When working in square roots, each component can be treated separately.

$\displaystyle \sqrt{50x^3y^4}=\sqrt{50}\sqrt{x^3}\sqrt{y^4}$

Now, we can simplify each term.

$\displaystyle \sqrt{50}=5\sqrt{2}$

$\displaystyle \sqrt{x^3}=x\sqrt{x}$

$\displaystyle \sqrt{y^4}=y^2$

Combine the simplified terms to find the answer. Anything outside of the square root is combined, while anything under the root is combined under the root.

$\displaystyle (5\sqrt{2})(x\sqrt{x})(y^2)=5xy^2\sqrt{2x}$

Remember that any term outside the radical will be in the denominator of the exponent.

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

Since $\displaystyle 13$ does not have any roots, we are simply raising it to the one-fourth power.

$\displaystyle \sqrt[4]{13}=13^{\frac{1}{4}}$

An exponent written as a fraction can be rewritten using roots.  $\displaystyle 9^ \frac{5}{2}$ can be reqritten as $\displaystyle \sqrt[2]{9^5}$.  The bottom number on the fraction becomes the root, and the top becomes the exponent you raise the number to. $\displaystyle \sqrt[2]{9^5}$ is the same as $\displaystyle (\sqrt[2]{9})^{5}=3^{5}$. This will give us the answer of 243.

To convert the radical to exponent form, begin by converting the integer:

$\displaystyle \sqrt{8x^3y^4}$

$\displaystyle =\sqrt{2^3x^3y^4}$ 

Now, divide each exponent by $\displaystyle 2$ to clear the square root:

$\displaystyle =2^{(\frac{3}{2})}x^{(\frac{3}{2})}y^{(\frac{4}{2})}$

Finally, simplify the exponents:

$\displaystyle =2^{(\frac{3}{2})}x^{(\frac{3}{2})}y^2$

To convert the radical to exponent form, begin by converting the integer:

$\displaystyle \sqrt[4]{96a^5b^7c^8}$

$\displaystyle =\sqrt[4]{3\cdot 2^5a^5b^7c^8}$

Now, divide each exponent by $\displaystyle 4$ to cancel the radical:

$\displaystyle =3^{\frac{1}{4}}2^{\frac{5}{4}}a^{\frac{5}{4}}b^{\frac{7}{4}}c^{\frac{8}{4}}$

Finally, simplify the exponents:

$\displaystyle =3^{\frac{1}{4}}2^{\frac{5}{4}}a^{\frac{5}{4}}b^{\frac{7}{4}}c^2$

Multiply the numerator and denominator by the compliment of the denominator:

$\displaystyle =\frac{x^{\frac{5}{2}}}{x^{\frac{1}{2}}+y^{\frac{1}{2}}} \cdot \frac{x^{\frac{1}{2}}-y^{\frac{1}{2}}}{x^{\frac{1}{2}}-y^{\frac{1}{2}}}$

Simplify the expression:

$\displaystyle =\frac{x^3-x^{\frac{5}{2}}y^\frac{1}{2}}{x-y}$

Multiply the numerator and denominator by the compliment of the denominator:

$\displaystyle =\frac{x^{\frac{2}{3}}}{x^{\frac{2}{3}}-x^{\frac{-1}{3}}} \cdot \frac{x^{\frac{1}{3}}}{x^{\frac{1}{3}}}$

Simplify the expression:

$\displaystyle =\frac{x}{x-1}$

Multiply the numerator and denominator by the compliment of the denominator:

$\displaystyle =\frac{x^{\frac{1}{2}}-y^{\frac{1}{2}}}{x^{\frac{1}{2}}+y^{\frac{1}{2}}}$ $\displaystyle \cdot \frac{x^{\frac{1}{2}}-y^{\frac{1}{2}}}{x^{\frac{1}{2}}-y^{\frac{1}{2}}}$

Simplify the expression:

$\displaystyle =\frac{x-2x^{\frac{1}{2}}y^{\frac{1}{2}}+y}{x-y}$

Multiply the numerator and denominator to the exponent:

$\displaystyle (\frac{a^{\frac{-2}{3}}}{2^{\frac{1}{2}}a^2})^{\frac{-1}{2}}$

$\displaystyle =\frac{a^{\frac{1}{3}}}{2^{\frac{-1}{4}}a^{-1}}$

Simplify the expression by combining like terms:

$\displaystyle =2^{\frac{1}{4}}a^{\frac{4}{3}}$

To convert the radical to exponent form, begin by converting the integer:

$\displaystyle \sqrt[3]{16x^5y^6z^8}$

$\displaystyle =\sqrt[3]{2^4x^5y^6z^8}$

Now, divide each exponent by $\displaystyle 3$ to remove the radical:

$\displaystyle =2^{\frac{4}{3}}x^{\frac{5}{3}}y^{\frac{6}{3}}z^{\frac{8}{3}}$

Finally, simplify the exponents:

$\displaystyle =2^{\frac{4}{3}}x^{\frac{5}{3}}y^2z^{\frac{8}{3}}$