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In order to access this I need to be confident with:

Negative numbers

Multiplication

Division

Converting between fractions and decimals

This topic is relevant for:

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.

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

COMING SOONNegative indices is part of the larger topic, laws of indices. It may be useful to explore the main topic before looking into the detailed individual lessons below:

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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