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,

2^{3} = 2 × 2 × 2

For your exam there are several “Rules of Indices” you need to remember.

## The Rules of Indices

### First rule:

a^{m} × a^{n} = a^{m+n}

If we have the product (multiplication) of the same number with different powers then we add the powers.

For example:

x^{2} × x^{3} = x^{2+3} = x^{5}

2^{2} × 2^{3} = 2^{2+3} = 2^{5} = 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.

3^{2} × 2^{3} ≠ 2^{2+3} ≠ 3^{2+3} ≠ 6^{2+3}

### Second rule:

a^{m} ÷ a^{n} = a^{m-n}

This is similar to the first rule but if we divide, we subtract the powers.

For example:

x^{5} ÷ x^{3} = x^{5-3} = x^{2}

4^{5} ÷ 4^{3} = 4^{5-3} = 4^{2} = 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.

### Third rule:

(a^{m})^{n} = a^{m×n} = a^{mn}

If we have a power of a power (like the cube of a number squared) we multiply the powers together.

For example:

(x^{2})^{3} = x^{2×3} = x^{6}

(2^{2})^{3} = 2^{2×3} = 2^{6} = 64

### Fourth rule:

a^{-1} = ^{1}⁄_{a} or more generally
a^{-m} = ^{1} ⁄ a^{m}

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.

For example:

x^{-1} = ^{1}⁄ x

3^{-1} = ^{1}⁄_{3}

x^{-2} = ^{1} ⁄ x^{2}

3^{-2} = ^{1} ⁄ 3^{2} = ^{1}⁄_{9}

### Fifth rule:

a^{1/2} = √a or more generally a^{1/n} = ^{n}√a

A number to the power of 1 over n is equal to the nth root of that number.

For example:

x^{1/2} = √x

9^{1/2} = √9 = 3

x^{1/3} = ^{3}√x

8^{1/3} = ^{3}√8 = 2

8^{2/3} = (^{3}√8) ^{2} = ( 2 )^{2} = 4

**Those are the main 5 rules but there are two more things to be aware of:**

x^{0} = 1

1^{a} = 1

23432^{0} = 1

1^{237} = 1

Anything to the power of zero is equal to 1.

1 to the power of anything is equal to 1.