## What Are Indices?

Indices (singular: index) is just another word for ‘powers’. It’s an extension of the ‘squared’ idea but with different numbers. So:

`2`

^{2} = 2 x 2 = 4

2^{3} = 2 x 2 x 2 = 8

2^{4} = 2 x 2 x 2 x 2 = 16

You can do this with algebra as well, so a^{3} just means ‘a’ times ‘a’ times ‘a’. I’m using ‘a’ because in the web ‘x’ could be confused for ‘times’.

There are two special indices you need to know: anything to the power of 1 is itself and anything to the power of 0 is 1. So:

`2`

^{1} = 2 17^{1} = 17 a^{1} = a

2^{0} = 1 17^{0} = 1 a^{0} = 1

## Negative and Fractional Indices

This is where it gets weird. Indices can be negative and/or fractions.

A negative index just means “one over” what it would have been. So:

`2`

^{-3} = 1/2^{3} = 1/8

5^{-2} = 1/5^{2} = 1/25

a^{-4} = 1/a^{4}

a^{-1} = 1/a^{1} = 1/a

A fractional index means a root:

`16`

^{1/2} = √16 = 4

1000^{1/3} = ^{3}√1000 = 10

a^{1/4} = ^{4}√a

These can be combined, so:

`25`

^{-1/2} = 1/25^{1/2} = 1/√25 = 1/5

a^{-1/6} = 1/^{6}√a

a^{-5/2} = 1/^{2}√a^{5}

## Index Rules

There are three basic rules when manipulating indices. These can *only* be used when you’re taking powers of the same thing.

1) When you multiply powers, add the indices

`a`

^{2} x a^{3} = a^{5}

a^{6} x a^{-2} = a^{4}

a^{3/2} x a^{-1/2} = a^{1} = a

2) When you divide powers, subtract the indices

`a`

^{7} ÷ a^{2} = a^{5}

a^{2} ÷ a^{8} = a^{-6}

a^{2/3} ÷ a^{1/3} = a^{1/3
}

3) When you do a power of a power, multiply the indices

`(a`

^{2})^{3} = a^{6}

(a^{-4})^{1/2} = a^{-2}

(a^{1/7})^{7} = a^{1} = a

## Multiplying Out Brackets

A common Nat5 question is to get you to expand an expression with indices. Just multiply ouy the terms as usual then use your power rules:

` a`

^{1/2}(a^{3} + a^{-1/2})

= a^{1/2}a^{3} + a^{1/2}a^{-1/2}

= a^{1/2}a^{6/2} + a^{1/2}a^{-1/2}

= a^{7/2} + a^{0}

= a^{7/2} + 1