Negative Indices

Negative Indices

Now these seem simple enough: you raise a number to a power and it usually means multiplying a number by itself n times.

a³ = a x a x a

All numbers to the power of 1 are themselves, and all numbers to the power 0 are one.

a⁰ = 1



Easy enough.


But then you can also have negative powers.

a^-3

What? You can’t multiply a number by itself a negative number of times!

Well, no, clearly. But you can see how negative powers come about.


It’s to do with the powers rule.

aⁿ x a^m = a^n+m

You can see this if you write out aⁿ and a^m in full.

aⁿ x a^-n = a^{n+ -n}

= a^n-n
=a¹
=a

Therefore, the negative powers are used to denote reciprocals (the number you multiply n by to get 1 – so 1/6 is the reciprocal of 6 – and it is always 1/n.

So.


3³ =27

3^-3 = 1/27 (or 1/33 – the reciprocal).


Negative powers are easy to manipulate, they just seem a bit weird until you think that positive powers are going up by multiplying the number by itself:

2⁴ is 2 x 2 x 2 x 2

But if you go down, towards 0, the same process is division by two.

64...32...16..8...4..2..

This continues as you go down below zero:

......2, 1, ½, ¼, 1/8, 1/16......

(2¹, 2⁰, 2^-1, 2^-2, 2^-3, 2^-4)

And you can see that these are reciprocals of the powers of 2.

The same applies for the powers of each number.