An expression like $$\displaystyle \sqrt{3}, \sqrt[3]{4}, \sqrt{a+b}, \sqrt[5]{3a-8b}$$ that has a radical sign ($$\displaystyle \sqrt{ }$$), is called a radical.
The "radical" word derives from the Latin word "radix", which means "root". Now we will learn how to deal with expressions that have radical signs.
Till now we were calculating powers of integer exponents. But what happens if the exponent is a fraction?
For example, $$\displaystyle 5^{\frac{2}{3}}$$. In this case, it means that we must do the cubic root of $$5$$ raised to $$2$$. That is, $$\displaystyle 5^{\frac{2}{3}}= \sqrt[3]{5^2}$$.
Namely, managing fractional exponents we will use of the following equality: $$\displaystyle a^{\frac{n}{m}}= \sqrt[m]{a^n}$$
For example:
$$\displaystyle 5^{\frac{23}{6}}=\sqrt[6]{5^23}$$ and $$\displaystyle 3^{\frac{9}{2}}=\sqrt{3^9}$$
The expression $$\displaystyle \sqrt[n]{a}$$ is a radical with index $$n$$: the number $$n$$ is the index of the radical and the number $$a$$ is the radicand. Therefore, a power of the fractional exponent is a radical.
The index of the root (except in case of a square root) is placed in the aperture of the radical symbol. The index says which root we are trying to extract from the radicand.
For instance $$\displaystyle \sqrt[5]{32}$$: the radicand is $$32$$ and the index of the root is $$5$$. The fifth root of $$32$$ is for what we are looking for. When the index is $$2$$ it is not written, but it is understood.
Remember that if it is possible to determine the square root of a number, then it is always possible to determine two of them.
Radicals that have the same index and the same radicand are similar.
Similar radicals can have different coefficients in front of the radical sign.
The common operations with powers continue to apply because we are still working with powers, but now with fractional exponents.
This way, for example,
$$\displaystyle 4^{\frac{4}{2}}\cdot 4^{\frac{6}{2}}=4^{\frac{3+6}{2}}=4^{\frac{9}{2}}=\sqrt{4^9}$$
therefore this way of passing from radicals to powers allows us to manage expressions that contain roots much more easily.
Let's see another example:
$$\displaystyle \sqrt[4]{5^{34}} \cdot \sqrt[2]{5^{12}}=5^{\frac{34}{4}}\cdot 5^{\frac{12}{2}}=5^{\frac{34+24}{4}}=5^{\frac{58}{4}}=5^{\frac{29}{2}}=\sqrt{5^{29}}$$