Index Form, Roots and Laws

An Index can sometimes also be referred to as a Power. An Index is the small number that floats beside a letter or number. Indices is the plural term for an Index.

The index number shows how many times a letter or number has been multiplied by itself, essentially telling you how many times the number or letter has been multiplied.

When you see 3⁴ 3 is the base number and ⁴ is the Index number. So this would indicate 3 x 3 x 3 x 3. With 3 multiplied 4 times.

Roots can be calculated as follows:

√25 = 5 this symbolises that the square root of 25 is 5, because 5 x 5 = 25.

³√8 = 2 this symbolises that the cube root of 8 is 2, because 2 x 2 x 2 = 8.

⁴√81 = 3 this symbolises that the fourth root of 81 is 3, because 3 x 3 x 3 x 3 = 81.

⁵√32 = 2 this symbolises that the fifth root of 32 is 2, because 2 x 2 x 2 x 2 x 2 = 32.

Index laws will only apply if the base numbers are the same and the normal rules will also apply for negative numbers.

To multiply indices you simply have to add the powers to get the final index, for example:

3⁴ x 3³ = 3^{4 + 3} = 3⁷

To divide indices you simply have to subtract the powers to get the final index, for example:

3⁵ ÷ 3³ = 3^{5 - 3} = 3²

To raise one power to another power we simply have to multiply the powers to give the final index for example:

(3²) ⁴ = 3^{2 x 4} = 3⁸

This video explains how to use Indices in Algebraic form.

You can find out more on working with Indices in Algebra here.