To work with expressions more comfortably there are processes that simplify the roots. Let's see what they are:

If we multiply (amplify) or divide (simplify) the index and the exponent of a radical by the same nonempty number, the radical obtained is equivalent to the first one.

That is, the radicals are equivalent because the exponents of the associated powers are equivalent fractions.

$$\displaystyle \sqrt[3]{4^2}= 4^{\frac{2}{3}}=4^{\frac{2}{3}\cdot \frac{2}{2}}=\sqrt[6]{4^4}$$

It is equivalent to divide the exponent of a power by the index of the root and to have the root as this number.

If a factor of the radicand has an exponent that is not a multiple of the index of the root, the factor can be separated in such a way that the exponent is divisible by the index.

For instance:

$$\displaystyle \sqrt{3^7}=\sqrt{3^6\cdot 3}=\sqrt{3^6}\cdot \sqrt{3}=3^{\frac{6}{2}}\cdot 3^{\frac{1}{2}}=3^3\cdot \sqrt{3}=27\sqrt{3}$$

Some radicals can turn into an equivalent form which is easier to deal with. A radical is in its simplest form when there is not any factor that can be extracted, when there is no fraction under the radical sign and when the index of the root cannot be reduced.

It is possible to extract a factor of the radical if this appears the same number of times as the index of the root.

The examples that follows show this:

$$\displaystyle \begin{array}{rcl} \sqrt{28}&=&\sqrt{2^2\cdot 7}=2^{\frac{2}{2}}\sqrt{7}=2\sqrt{7} \\ \sqrt[5]{160}&=&\sqrt[5]{2^5\cdot 5}=2^{\frac{5}{5}}\sqrt[5]{5}=2\sqrt{5}\end{array}$$

To be able to simplify radicals easily, it is convenient to know the squares of the integers up to $$25$$ and some of the smallest powers of the numbers $$2$$, $$3$$, $$4$$ and $$5$$. The following tables are very usefull.

$$\begin{array}{rclrclrcl} 1^2&=&1&11^2&=&121&21^2&=&441 \\\\ 2^2&=&4&12^2&=&144&22^2&=&484\\\\ 3^2&=&9&13^2&=&169&23^2&=&529 \\\\ 4^2&=&16&14^2&=&196&24^2&=&576 \\\\ 5^2&=&25&15^2&=&225&25^2&=&625 \\\\ 6^2&=&36 & 16^2&=& 256 &&&\\\\7^2&=&49& 17^2&=& 289 &&& \\\\ 8^2&=&64& 18^2&=& 324 &&& \\\\ 9^2&=&81& 19^2&=& 361 &&& \\\\ 10^2&=&100& 20^2&=& 400 &&&\end{array}$$$$\begin{array}{rclrclrclrcl} 2^3&=&8 & 3^3&=&27 & 4^3&=&64 & 5^3&=&125 \\\\2^4&=&16 & 3^4&=&81 & 4^4&=&256 & 5^4&=&625 \\\\ 2^5&=&32 & 3^5&=&243 & 4^5&=&1024 & 5^5&=&3125 \end{array}$$