### Numbers: Powers and roots

### Rules of calculation for higher roots

For higher roots, we have similar rules of calculation as for square roots.

Powers and roots

When we calculate #\left(\sqrt[\green3]{\blue8}\right)^\green3#, we get:

\[\left(\sqrt[\green3]{\blue8}\right)^\green3=2^\green3=\blue8\]

Similarly, we can also do this for roots with different indices. In general we can state:

*A higher root raised to the index of the root is equal to the number inside the radical sign.*

Similar to square roots, it also holds that:

\[\begin{array}{rcrrcr}\sqrt[\green3]{\blue8^\green3}&=& \sqrt[\green3]{512}&=&\blue8\\\sqrt[\green3]{(\blue{-8})^\green3}&=&\sqrt[\green3]{-512}&=&\blue{-8}\\\sqrt[\green4]{\blue2^\green4}&=&\sqrt[\green4]{16}&=&\blue2\\\sqrt[\green4]{(\blue{-2})^\green4}&=&\sqrt[\green4]{16}&=&\blue2\end{array}\]

In general we can state:

*For odd indices: the higher root of a number that is raised to the index of the root is equal to the number that is exponentiated.*

*For even indices: the higher root of a number that is raised to the index of the root is equal to the absolute value of the number that is exponentiated.*

**Examples**

\[\begin{array}{rclrcl}\left(\sqrt[\green4]{\blue2}\right)^\green4&=&\blue2 \\ \\ \left(\sqrt[\green5]{\blue{20}}\right)^\green5&=&\blue{20} \\\\ \left(\sqrt[\green5]{\blue{-3}}\right)^\green5&=&\blue{-3}\\\\\\ \\ \sqrt[\green5]{\blue2^\green5}&=&\blue2 \\ \\ \sqrt[\green4]{\blue{20}^\green4}&=&\abs{\blue{20}}&=&\blue{20} \\ \\ \sqrt[\green3]{\left(\blue{-4}\right)^\green3}&=&\blue{-4} \\ \\ \sqrt[\green4]{\left(\blue{-5}\right)^\green4}&=&\abs{\blue{-5}}&=&\blue{5} \end{array}\]

Products of roots

When we multiply #\sqrt[\green3]{\blue8}# by #\sqrt[\green3]{\orange{64}}#, we get:

\[\sqrt[\green3]{\blue8} \times \sqrt[\green3]{\orange{64}}=2 \times 4=8=\sqrt[\green3]{512}=\sqrt[\green3]{\blue8 \times \orange{64}}\]

The following generally applies:

*The product of two higher roots with the same index equals the higher root of the product of the numbers inside the radical symbols.*

We can use rule the other way around as well:

*The higher root of a product equals the product of the higher roots with the same index.*

**Examples**

\[\begin{array}{rcl}\sqrt[\green4]{\blue3}\times \sqrt[\green4]{\orange{27}}&=&\sqrt[\green4]{\blue3 \times \orange{27}} \\ &=& \sqrt[\green4]{81} \\ &=& 3 \\ \\ \sqrt[\green3]{32}&=&\sqrt[\green3]{\blue8 \times \orange4} \\&=& \sqrt[\green3]{\blue8} \times \sqrt[\green3]{\orange4}\\&=&2\sqrt[\green3]{\orange4} \end{array}\]

Roots of fractions

If we want to take the higher root of a fraction, we should look for a number that is equal to this fraction, when exponentiated. For #\sqrt[\green3]{\orange{\frac{8}{27}}}# we try to find a number that, when raised to the power of #\green3#, equals #\orange{\frac{8}{27}}#. This is #\blue{\frac{2}{3}}# because \[\left(\blue{\frac{2}{3}}\right)^\green3=\frac{\blue2^\green3}{\blue3^\green3}=\orange{\frac{8}{27}}\]

So, we see:

\[\sqrt[\green3]{\orange{\frac{8}{27}}}=\frac{\sqrt[\green3]{\orange8}}{\sqrt[\green3]{\orange{27}}}=\frac{\blue2}{\blue3}\]

In general we can state:

*The higher root of a fraction equals the higher root with the same index of the numerator divided by the higher root with the same index of the denominator.*

**Examples**

\[\begin{array}{rcl}\displaystyle \sqrt[\green4]{\orange{\frac{1}{16}}}&=&\displaystyle \frac{\sqrt[\green4]{\orange1}}{\sqrt[\green4]{\orange{16}}} \\ &=& \displaystyle \blue{\frac{1}{2}} \\ \\ \displaystyle \sqrt[\green5]{\orange{\frac{3}{32}}}&=&\displaystyle \frac{\sqrt[\green5]{\orange3}}{\sqrt[\green5]{\orange{32}}} \\ &=& \displaystyle \frac{\blue{\sqrt[\green5]{3}}}{\blue2} \\ \\ \displaystyle \sqrt[\green3]{\orange{\frac{2}{5}}}&=&\displaystyle \frac{\blue{\sqrt[\green3]{2}}}{\blue{\sqrt[\green3]{5}}} \end{array}\]

The calculation rule states that a higher root raised to the index of the root is equal to the number inside the radical symbol. This means that:

\[\left(\sqrt[4]{75}\right)^4=75\]

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