Simplify $\sqrt[8]{x^6y^2}$.

$\begin{array}{ll} \text{First rewrite the radical as a fractional exponent}& (x^6y^2)^{\frac{1}{8}} \\ \\ \text{Multiply all exponents}& x^{6\cdot \frac{1}{8}}y^{2\cdot \frac{1}{8}} \\ \\ \text{This yields} & x^{\frac{6}{8}}y^{\frac{2}{8}}\\ \\ \text{Reducing this gives} & x^{\frac{3}{4}}y^{\frac{1}{4}}\\ \\ \text{Rewrite as} & \sqrt[4]{x^3y} \end{array}$

Simplify $\sqrt[24]{a^6b^9c^{15}}$.

For this radical, notice that each exponent has the common factor 3.

The solution is to divide each exponent by 3, which yields $\sqrt[8]{a^2b^3c^5}$.

Simplify $\sqrt[3]{4x^2y}\cdot \sqrt[4]{8xy^3}$.

First, convert each radical to a complete exponential form.

$\begin{array}{ll} \text{This looks like} & (4x^2y)^{\frac{1}{3}}(8xy^3)^{\frac{1}{4}} \\ \\ \text{Multiply all exponents} & 4^{\frac{1}{3}}x^{2\cdot \frac{1}{3}}y^{\frac{1}{3}}8^{\frac{1}{4}}x^{\frac{1}{4}}y^{3\cdot \frac{1}{4}} \\ \\ \text{This yields} & 4^{\frac{1}{3}}x^{\frac{2}{3}}y^{\frac{1}{3}}8^{\frac{1}{4}}x^{\frac{1}{4}}y^{\frac{3}{4}} \\ \\ \text{Combining like variables leaves} & 4^{\frac{1}{3}}8^{\frac{1}{4}}x^{\frac{2}{3}}x^{\frac{1}{4}}y^{\frac{1}{3}}y^{\frac{3}{4}} \\ \\ (\text{Note: }& 4^{\frac{1}{3}}8^{\frac{1}{4}}=2^{2\cdot \frac{1}{3}}2^{3\cdot \frac{1}{4}}=2^{\frac{2}{3}}2^{\frac{3}{4}}) \\ \\ \text{Accounting for this yields} & 2^{\frac{2}{3}}2^{\frac{3}{4}}x^{\frac{2}{3}}x^{\frac{1}{4}}y^{\frac{1}{3}}y^{\frac{3}{4}}\\ \\ \text{Reducing this yields} & 2^{\frac{2}{3}+\frac{3}{4}}x^{\frac{2}{3}+\frac{1}{4}}y^{\frac{1}{3}+\frac{3}{4}} \\ \\ \text{Which further reduces to} & 2^{\frac{17}{12}}x^{\frac{11}{12}}y^{\frac{13}{12}} \\ \\ \text{Reduce this} & 2y\cdot 2^{\frac{5}{12}}x^{\frac{11}{12}}y^{\frac{1}{12}} \\ \\ \text{Convert this back into a radical} & 2y(2^5x^{11}y)^{\frac{1}{12}} \\ \\ \text{Which leaves } & 2y \sqrt[12]{2^5x^{11}y} \end{array}$

Simplify $\sqrt{3x(y+z)}\cdot \sqrt[3]{9x(y+z)^2}$.

First, convert each radical to a complete exponential form.

$\begin{array}{ll} \text{This looks like} & 3^{\frac{1}{2}}x^{\frac{1}{2}}(y+z)^{\frac{1}{2}}9^{\frac{1}{3}}x^{\frac{1}{3}}(y+z)^{\frac{2}{3}} \\ \\ (\text{Note: } & 9^{\frac{1}{3}}=3^{\frac{2}{3}})\\ \\ \text{Combining like variables leaves} & 3^{\frac{1}{2}}3^{\frac{2}{3}}x^{\frac{1}{2}}x^{\frac{1}{3}} (y+z)^{\frac{1}{2}}(y+z)^{\frac{2}{3}}\\ \\ \text{Reducing this yields} & 3^{\frac{1}{2}+\frac{2}{3}}x^{\frac{1}{2}+\frac{1}{3}}(y+z)^{\frac{1}{2}+\frac{2}{3}} \\ \\ \text{Which further reduces to} & 3^{\frac{7}{6}}x^{\frac{5}{6}}(y+z)^{\frac{7}{6}} \\ \\ \text{Reduce this} & 3(y+z)3^{\frac{1}{6}}x^{\frac{5}{6}}(y+z)^{\frac{1}{6}} \\ \\ \text{Convert this back into a radical} & 3(y+z)[3x^5(y+z)]^{\frac{1}{6}} \\ \\ \text{Which leaves} & 3(y+z) \sqrt[6]{3x^5(y+z)} \end{array}$