Negative Exponents (powers) and Reciprocals | KS3 Maths Resources

## What you need to know

Things to remember:

• We can write negative exponents as fractions with a numerator of 1, and a denominator of the original number and power without the negative.

Powers can come in all shapes and sizes, from integers to fractions and decimals, positive numbers to negative numbers, and real numbers to imaginary…………………. but you don’t need to worry about imaginary numbers…

For example, we could see powers like this:

$$5^4$$

$$5^{-4}$$

$$5^{0.4}$$

$$5^{\dfrac{1}{4}}$$

Before we can look at negative powers, we need to remember what a reciprocal of a number is. The reciprocal of a number is the number multiply it by to make 1.

$$2\times\frac{1}{2}=1$$

$$6\times\frac{1}{6}=1$$

$$10\times\frac{1}{10}=1$$

So, to make the reciprocal of a number, we just write it as a fraction with a 1 on top!

Changing the negative exponent into a nicer form takes two steps:

Step 1: Remove the negative symbol.

Step 2: Write as a reciprocal of the number and power.

Exponent Step 1 Step 2

$5^{-3}$ $5^{3}$ $\dfrac{1}{5^3}$

$6^{-7}$ $6^{7}$ $\dfrac{1}{6^7}$

$2^{-9}$ $2^{9}$ $\dfrac{1}{2^9}$

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## Example Questions

Exponent

$11^{-19}$

Step 1

$11^{19}$

Step 2

$\dfrac{1}{11^19}$

$$11^{-19}=\frac{1}{11^{19}}$$

Exponent

$27^{-4}$

Step 1

$27^{4}$

Step 2

$\dfrac{1}{27^4}$

$$27^{-4}=\frac{1}{27^4}$$

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