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:
22 = 2 x 2 = 4
23 = 2 x 2 x 2 = 8
24 = 2 x 2 x 2 x 2 = 16

You can do this with algebra as well, so a3 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:
21 = 2    171 = 17    a1 = a
20 = 1    170 = 1     a0 = 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 = 1/23 = 1/8
5-2 = 1/52 = 1/25
a-4 = 1/a4
a-1 = 1/a1 = 1/a

A fractional index means a root:
161/2 = ­√16 = 4
10001/3 = 3­√1000 = 10
a1/4 = 4­√a

These can be combined, so:
25-1/2 = 1/251/2 = 1/­√25 = 1/5
a-1/6 = 1/6­√a
a-5/2 = 1/2­√a5

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
a2 x a3 = a5
a6 x a-2 = a4
a3/2 x a-1/2 = a1 = a

2) When you divide powers, subtract the indices
a7 ÷ a2 = a5
a2 ÷ a8 = a-6
a2/3 ÷ a1/3 = a1/3

3) When you do a power of a power, multiply the indices
(a2)3 = a6
(a-4)1/2 = a-2
(a1/7)7 = a1 = 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:
a1/2(a3 + a-1/2)
= a1/2a3 + a1/2a-1/2
= a1/2a6/2 + a1/2a-1/2
= a7/2 + a0
= a7/2 + 1

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