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Negative numbers

Multiplication

Division

Converting between fractions and decimals

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Here we will learn everything you need to know about multiplying indices for GCSE maths (Edexcel, AQA and OCR). You’ll learn how to use the laws of indices to multiply indices and how to multiply indices that have different bases.

Look out for the laws of indices worksheets and exam questions at the end.

Multiplying indices is where we multiply terms that involve indices or powers.

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

- When the bases are the same

E.g.

\[a^{3} \times a^{4}=a^{3+4}=a^{7}\]

These questions usually ask you ‘simplify’ the calculation

2When the bases are different

E.g.

\[2^{3} \times 3^{2}=8 \times 9=72\]

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

This lesson is part of our series on laws of indices. You may find it helpful to start with the main laws of indices lesson for a summary of what to expect and then also work through the following:

- Laws of indices
- Dividing indices
- Fractional indices
- Negative indices
- Power of 0

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

COMING SOONGet your free multiplying indices worksheet of 20+ questions and answers. Includes reasoning and applied questions.

COMING SOONIn order to multiply indices when the bases are the same we can use the one of the laws of indices.

E.g.

To simplify the following expression:

\[8^{3} \times 8^{4}\]

We can write out each power in its expanded form.

\[\begin{aligned}
8^{3} \times 8^{4} &=8 \times 8 \times 8 \times 8 \times 8 \times 8 \times 8 \\
&=8^{7}
\end{aligned}\]

Is there a quicker way to work this out?

Simplifying

\[8^{3} \times 8^{4} \text { to } 8^{3+4}\]

we can work out the simplified answer is

\[8^{7}\]

\[8^{3} \times 8^{4}=8^{3+4}=8^{7}\]

The base has stayed the same and we have added the indices together.

This is the multiplication law of indices.

1**Simplify and leave your answer in index form**

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

\[5 + 2 = 7\]

**Answer: **a^{7}

Simplify and leave your answer in index form.

\[4 a^{3} \times 7 a^{2}\]

**This can be written as**

\[4 a^{3} \times 7 a^{2}=4 \times a \times a \times a \times 7 \times a \times a\]

**Add together the indices 3 and the 2.**

\[3 + 2 = 5\]

**Multiply 4 and 7 together.**

\[4 \times 7 =28\]

\[=28a^{5}\]

Simplify and leave your answer in index form.

\[7 a^{-6} \times 9 a^{-2}\]

**Add together the indices -6 and the -2.**

\[-6 +- 2 = -6-2=-8\]

**Multiply the 7 and 9 together**.

\[7 \times 9=63\]

\[=63a^{-8}\]

Simplify and leave your answer in index form.

\[6 a^{6 b} \times-2 a^{-7 b}\]

**Add together the indices 6b and -7b.**

\[6b +- 7b =6b-7b= -b\]

**Multiply the 6 and -2 together**.

\[6 \times-2=-12\]

\[ =12a^{-b}\]

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

Step-by-Step: laws of Indices

Work out the exact value of

\[8^{\frac{1}{3}} \times 2^{n}=4^{3}\]

Although it appears that the bases are different, we can rewrite each term so that it has a base of 2.

**Rewrite each term so that it has the same base.**

8 can be written as 2^{3}, so

\[8^{\frac{1}{3}}=\left(2^{3}\right)^{\frac{1}{3}}=2^{3 \times \frac{1}{3}}=2^{1}\]

4 can be written as 2^{2}, so

\[4^{3}=\left(2^{2}\right)^{3}=2^{2 \times 3}=2^{6}\]

So,

\[\begin{aligned}
8^{\frac{1}{3}} \times 2^{n}&=4^{3} \\
2^{1} \times 2^{n}&=2^{6}
\end{aligned}\]

**Use the multiplication law of indices to work out the unknown power.**

\[2^{1} \times 2^{n}=2^{1+n}=2^{6}\]

So,

\[\begin{aligned}
1+n &=6 \\
n &=6-1 \\
n &=5
\end{aligned}\]

1. Simplify and leave your answer in index form.

x^{5} \times x^{2}

Show answer

=x^{7}

2. Simplify and leave your answer in index form.

4 b^{3} \times b^{-7}

Show answer

=4 b^{-4}

3. Simplify and leave your answer in index form.

6 a^{5} \times 4 a^{-3}

Show answer

=24 a^{2}

4. Simplify and leave your answer in index form.

-8 a^{7 b} \times 3 a^{3 b}

Show answer

=-24 a^{10 b}

5. Work out the exact value of

27^{\frac{2}{3}} \times 3^{n}=9^{5}

Show answer

n=8

In order to multiply indices when the bases are different we need to write out each term and multiply them together.

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

- Write out each term without the indices
- Work out the calculation

E.g.

To evaluate the following expression:

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

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

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

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

We can the work out the final answer by multiplying these numbers together

\[\begin{aligned}
2^{3} \times 3^{2} &=8 \times 9 \\
&=72
\end{aligned}\]

Evaluate:

\[4^{2} \times 2^{4}\]

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

\[4^{2} \times 2^{4}=4 \times 4 \times 2 \times 2 \times 2 \times 2\]

2**Work out the calculation.**

\[4^{2}=4 \times 4=16 \\
2^{4}=2 \times 2 \times 2 \times 2=16\]

So,

\[\begin{aligned}
4^{2} \times 2^{4} &=16 \times 16 \\
&=256
\end{aligned}\]

Evaluate:

\[3^{2} \times 2^{3} \times 10^{2}\]

Write out each term without the indices

\[3^{2} \times 2^{3} \times 10^{2}=3 \times 3 \times 2 \times 2 \times 2 \times 10 \times 10\]

Work out the calculation.

\[3^{2}=3 \times 3=9 \\
2^{3}=2 \times 2 \times 2=8 \\
10^{2}=10 \times 10=100\]

So,

\[\begin{aligned}
3^{2} \times 2^{3} \times 10^{2} &=9 \times 8 \times 100 \\
&=7200
\end{aligned}\]

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

**Step by step guide: **Laws of indices** (coming soon)**

Evaluate:

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

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.

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

1. Evaluate

3^{3} \times 2^{2} Show answer

=108

2. Evaluate

8^{2} \times 4^{-2}

Show answer

=4

3. Evaluate

2^{4} \times 5^{2} \times 3^{3}

Show answer

=10800

4. Evaluate

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

Show answer

=8

1. Simplify

x^{2} \times x^{3}

**(1 mark)**

Show answer

x^{5}

**(1)**

2. Simplify

6 h^{3} m^{6} \times 4 h^{4} m^{5}

**(2 marks)**

Show answer

h^{7} \text { or } m^{11} seen (evidence of adding powers)

**(1)**

24 h^{7} m^{11}

**(1)**

3. Simplify

4 x^{-3} y^{5} \times 3 x^{-2} y^{-3}

**(2 marks)**

Show answer

x^{-5} \text { or } y^{2} seen (evidence of adding powers)

**(1)**

12 x^{-5} y^{2}

**(1)**

**The multiplying indices law can only be used for terms with the same base**

E.g. We cannot simplify

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

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

- Solving algebraic equations
- Standard form
- Surds
- Quadratic equations

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