Why does $x^{\frac{1}{a}} = \sqrt[a]{x}$? I was attempting to answer this question, and came across the fact that $x$ to a non integer power, $x^\frac{1}{b}$ is equal to $\sqrt[b]{x}$. Why does this work?

Similarly, does $x^{\frac{a}{b}} = a\sqrt[b]{x}$  ?
It's because however we define fractional exponents, we want to make sure they behave according to the same algebraic rules as integer exponents.

For example, for integer exponents we have the property
$$(x^a)^b = x^{ab}$$
and we want to extend that property to make sure it also holds when one of the exponents is a fraction.  That means we want
$$(x^{1/a})^b = x^{a/b}$$
for all $a,b$ (with $a \ne 0$).  In particular, we need
$$(x^{1/b})^b = x^{b/b} = x^1 = x$$
which means (by definition) that $x^{1/b}$ is a $b$th root of $x$, i.e. that $x^{1/b} = \sqrt[b]{x}$.
It basically is a definition;

Consider $\sqrt{2}$, this can be written as $2^{\frac{1}{2}}$ which is same as $\sqrt[2]{2}$