Powers (Indicies)

To write 3+3+3+3+3+3+3, you write 7x3

that is 3 added to itself 7 times.

To multiply 3 by itself 7 times, that is 3x3x3x3x3x3x3

you write 3⁷

The little number is called a power or an index (pleural indicies)

3⁵ x 3⁴ = 3x3x3x3x3 x 3x3x3x3 = 3⁹

So 3⁵ x 3⁴ = 3⁵⁺⁴ = 3⁹

And this leads to the general rule 3^a x 3^b = 3^a+b

Or more generally x^a x x^b = x^a+b

What about 3⁵ ÷ 3⁴ ?

Writing this in the form the four 3’s at the bottom can be cancelled with four of the 3’s on the top leaving 3

So 3⁵ ÷ 3⁴ = 3^5-4 = 3¹ = 3

And the general rule is x^a ÷ x^b = x^a-b

Using this rule, 104 ÷ 107 = 10-3 What does this mean?

Again writing this out in the form

Cancelling the four 10’s on the top with four of the 10’s on the bottom leaves

This must be the same answer as given by the subtracting the indicies method and so

And generally

Following from this 3¹ = 3^½+^½ = 3^½ x 3^½

So 3^½ is the number which multiplied by itself is 3, that is

And so x^½ is just another way of writing

And generally so, for example

You may or may not need a calculator to work out:-

A number written using the symbol is called a surd. For more about surds click here.

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