You can find questions, worked solutions and a crib sheet at the bottom of the page.
Indices tell us how many times to multiply a number. It is also known as a power. For example,
23 = 2 × 2 × 2
For your exam there are several “Rules of Indices” you need to remember.
The Rules of Indices
am × an = am+n
If we have the product (multiplication) of the same number with different powers then we add the powers.
x2 × x3 = x2+3 = x5
22 × 23 = 22+3 = 25 = 32
But the subject number (a in the rule) must be the same in both cases. We cannot combine the powers of two different numbers.
32 × 23 ≠ 22+3 ≠ 32+3 ≠ 62+3
am ÷ an = am-n
This is similar to the first rule but if we divide, we subtract the powers.
x5 ÷ x3 = x5-3 = x2
45 ÷ 43 = 45-3 = 42 = 16
But as before the subject number (a in the rule) must be the same in both cases. We cannot combine the powers of two different numbers.
(am)n = am×n = amn
If we have a power of a power (like the cube of a number squared) we multiply the powers together.
(x2)3 = x2×3 = x6
(22)3 = 22×3 = 26 = 64
a-1 = 1⁄a or more generally a-m = 1 ⁄ am
A number to the power of -1 becomes one over that number (a fraction). Using the third rule, it makes sense that a number to the power of -2 becomes one over that number squared and a number to the power of -3 is one over that number cubed and so on.
x-1 = 1⁄ x
3-1 = 1⁄3
x-2 = 1 ⁄ x2
3-2 = 1 ⁄ 32 = 1⁄9
a1/2 = √a or more generally a1/n = n√a
A number to the power of 1 over n is equal to the nth root of that number.
x1/2 = √x
91/2 = √9 = 3
x1/3 = 3√x
81/3 = 3√8 = 2
82/3 = (3√8) 2 = ( 2 )2 = 4
Those are the main 5 rules but there are two more things to be aware of:
x0 = 1
1a = 1
234320 = 1
1237 = 1
Anything to the power of zero is equal to 1.
1 to the power of anything is equal to 1.