m x n
Zero and Negative Indices
Example
What is the value of b2 ÷ b2?
= b2 - 2 = b2/b2
= b0 = 1
==> a0 = 1
Example
What is the value of a2 ÷ a5 ?
= a x a = a 2 - 5
a x a x a x a x a. = a-3
= 1
a3
==> a-m = 1/am
Fractional Indices
The numerator is the power and the denominator is the root.
Example
a½ = (√a)1
Example
What is the value of 8⅔ ?
8 = (3√8)2
= (2)2 = 4
==> am/n = (n√a)m
Indices Rules
Positive Indices
(1) am x an= am+n
(2) am ÷ an = am-n
(3) (am)n= am x n
Zero and Negative Indices
(4) a-m = 1/ am
(5) a0 = 1
Fractional Indices
(6) a1/2 = √a
(7) am/n = n√am
Example:
Simplify (xy2)3x (-3x2y)4
Step1: Open and Expand the bracket using (am)n= amxn
= x3y2x3x (-3)4x2x4y4
= x3y6x 81x8y4
Step2: Group numbers and variables
= 81 x3+8y6+4
= 81 x11y10
Example:
Solve for z: 9z = 27
(32)z= 33 (Step 1 : the lowest common no, 9=3x3, 27=3x3x3 => 3)
32z= 33 (Step 2 : Simplify)
2z = 3 (Step 3 : Solve by equating the power)
z = 3/2
Example
Find a
5 a+2 = 5 √5 x 53
5 a+2 = 5 √5 x 53 (Step1: Change equation to xm = xn)
5 a+2 = 5 √5 x 53
5 a+2 = 51+½+3
a + 2 = 1 + ½ + 3 (Step2: Solve by Equating power )
a + 2 = 4 ½
a = 4 ½ - 2
a = 2 ½
Example
Find x.
4x = 3√22 x 82
Step1: Find the lowest Common Value for the numbers.
[look at the smallest number (2) and try]
4x = 3√22 x 82 [ 4 = 22, 8 = 23 ]
(22)x = (22)1/3 x (23)2
22x = 2 2/3 x 23x2 (Step2: Change equation to xm = xn)
22x = 2 2/3 x 26
22x = 2 2/3 + 6
2x = 6 2/3 (Step3: Solve by equating the power)
x = 20 x 1
3 2
x = 10/3
Indices Rules and Example of ‘Simplifying Indices Question’
