1. [latex]{x}^{\frac{1}{7}}[/latex], the numerator is [latex]1[/latex] and the denominator is [latex]7[/latex], therefore we will have the seventh root of [latex]x[/latex], [latex]\sqrt[7]{x}.[/latex]
2. [latex]{5}^{\frac{4}{7}}[/latex], the numerator is [latex]4[/latex] and the denominator is [latex]7[/latex], so we will have the seventh root of [latex]5[/latex] raised to the fourth power. [latex]\sqrt[7]{5^4}.[/latex]
3. [latex]{9}^{\frac{3}{2}}[/latex], the numerator is [latex]3[/latex] and the denominator is [latex]2[/latex], so we will have the square root of [latex]9[/latex] raised to the fourth power, [latex]\sqrt{9^3}.[/latex] Note that this is still tricky to evaluate without a calculator, so it is easier in this case to use the alternative form of the radical with the power on the outside, [latex]\left(\sqrt{9}\right)^3.[/latex] Now simplify, [latex]\left(\sqrt{9}\right)^3=3^3=27.[/latex]
4. Rewrite the expression with the fractional exponent as a radical. The denominator of the fraction determines the root, in this case the cube root.

   [latex]\sqrt[3]{2x}[/latex]

   The parentheses in [latex]{{\left( 2x \right)}^{\frac{1}{3}}}[/latex] indicate that the exponent refers to everything within the parentheses.

5. Rewrite the expression with the fractional exponent as a radical. The denominator of the fraction determines the root, in this case the cube root.

   [latex]2\sqrt[3]{x}[/latex]

   The exponent refers only to the part of the expression immediately to the left of the exponent, in this case [latex]x[/latex], but not the [latex]2[/latex].

1. The fourth root can be rewritten as the exponent [latex]\dfrac{1}{4}[/latex]. Remove the radical and place the exponent next to the base. 

   [latex]\sqrt[4]{81}=81^{1/4}[/latex]

2. Rewrite the radical using a rational exponent. The index determines the denominator. In this case, the index of the radical is [latex]3[/latex], so the rational exponent will be [latex]\dfrac{1}{3}[/latex].

   [latex]4\sqrt[3]{xy}=4{{(xy)}^{1/3}}[/latex]

   Since [latex]4[/latex] is outside the radical, it is not included in the grouping symbol and the exponent does not refer to it.

3. [latex]\sqrt[n]{a^{m}}[/latex] can be rewritten as [latex]a^{\frac{m}{n}}[/latex], so in this case [latex]n=6,\text{ and }m=3[/latex], therefore

   [latex]\sqrt[6]{{{a}^{3}}}={{a}^{3/6}}={{a}^{1/2}}[/latex]

   Note that this final answer is equivalent to [latex]\sqrt{a}.[/latex] This process is sometimes called reducing the index.

4. [latex]\sqrt[n]{a^{m}}[/latex] can be rewritten as [latex]a^{\frac{m}{n}}[/latex], so in this case [latex]n=12,\text{ and }m=3[/latex], therefore

   [latex]\sqrt[12]{16^3}={16}^{3/12}={16}^{1/4}[/latex]

   This is equivalent to [latex]\sqrt[4]{16},[/latex] which evaluates as [latex]2.[/latex]

1. There are many ways to approach problems of simplifying exponents. Each approach is correct as long as you apply valid exponent rules at each step. In this example, we begin with using quotient to a power:

[latex]\begin{align}\left(\frac{x^{2/5}}{y^{4/3}}\right)^{-1/3}=&\quad\frac{(x^{2/5})^{-1/3}}{(y^{4/3})^{-1/3}}&&\color{blue}{\textsf{Power of a Quotient}}\\=&\quad\frac{x^{\frac{2}{5}\cdot -\frac{1}{3}}}{y^{\frac{4}{3}\cdot-\frac{1}{3}}}&&\color{blue}{\textsf{Power Rule}}\\=&\quad\frac{x^{-2/15}}{y^{-4/9}}\\=&\quad\frac{y^{4/9}}{x^{2/15}}\end{align}[/latex]

In the last step we handled the negative exponents by switching them to the opposite side of the fraction bar.

2. Again, there are different ways we can simplify the expression. We start this example by applying the power in the numerator.

[latex]\begin{align} \frac{x^{1/5}y^{1/2}(x^{1/2}y^{-1})^{-7/5}}{x^{3/2}y^{8/5}}=&\quad\frac{x^{1/5}y^{1/2}(x^{1/2}y^{-1})^{-7/5}}{x^{3/2}y^{8/5}}\\ =&\quad\frac{x^{1/5}y^{1/2}(x^{1/2})^{-7/5}(y^{-1})^{-7/5}}{x^{3/2}y^{8/5}}&&\color{blue}{\textsf{Power of a Product}}\\ =&\quad\frac{x^{1/5}y^{1/2}x^{-7/10}y^{7/5}}{x^{3/2}y^{8/5}}&&\color{blue}{\textsf{Power Rule}}\\ =&\quad\frac{x^{\frac{1}{5}-\frac{7}{10}}y^{\frac{1}{2}+\frac{7}{5}}}{x^{3/2}y^{8/5}}&&\color{blue}{\textsf{Product Rule}}\\ =&\quad\frac{x^{-5/10}y^{\frac{19}{10}}}{x^{3/2}y^{8/5}}\\ =&\quad x^{-\frac{1}{2}-\frac{3}{2}}y^{\frac{19}{10}-\frac{8}{5}}&&\color{blue}{\textsf{Quotient Rule}}\\ =&\quad x^{-\frac{4}{2}}y^{\frac{3}{10}}\\ =&\quad \frac{y^{\frac{3}{10}}}{x^2}&&\color{blue}{\textsf{Negative Exponent Rule}}\\ \end{align}[/latex]

We first convert both radicals to exponent notation, and then use properties of exponents to simplify.

[latex]\sqrt[3]{\sqrt[4]{x}}=\sqrt[3]{x^{1/4}}=\left(x^{1/4}\right)^{1/3}=x^{\frac{1}{4}\cdot\frac{1}{3}}=x^{1/12}[/latex]

Now, convert back to radical notation, [latex]\sqrt[12]{x}.[/latex]