Basic radicals and exponents

Roots and powers are two ways of writing the same thing. Taking a root is the same as raising to a fractional power. The $n$th root of $x$ is $x^{1/n}$.

So the square root is a power of one half, $\sqrt{x} = x^{1/2}$, and the cube root is a power of one third, $\sqrt[3]{x} = x^{1/3}$.

When there is a power inside the root, the two combine into a single fraction. The $n$th root of $x^m$ is $x^{m/n}$.
The power $m$ becomes the top of the fraction, and the root $n$ becomes the bottom.
For example $\sqrt{x^3} = x^{3/2}$ and $\sqrt[3]{x^5} = x^{5/3}$.

A number multiplied in front simply stays in front: $5\sqrt{6^3 x^4} = 5(6^3 x^4)^{1/2}$. Rewriting roots as fractional exponents like this lets you use the ordinary exponent rules to simplify or combine them.

Write $\sqrt{x}$ and $\sqrt[3]{x}$ as powers.
The square root is the $2$nd root, so $\sqrt{x} = x^{1/2}$.
The cube root is the $3$rd root, so $\sqrt[3]{x} = x^{1/3}$.

Write $\sqrt{x^3}$ as a power of $x$.
The square root is the $2$nd root, and the power inside is $3$.
Power on top, root on the bottom: $\sqrt{x^3} = x^{3/2}$.

Write $\sqrt[3]{x^5}$ as a power of $x$.
The root is $3$ and the power inside is $5$.
So $\sqrt[3]{x^5} = x^{5/3}$.

Rewrite $3\sqrt{a} = 2\sqrt[3]{b}$ using fractional exponents.
$\sqrt{a} = a^{1/2}$ and $\sqrt[3]{b} = b^{1/3}$.
So the equation becomes $3a^{1/2} = 2b^{1/3}$.