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}

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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