### Numbers: Powers and roots

### Standard notation of higher roots

As with *square roots,* we can write higher roots in standard notation using the *calculation rules*.

When there is a higher root in an answer, we simplify it to the **standard notation.** This means that the higher root needs to meet the following requirements:

*We bring the greatest possible factor in front of the radical sign.*

This means that we remove any numbers to the power of the index of the root, from the number inside the radical sign.

\[\sqrt[\green3]{16}=\sqrt[\green3]{2^\green3 \times 2}=\sqrt[\green3]{2^\green3} \times \sqrt[\green3]{2}=2\sqrt[\green3]{2}\]*We make sure there are no higher roots in the denominator of the fraction.*

We do this by multiplying the numerator and the denominator of the fraction by a number to make sure the denominator does not contain a higher root anymore. A smart choice for this number is the higher power from the denominator, exponentiated to the index of the root minus #1#.

\[\frac{1}{\sqrt[\green4]{3}}=\frac{\left(\sqrt[\green4]{3}\right)^{\green4-1}}{\sqrt[\green4]{3} \times \left(\sqrt[\green4]{3}\right)^{\green4-1}}=\frac{\left(\sqrt[\green4]{3}\right)^3}{3}\]*We make sure the index of the higher root is as small as possible.*

We do this by checking if the number inside the radical sign can be written as a power with an exponent that is a divisor of the index of the root.

\[\sqrt[\green8]{324}=\sqrt[\green8]{18^2}=\sqrt[4]{18}\]

**Examples**

\[\begin{array}{rcl}\sqrt[\green4]{96}&=&\sqrt[\green4]{2^\green4 \times 6} \\ &=& \sqrt[\green4]{2^\green4} \times \sqrt[\green4]{6} \\ &=& 2 \sqrt[\green4]{6} \\ \\\sqrt[\green5]{15552}&=&\sqrt[\green5]{2^\green5 \times 3^\green5 \times 2} \\&=& \sqrt[\green5]{2^\green5} \times \sqrt[\green5]{3^\green5} \times \sqrt[\green5]{2} \\ &=& 2 \times 3 \times \sqrt[\green5]{2} \\ &=& 6 \sqrt[\green5]{2} \\ \\ \dfrac{2}{\sqrt[\green3]{5}}&=& \dfrac{2 \left(\sqrt[\green3]{5}\right)^2}{\sqrt[\green3]{5} \times \left(\sqrt[\green3]{5}\right)^2}\\ &=& \dfrac{2 \left(\sqrt[\green3]{5}\right)^2}{5} \\ \\ \sqrt[\green6]{225}&=& \sqrt[\green6]{15^2}\\&=&\sqrt[3]{15}\end{array}\]

#\begin{array}{rcl}

\sqrt[3]{256608}&=& \sqrt[3]{{2}^{5}\times {3}^{6}\times {11}} \\ &&\phantom{xxx}\blue{\text{number inside the radical symbol written in prime factorization}} \\

&=& \sqrt[3]{2^{3} \times {2}^{2}\times \left({3}^{2}\right)^{3} \times {11}} \\ &&\phantom{xxx}\blue{\text{powers of }3\text{ isolated using the calculation rules for powers}}\\

&=& \sqrt[3]{2^{3} } \times \sqrt[3]{\left({3}^{2}\right)^{3}} \times \sqrt[3]{ {{2}^{2}\times } {} 11 } \\ &&\phantom{xxx}\blue{\text{calculation rule: higher root of a product equals the product of higher roots}} \\

&=& 2\times {3}^{2}\times \sqrt[3]{ {{2}^{2}\times } {} 11 } \\ &&\phantom{xxx}\blue{\text{eliminated root(s)}} \\

&=& 18 \sqrt[3]{44}\\ &&\phantom{xxx}\blue{\text{calculated}}

\end{array}#

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