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Here we will learn how to simplify and evaluate with negative indices for GCSE & iGCSE maths (Edexcel, AQA and OCR).

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

Negative indices are powers (also called exponents) with a minus sign in front of them.

E.g.

\[x^{-2}\]

\[3^{-4}\]

\[2b^{-\frac{1}{2}}\]

We get negative indices by dividing two terms with the same base where the first term is raised to a power that is smaller than the power that the second term is raised to.

E.g.

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

When we cancel the common factors of

\[\frac{\not x \times \not x \times \not x}{\not x \times \not x \times \not x \times x}\]

We are left with

\[\frac{1}{x}\]

But using the division law of indices we know that

\[x^{3} \div x^{4}=x^{3-4}=x^{-1}\]

So,

\[x^{-1}=\frac{1}{x}\]

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

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

DOWNLOAD FREE**Negative indices** is part of our series of lessons to support revision 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, or use the step by step guides below for further detail on individual topics. Other lessons in this series include:

To make the negative index positive we put the term over 1 and flip it.

It is known as finding the **reciprocal **of the term.

E.g.

\[x^{-2}=\frac{x^{-2}}{1}=\frac{1}{x^{2}}\]

\[\frac{1}{x^{2}}\]

is the same as

\[x^{-2}\]

E.g.

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

Negative exponents will often be used in conjunction with other index laws, including the division, brackets and multiplication laws.

Simplify and leave your answer in index form.

\[a^{-4}\]

**Put the term from the question over 1.**

\[\frac{a^{-4}}{1}\]

2 **Flip and change the power from -4 to +4.**

\[\frac{1}{a^{4}} \]

Simplify and leave your answer in index form.

\[(10 a)^{-3}\]

Notice how the index affects the entire bracket.

**Put the term over 1.**

\[\frac{(10 a)^{-3}}{1}\]

**Flip and change the power -3 to +3.**

\[\frac{1}{(10 a)^{3}}\]

**Simplify the denominator.**

\[=\frac{1}{1000 a^{3}}\]

Simplify and leave your answer in index form.

\[3b^{-2}\]

Notice how the index only affects the variable

**Put the term over 1.**

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

**Flip and change the power -2 to +2.**

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

The index only applies to the variable

When dealing with fractions it is easier to skip to step 2.

Evaluate

\[\left(\frac{4}{3}\right)^{-2}\]

**Flip and change the power -2 to +2.**

\[\left(\frac{4}{3}\right)^{-2}=\left(\frac{3}{4}\right)^{2}\]

**Simplify the numerator and denominator.**

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

**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.

**Not turning a negative index into a positive index when flipping the term**

**Making a mistake when writing one over a fraction**

E.g

When we find the reciprocal of

\[\left(\frac{2}{3}\right)^{-2}\]

we can follow the steps as outlined above

**1 Put the term over 1**

\[\frac{\left(\frac{2}{3}\right)^{-2}}{1}\]

**2 Flip and change the power -2 to +2**

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

**3 Simplify the denominator**

\[\frac{1}{\left(\frac{2}{3}\right)^{2}}=\frac{1}{\frac{2^{2}}{3^{2}}}=\frac{1}{\frac{4}{9}}\]

However, the fraction needs to be simplified further.

\[\frac{1}{\frac{4}{9}}=1 \div \frac{4}{9}=1 \times \frac{9}{4}=\frac{9}{4}\]

This is why when dealing with fractions it is easier to skip to step 2 and just flip the fraction:

\[\left(\frac{2}{3}\right)^{-2}=\left(\frac{3}{2}\right)^{2}=\frac{3^{2}}{2^{2}}=\frac{9}{4}\]

1. Simplify. Express your answer in index form.

x^{-5}

x^{5}

\frac{1}{x^{-5}}

\frac{1}{x^{5}}

\frac{1}{5x}

The negative index number means we need to find the reciprocal, so

x^{-5}=\frac{1}{x^{5}}

2. Simplify. Express your answer in index form.

(2 b)^{-4}

16b^{-4}

\frac{1}{2 b^{4}}

\frac{1}{8 b^{4}}

\frac{1}{16 b^{4}}

The negative index number means we need to find the reciprocal, so

\begin{aligned} &a(2 b)^{-4}\\ &=\frac{1}{(2b)^{4}}\\ &=\frac{1}{2b\times2b\times2b\times2b}\\ &=\frac{1}{16 b^{4}} \end{aligned}

3. Simplify. Express your answer in index form.

3 b^{-3}

\frac{1}{b^{3}}

\frac{3}{b^{3}}

\frac{3}{3b}

\frac{1}{3b}

The negative index number means we need to find the reciprocal, so

3 b^{-3}=\frac{3}{b^{3}}

4. Simplify. Express your answer in index form.

\left(\frac{5}{8}\right)^{-1}

-\frac{8}{5}

\frac{8}{5}

-\frac{3}{5}

\frac{3}{5}

The negative index number means we need to find the reciprocal, which means inverting the fraction, so

\left(\frac{5}{8}\right)^{-1}=\frac{8}{5}

1. Evaluate

9^{-2}

** (1 mark)**

Show answer

\frac{1}{81}

** (1)**

2. Evaluate

\left(\frac{4}{5}\right)^{-2}

** (2 marks)**

Show answer

\left(\frac{5}{4}\right)^{2}

**(1)**

\frac{25}{16}

**(1)**

3. Evaluate

\left(\frac{8}{125}\right)^{-\frac{2}{3}}

** (3 marks)**

Show answer

\left(\sqrt[3]{\frac{125}{8}}\right)^{2}

**(1)**

\left(\frac{5}{2}\right)^{2}

**(1)**

\frac{25}{4}=6.25

**(1)**

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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