# Negative Indices

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.

## What are negative indices?

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 x

$\frac{\not x \times \not x \times \not x}{\not x \times \not x \times \not x \times x}$

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,

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

## How to use negative indices

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.

## Negative indices examples

### Example 1: no coefficient in front of the base

$a^{-4}$

1. Put the term from the question over 1.

$\frac{a^{-4}}{1}$

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

$\frac{1}{a^{4}}$

### Example 2: with a coefficient in front of base

$(10 a)^{-3}$

Notice how the index affects the entire bracket.

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

$\frac{1}{(10 a)^{3}}$

$=\frac{1}{1000 a^{3}}$

### Example 3: with a coefficient in front of base

$3b^{-2}$

Notice how the index only affects the variable b.

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

\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 b and not the coefficient 3.

### Example 4: with fractional exponents

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

Evaluate

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

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

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

### Common misconceptions

• 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.

$a^{\frac{1}{3}}=\sqrt{a}$

• 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:

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

### Practice negative indices questions

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 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 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}}

\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}

### Negative indices GCSE questions

1. Evaluate

9^{-2}

(1 mark)

\frac{1}{81}

(1)

2. Evaluate

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

(2 marks)

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

(1)

\frac{25}{16}

(1)

3. Evaluate

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

(3 marks)

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

(1)

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

(1)

\frac{25}{4}=6.25

(1)

## Learning checklist

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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