### Numbers: Powers and roots

### Roots of fractions

We can also extract *roots* of fractions. For this, we have a useful calculation rule.

Roots of fractions

When we want to calculate the root of a fraction, we need to find a number that is equal to this fraction when squared. For #\sqrt{\green{\tfrac{4}{9}}}#, we are looking for a number that, when squared, equals #\green{\tfrac{4}{9}}#. This is #\blue{\tfrac{2}{3}}# because \[\left(\blue{\frac{2}{3}}\right)^2=\frac{\blue2^2}{\blue3^2}=\green{\frac{4}{9}}\]

We can therefore see that:

\[\sqrt{\green{\frac{4}{9}}}=\frac{\sqrt{\green4}}{\sqrt{\green9}}=\blue{\frac{2}{3}}\]

In general we can state:

*The square root of a fraction is equal to the square root of the numerator divided by the square root of the denominator.*

**Examples**

\[\begin{array}{rcl}\displaystyle \sqrt{\green{\frac{1}{4}}}&=&\displaystyle \frac{\sqrt{\green1}}{\sqrt{\green4}} \\ &=& \displaystyle \blue{\frac{1}{2}} \\ \\ \displaystyle \sqrt{\green{\frac{3}{4}}}&=&\displaystyle \frac{\sqrt{\green3}}{\sqrt{\green4}} \\ &=& \displaystyle \frac{\blue{\sqrt{3}}}{\blue2} \\ \\ \displaystyle \sqrt{\green{\frac{2}{3}}}&=&\displaystyle \frac{\blue{\sqrt{2}}}{\blue{\sqrt{3}}} \end{array}\]

#\begin{array}{rcl}\sqrt{\dfrac{9}{36}}&=&\dfrac{\sqrt{9}}{\sqrt{36}} \\ &&\phantom{xxx}\blue{\text{calculation rule: the square root of a fraction is equal to }} \\ &&\phantom{xxx}\blue{\text{the square root of the numerator divided by the square root of the denominator}}\\

&=& \dfrac{3}{6} \\ &&\phantom{xxx}\blue{\text{calculated roots}}\\

&=& \displaystyle \frac12 \\ &&\phantom{xxx}\blue{\text{simplified fraction}}

\end{array}#

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