One to one maths interventions built for GCSE success

Weekly online one to one GCSE maths revision lessons available in the spring term

Find out more

Dividing Indices

Here we will learn about dividing indices including how to use the laws of indices to divide indices and how to divide indices that have different bases.

There are also laws of indices worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.

What is dividing indices?

Dividing indices is where we divide terms that involve indices or powers.

There are two methods we can use to divide terms involving indices.

1 When the bases are the same:

E.g

\[a^{5} \div a^{3}=a^{5-3}=a^{2}\]

These questions usually ask you to ‘simplify’ the calculation.

2 When the bases are different:

E.g

\[2^{5} \div 4^{2}=32 \div 16=2\]

These questions usually ask you to ‘evaluate’ (work out) the calculation

What do we mean by dividing indices?

What do we mean by dividing indices?

How to divide indices when the bases are the same

When dividing indices with the same base, subtract the powers.

\[a^{m} \div a^{n}=a^{m-n}\]

  1. Subtract the indices
  2. Divide any coefficients of the base number or letter

In order to divide indices when the bases are the same we can use one of the laws of indices.

E.g.

To simplify the following expression:

\[8^{7} \div 8^{4}\]

We can write out each power in its expanded form:

\[8^{7} \div 8^{4}=\frac{8 \times 8 \times 8 \times 8 \times 8 \times 8 \times 8}{8 \times 8 \times 8 \times 8}\]

We know that 8 ÷ 8 = 1, so we can keep canceling an 8 on the top with an 8 on bottom until we are left with:

\[8 \times 8 \times 8=8^{3} \]

We can simplify this process by simplifying 87 ÷ 84 to 87-4 = 83:

\[8^{7} \div 8^{4}=8^{7-4}=8^{3}\]

The base has stayed the same and we have subtracted the indices.

This is the division law of indices.

Explain how to divide indices when the bases are the same

Explain how to divide indices when the bases are the same

Dividing indices worksheet

Dividing indices worksheet

Dividing indices worksheet

Get your free dividing indices worksheet of 20+ questions and answers. Includes reasoning and applied questions.

DOWNLOAD FREE
x
Dividing indices worksheet

Dividing indices worksheet

Dividing indices worksheet

Get your free dividing indices worksheet of 20+ questions and answers. Includes reasoning and applied questions.

DOWNLOAD FREE

Dividing indices examples (with the same base)

Example 1: no coefficient in front of base

Simplify and leave your answer in index form.

\[a^{5} \div a^{2}\]

  1. Subtract the indices 5 and 2.

\[5 – 2 = 3\]

So,

\[a^{5} \div a^{2}=a^{3}\]

Example 2: with a coefficient in front of base

Simplify and leave your answer in index form.

\[21 a^{9} \div 7 a^{2}\]

\[9 – 2 = 7\]

\[21 ÷ 7 =3\]


So,

\[21 a^{9} \div 7 a^{2}=3a^{7}\]

Example 3: dividing indices (negative Indices)

Simplify and leave your answer in index form.

\[18 a^{-6} \div 9 a^{-4}\]

\[-6 -(-4) = -6+4=-2\]

\[18 ÷ 9 = 2\]


So,

\[18 a^{-6} \div 9 a^{-4}=2 a^{-2}\]

Example 4: algebraic expressions

Simplify and leave your answer in index form.

\[100 a^{6 b} \div-10 a^{-7 b}\]

\[6b -(- 7b) = 6b+7b=13b\]

\[100 \div-10=-10\]


So,

\[100 a^{6 b} \div-10 a^{-7 b}=-10 a^{13 b}\]

Example 5: with fractional powers

Simplify and leave your answer in index form.

\[x^{\frac{4}{5}} \div x^{\frac{2}{3}}\]

\[\frac{4}{5}-\frac{2}{3}\]


Remember, to subtract fractions we need to find the lowest common denominator.


Multiples of 5           Multiples of 3

\[5, 10, 15, 20 \quad \quad \quad 3, 6, 9, 12, 15, 18\]


We can see that 15 is the lowest common denominator.


To change the denominator 5 into 15 we need to multiply by 3, so we multiply the numerator by 3 too.


