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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.

**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

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

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

**Subtract the indices****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

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

We can simplify this process by simplifying ^{7} ÷ 8^{4}^{7-4}^{3}

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

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

This is the division law of indices.

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

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

DOWNLOAD FREESimplify and leave your answer in index form.

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

**Subtract the indices 5 and 2.**

\[5 – 2 = 3\]

So,

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

Simplify and leave your answer in index form.

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

**Subtract the indices 9 and 2.**

\[9 – 2 = 7\]

**Divide the coefficients 21 and 7.**

\[21 ÷ 7 =3\]

So,

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

Simplify and leave your answer in index form.

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

**Subtract the powers -6 and the -4.**

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

**Divide the coefficients 18 and the 9.**

\[18 ÷ 9 = 2\]

So,

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

Simplify and leave your answer in index form.

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

**Subtract the indices 6b and -7b.**

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

**Divide the coefficients 100 and -10.**

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

So,

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

Simplify and leave your answer in index form.

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

**Subtract the indices ⅘ and ⅔.**

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

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

Multiples of

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

We can see that ** 15 **is the

To change the denominator

To change the denominator

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

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,

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

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

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

x^{11}

x^{4.5}

7x

x^{7}

9-2=7 so

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

2. Simplify

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

3b^{6}

8b^{6}

3b^{\frac{13}{7}}

3^{6}

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}}

56a

9a

9a^{-5}

-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}

20c^{4}d

6c^{4}

6c^{4}d

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}

-10 a^{8 x-b}

10 a^{8 x+5b}

90 a^{8 x-b}

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}}

2 x^{\frac{5}{12}}

0.5 x^{\frac{5}{12}}

2 x^{\frac{9}{4}}

\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}}

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}

Evaluate:

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

**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.**

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

Evaluate:

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

**Write out each term without the indices.**

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

**Work out the calculation and simplify.**

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

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:

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

**Write out each term without the indices.**

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

**Work out the calculation and simplify.**

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

1. Evaluate

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

\frac{25}{27}

2^{-1}

\frac{25}{81}

(\frac{5}{3})^{-1}

\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}

\frac{2}{27}

\frac{4}{27}

\frac{8}{27}

\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

108

\frac{81}{\sqrt{2}}

324

\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

9

\frac{1}{4}

\frac{9}{4}

\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}

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)**

**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.

You have now learned how to:

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

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