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:

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