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 the 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

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

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

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