Evaluating Exponents with Fractional Bases | KS3 Maths Resources

## What you need to know

Things to remember:

• When we put an exponent on a fraction, we can just put the exponent on the top and bottom numbers.

An exponent is a number written to the top right of a number and tells us how many times we multiply a number by itself.

$$6^2 = 6\times 6$$

$$6^3 = 6\times 6\times 6$$

$$6^4= 6\times 6\times 6\times 6$$

So, all we’re doing when we have exponents is lots of multiplications. When we do these multiplications, we refer to it as “evaluating the exponent”.

Evaluate the following exponents:

$$5^2=5\times5=25$$

$$7^3=7\times7\times7=343$$

$$8^4=8\times8\times8\times8=4096$$

This topic will focus on fractions with powers, meaning we will be multiplying lots of fractions, which we do by multiplying the tops and multiplying the bottoms.

$$\frac{2}{5}\times\frac{7}{3}=\frac{2\times7}{5\times3}=\frac{14}{15}$$

$$\frac{8}{9}\times\frac{1}{4}=\frac{8\times1}{9\times4}=\frac{8}{36}=\frac{2}{9}$$

We can use all these facts to help us evaluate exponents with fractional bases.

Evaluate $\left(\frac{2}{7}\right)^3$

We know that when we have a power of 3, we multiply the number by itself 3 times. This is the same with fractions.

$$\left(\frac{2}{7}\right)^3 =\frac{2}{7}\times\frac{2}{7}\times \frac{2}{7}$$

But, we know that we just multiply the tops and bottoms.

$$\frac{2}{7}\times\frac{2}{7}\times\frac{2}{7}=\frac{2\times2\times2}{7\times7\times7}$$

And now, we can just do these multiplications.

$$\frac{2\times2\times2}{7\times7\times7}=\frac{8}{343}$$

$$\left(\frac{2}{7}\right)^3=\frac{8}{343}$$

If we look back a step, we can learn an important fact:

$$\left(\frac{2}{7}\right)^3=\frac{2\times2\times2}{7\times7\times7}$$

We can write these multiplications as powers.

$$\left(\frac{2}{7}\right)^3=\frac{2^3}{7^3}$$

So, when we have an exponent on a fraction, it is the same as putting that exponent on both numbers in the fraction.

$$\left(\frac{4}{3}\right)^2=\frac{4^2}{3^2}=\frac{16}{9}$$

$$\left(\frac{2}{5}\right)^3=\frac{2^3}{5^3}=\frac{8}{125}$$

$$\left(\frac{3}{7}\right)^4=\frac{3^4}{7^4}=\frac{81}{2401}$$

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## Example Questions

$$\left(\dfrac{1}{6}\right)^3=\dfrac{1^3}{6^3}=\dfrac{1}{216}$$

$$\left(\dfrac{5}{2}\right)^2=\dfrac{5^2}{2^2}=\dfrac{25}{4}$$

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