Laws of indices

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Learn about the laws of indices.

This lesson includes:

  • a learning summary
  • one activity sheet

Learn

Students looking to achieve grade 4 in GCSE Maths must understand what indices are and how they are used in calculations.

Read pages 1 to 4 of our 'Laws of indices' Bitesize revision guide:

Revise - laws of indices

Using an index or power

An index, or power, is the small floating number that goes next to a number or letter. The plural of index is indices.

Indices show how many times a number or letter has been multiplied by itself.

Here is a number written in index form:

In this example, 2 is the base number and 4 is the index or power.

This is a short way of writing 2 × 2 × 2 × 2.

(read as 'a cubed') is a short way of writing a × a × a. In other words, a has been multiplied by itself three times.

Multiplying indices

Worked example

Simplify c³ × c².

To answer this question, you could write c³ and c² out in full:

c³ = c × c × c and c² = c × c.

Therefore, c³ × c² = c × c × c × c × c.

You can see that c has now been multiplied by itself 5 times.This means c³ × c² can be simplified to c⁵.

Another way to multiply indices is simply to add each index value to get a total.

In this example, 3 + 2 = 5. So, c³ × c² = c⁵

Note that this is is only true when the base number is the same. So, d³ × e² cannot be simplified because d and e are different values.

Dividing indices

Worked example

Simplify b⁵ ÷ b³.

One way to solve this is to consider that b⁵ ÷ b³ can be written as b⁵ over b³ (b⁵⁄ b³). Writing out the denominator (the bottom part of the fraction) and numerator (the top part of the fraction) in full gives:

There are common factors of b in the numerator and denominator and these can be cancelled out, which leaves b × b = b².

This means b⁵ ÷ b³ can be simplified to b².

A quicker way to divide indices with the same base number is to subtract the index values.

So, in this example 5-3=2 so b⁵ ÷ b³ = b²

Raising a power to a power

Worked example

Simplify (k³)²

This means that k³ is to be squared, or multiplied by k³ again: (k³)² = k³ × k³

Add the powers together, so (k³)² = k³ × k³ = k⁶, so (k³)² can be simplified to k⁶.

Negative indices

A negative power, or negative index, is often referred to as a reciprocal.

The reciprocal of a number is 1 divided by that number. For example, the reciprocal of the number 4 is ¼.

The reciprocal of a fraction is that fraction turned upside down. For example, the reciprocal of ³⁄₄ is ⁴⁄₃.

Worked example

Simplify d⁴ ÷ d⁵

Dividing indices means subtracting the index values, so:

d⁴ ÷ d⁵ = d⁴⁻⁵ = d⁻¹

This is an example of a negative index.

But you can write d⁴ ÷ d⁵ out and see that it is also equal to:

Cancelling common factors gives:

which shows that d⁴ ÷ d⁵ = ¹⁄d

So, d⁻¹ = ¹⁄d

To learn more about indices look at the Law of indices Bitesize guide here.

Practise

Activity 1

Index laws

Complete the activity sheet from Beyond by Twinkl on index laws to test your knowledge. You can print it out or write your answers on a piece of paper.

Index laws

Click here to see the right answers.

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