When dealing with fractional exponents, we rewrite as such: $\displaystyle x^{\frac{a}{b}}=\sqrt[b]{x^a}$ which $\displaystyle b$ is the index of the radical and $\displaystyle a$ is the exponent raising base $\displaystyle x$.

$\displaystyle 9^{\frac{1}{9}}=\sqrt[9]{9}$

When dealing with fractional exponents, we rewrite as such: $\displaystyle x^{\frac{a}{b}}=\sqrt[b]{x^a}$ which $\displaystyle b$ is the index of the radical and $\displaystyle a$ is the exponent raising base $\displaystyle x$. When dealing with negative exponents, we convert to fractions as such: $\displaystyle x^{-a}=\frac{1}{x^a}$ which $\displaystyle a$ is the positive exponent raising base $\displaystyle x$.

$\displaystyle 16^{-\frac{2}{3}}=\frac{1}{\sqrt[3]{16^2}}=\frac{1}{\sqrt[3]{256}}$

When dealing with fractional exponents, we rewrite as such: $\displaystyle x^{\frac{a}{b}}=\sqrt[b]{x^a}$ which $\displaystyle b$ is the index of the radical and $\displaystyle a$ is the exponent raising base $\displaystyle x$. When dealing with negative exponents, we convert to fractions as such: $\displaystyle x^{-a}=\frac{1}{x^a}$ which $\displaystyle a$ is the positive exponent raising base $\displaystyle x$.

$\displaystyle 20^{-\frac{2}{5}}=\frac{1}{\sqrt[5]{20^2}}=\frac{1}{\sqrt[5]{400}}$

When dealing with fractional exponents, we rewrite as such: $\displaystyle x^{\frac{a}{b}}=\sqrt[b]{x^a}$ which $\displaystyle b$ is the index of the radical and $\displaystyle a$ is the exponent raising base $\displaystyle x$.

$\displaystyle \frac{1}{4}^{\frac{1}{2}}=\sqrt{\frac{1}{4}}=\frac{1}{2}$

When dealing with fractional exponents, remember that the numerator of the fraction represents the power to which we are taking the term that has the exponent, and the denominator represents the degree of the root we are taking of that term.

For our expression, the numerator is 1, which means we raise a to the first power. The denominator is 4, which means we are taking the fourth root of the term:

$\displaystyle \sqrt[4]{(3x^4y^8)^1}$

We can only move the cubes out of the radical, and when we do so, we get

$\displaystyle xy^2\sqrt[4]{3}$

When dealing with fractional exponents, we rewrite as such: 

$\displaystyle x^\frac{a}{b}=\sqrt[b]{x^a}$ 

in which $\displaystyle b$ is the index of the radical and $\displaystyle a$ is the exponent raising base $\displaystyle a$.

$\displaystyle 17^\frac{1}{2}=\sqrt{17}$

When dealing with fractional exponents, we rewrite as such: 

$\displaystyle x^\frac{a}{b}=\sqrt[b]{x^a}$ 

in which $\displaystyle b$ is the index of the radical and $\displaystyle a$ is the exponent raising base $\displaystyle a$.

$\displaystyle 18^\frac{2}{3}=\sqrt[3]{18^2}=\sqrt[3]{324}$

When dealing with fractional exponents, we rewrite as such: 

$\displaystyle x^\frac{a}{b}=\sqrt[b]{x^a}$ 

in which $\displaystyle b$ is the index of the radical and $\displaystyle a$ is the exponent raising base $\displaystyle a$.

$\displaystyle 24^\frac{2}{5}=\sqrt[5]{24^2}=\sqrt[5]{576}=2\sqrt[5]{18}$ 

We were able to simplify it by factoring out perfect fifth root.

In this case, it was $\displaystyle 32 =2^5$.

When dealing with fractional exponents, we rewrite as such: 

$\displaystyle x^\frac{a}{b}=\sqrt[b]{x^a}$ 

in which $\displaystyle b$ is the index of the radical and $\displaystyle a$ is the exponent raising base $\displaystyle a$.

$\displaystyle \frac{1}{2}^\frac{1}{2}=\sqrt{\frac{1}{2}}=\frac{1}{\sqrt{2}}=\frac{\sqrt{2}}{2}$ 

Remember when getting rid of radicals, we just multiply top and bottom by the radical.

To simplify fractional exponents, remember that the numerator of the fraction corresponds to the power the term is taken, and the denominator indicates what root we are taking of that term (i.e. 2 means square root, 3 means cube root and so on).

Doing this, we get

$\displaystyle \sqrt[3]{(9x)^2}+\sqrt{(10x)^3}$

Expanding the inside of the roots, we get

$\displaystyle \sqrt[3]{81x^2}+\sqrt{1000x^3}$

We can now factor the terms on the inside, pulling out any cubes or squares, respectively:

$\displaystyle \sqrt[3]{27\cdot 3\cdot x^2} +\sqrt{100\cdot 10\cdot x^2\cdot x}=3\sqrt[3]{3x^2}+10x\sqrt{10x}$