To change the denominator 3 into 15 we need to multiply by 5, so we multiply the numerator by 5 too.



\[\frac{12}{15}-\frac{10}{15} \]


Now that we have a common denominator we can subtract the numerators:

\[\frac{12}{15}-\frac{10}{15}=\frac{12-10}{15}=\frac{2}{15}\]


So,

\[x^{\frac{4}{5}} \div x^{\frac{2}{3}}=x^{\frac{2}{15}}\]

Practice dividing indices questions (with the same base)

1. Simplify the expression. Express your answer in index form.

 

x^{9} \div x^{2}

 

x^{11}
GCSE Quiz False

x^{4.5}
GCSE Quiz False

7x
GCSE Quiz False

x^{7}
GCSE Quiz True

9-2=7 so

 

x^{9} \div x^{2} = x^{7}

2. Simplify

 

12 b^{13} \div 4 b^{7}

 

3b^{6}
GCSE Quiz True

8b^{6}
GCSE Quiz False

3b^{\frac{13}{7}}
GCSE Quiz False

3^{6}
GCSE Quiz False

13-7=6

 

12 \div 4 = 3

 

12 b^{13} \div 4 b^{7}=3b^{6}

3. Simplify

 

63^{-2} \div 7 a^{-3}

9a^{\frac{2}{3}}
GCSE Quiz False

56a
GCSE Quiz False

9a
GCSE Quiz True

9a^{-5}
GCSE Quiz False

-2 – (-3) = -2+3 = 1

 

63 \div 7=9

 

63a^{-2} \div 7 a^{-3} = 9a

 

(When the power is 1 we do not need to write it)

4. Simplify

 

\frac{24c^{6}d^{3}}{4c^{2}d^{2}}

 

6cd^{5}
GCSE Quiz False

20c^{4}d
GCSE Quiz False

6c^{4}
GCSE Quiz False

6c^{4}d
GCSE Quiz True

For c: 6-2=4

 

For d: 3-2=1

 

24 \div 4 = 6

 

\frac{24c^{6}d^{3}}{4c^{2}d^{2}} = 6c^{4}d

5. Simplify

 

-100 a^{8 x+2 b} \div 10 a^{3 b}

 

10 a^{8 x-b}
GCSE Quiz False

-10 a^{8 x-b}
GCSE Quiz True

10 a^{8 x+5b}
GCSE Quiz False

90 a^{8 x-b}
GCSE Quiz False

8x+2b-3b=8x-b

 

-100 \div 10 = -10

 

-100 a^{8 x+2 b} \div 10 a^{3 b}=-10 a^{8 x-b}

6. Simplify

 

4 x^{\frac{3}{4}} \div 2 x^{\frac{1}{3}}

4 x^{\frac{5}{12}}
GCSE Quiz False

2 x^{\frac{5}{12}}
GCSE Quiz True

0.5 x^{\frac{5}{12}}
GCSE Quiz False

2 x^{\frac{9}{4}}
GCSE Quiz False

\begin{aligned} \frac{3}{4} – \frac{1}{3} &= \frac{9}{12} – \frac{4}{12}\\\\ &= \frac{5}{12} \end{aligned}

 

4 \div 2 = 2

 

4 x^{\frac{3}{4}} \div 2 x^{\frac{1}{3}} = 2 x^{\frac{5}{12}}

How to divide indices when the bases are different

In order to divide indices when the bases are different we need to write out each term and calculate the answer.

We cannot simplify them using the laws of indices as the bases are not the same.

1 Write out each term without the indices.

2 Work out the calculation and simplify.

E.g.

To evaluate the following expression:

\[2^{3} \div 3^{2}\]

We need to write each term of the calculation without using index notation:

\[2^{3}=2 \times 2 \times 2=8\] \[3^{2}=3 \times 3=9\]

So we have:

\[2^{3} \div 3^{2}=\frac{2 \times 2 \times 2}{3 \times 3}\]

Now we need to work out the calculation:

\begin{aligned} 2^{3} \div 3^{2} &=\frac{2 \times 2 \times 2}{3 \times 3} \\ &=\frac{8}{9} \end{aligned}

Explain how to divide indices when the bases are different

Explain how to divide indices when the bases are different

Dividing indices examples (with different bases)

Example 1: with positive indices

Evaluate:

\[4^{2} \div 2^{3}\]

  1. Write out each term without the indices.

\[4^{2} \div 2^{3}=\frac{4 \times 4}{2 \times 2 \times 2}\]

2 Work out the calculation and simplify.

\[\begin{aligned} 4^{2} \div 2^{3} &=\frac{4 \times 4}{2 \times 2 \times 2} \\\\ &=\frac{16}{8} \\\\ &=2 \end{aligned} \]

Example 2: with three terms

Evaluate:

\[8^{2} \div 2^{4} \div 3^{2}\]

\[8^{2}=8 \times 8=64\] \[2^{4}=2 \times 2 \times 2 \times 2=16\] \[3^{2}=3 \times 3=9\]

\[ \begin{aligned} 8^{2} \div 2^{4} \div 3^{2} &=64 \div 16 \div 9 \\\\ &=\frac{4}{9} \end{aligned} \]

Example 3: with negative powers and fractional powers

This example uses the Negative and Fractional indices. It is a good idea to check our Laws of Indices page for more information before attempting this question.

Evaluate:

\[8^{\frac{1}{3}} \div 2^{-2}\]

\[8^{\frac{1}{3}}=(\sqrt[3]{8})^{1}=2\] \[2^{-2}=\frac{1}{2^{2}}=\frac{1}{4}\]

\[\begin{aligned} 8^{\frac{1}{3}} \div 2^{-2} &=2 \div \frac{1}{4} \\\\ &=2 \times 4 \\\\ &=8 \end{aligned}\]

Practice dividing indices questions (with different bases)

1. Evaluate

 

5^{2} \div 3^{3}

 

\frac{25}{27}
GCSE Quiz True

2^{-1}
GCSE Quiz False

\frac{25}{81}
GCSE Quiz False

(\frac{5}{3})^{-1}
GCSE Quiz False
\begin{aligned} 5^{2} \div 3^{3} &= \frac{5 \times 5}{3 \times 3 \times 3}\\\\ &=\frac{25}{27} \end{aligned}

2. Evaluate

 

2^{6} \div 4^{2} \div 3^{3}

 

\frac{5}{30}
GCSE Quiz False

\frac{2}{27}
GCSE Quiz False

\frac{4}{27}
GCSE Quiz True

\frac{8}{27}
GCSE Quiz False
\begin{aligned} 2^{6} &= 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 64\\\\ 4^{2} &= 4 \times 4 = 16\\\\ 3^{3} &= 3 \times 3 \times 3 = 27 \end{aligned}

 

\begin{aligned} 2^{6} \div 4^{2} \div 3^{3} &= 64 \div 16 \div 27\\\\ &=\frac{4}{27} \end{aligned}

3. Evaluate

 

3^{4} \div 2^{-2}

 

20.25
GCSE Quiz False

108
GCSE Quiz False

\frac{81}{\sqrt{2}}
GCSE Quiz False

324
GCSE Quiz True
\begin{aligned} 3^{4}&=3 \times 3 \times 3 \times 3=81\\\\ 2^{-2}&=\frac{1}{-2}=\frac{1}{4} \end{aligned}

 

\begin{aligned} 3^{4} \div 2^{-2} &= 81 \div \frac{1}{4}\\\\ &=81 \times 4\\\\ &=324 \end{aligned}

4. Evaluate

 

27^{\frac{2}{3}} \div 2^{-2} \div 3^{2}

 

4
GCSE Quiz True

9
GCSE Quiz False

\frac{1}{4}
GCSE Quiz False

\frac{9}{4}
GCSE Quiz False
\begin{aligned} 27^{\frac{2}{3}}&=9\\\\ 2^{-2}&=\frac{1}{4}\\\\ 3^{2}&=9 \end{aligned}

 

\begin{aligned} 27^{\frac{2}{3}} \div 2^{-2}\div 3^{2} & = 9 \div \frac{1}{4} \div 9\\\\ &=4 \end{aligned}

Dividing indices GCSE questions

1. Simplify

x^{2} \div x^{3}

(1 mark)

Show answer
x^{-1}

            (1)

2. Simplify

8 h^{3} m^{6} \div 2 h^{4} m^{2}

(2 marks)

Show answer

h^{-1} \text { or } m^{4} seen (evidence of subtracting powers)

            (1)

 

4 h^{-1} m^{4}

            (1)

3. Simplify

\frac{30 x^{3} y^{-4}}{5 x^{-2} y^{-3}}

(2 marks)

Show answer

x^{5} \text { or } y^{-1} seen (evidence of subtracting powers)

            (1)

 

6 x^{5} y^{-1}

            (1)

Common misconceptions

  • The dividing indices law can only be used for terms with the same base

E.g. We cannot simplify

\[a^{4} \div b^{3}\]

as the bases are different

  • Confusing integer and fractional powers

Raising a term to the power of 2 means we square it

E.g.

\[a^{2}=a \times a\]

Raising a term to the power of ½ means we find the square root of it

E.g.

\[a^{\frac{1}{2}}=\pm \sqrt{a}\]

Raising a term to the power of 3 means we cube it

E.g.

\[a^{3}=a \times a \times a\]

Raising a term to the power of ⅓ means we find the cube root of it

E.g.

\[a^{\frac{1}{3}}=\sqrt[3]{a}\]

  • Indices, powers or exponents

Indices can also be called powers or exponents.

Learning checklist

You have now learned how to:

  • Simplify expressions involving the laws of indices
  • Calculate with roots, and with integer and fractional indices

Still stuck?

Prepare your KS4 students for maths GCSEs success with Third Space Learning. Weekly online one to one GCSE maths revision lessons delivered by expert maths tutors.

Find out more about our GCSE maths revision programme.