The initial expression is
[tex]\sqrt[]{\frac{16x^4y^8}{5z}}[/tex]
This can be simplified as shown below
[tex]\begin{gathered} \sqrt[]{\frac{16x^4y^8}{5z}}=\frac{\sqrt[]{16x^4y^8}}{\sqrt[]{5z}} \\ =\frac{\sqrt[]{16}\sqrt[]{x^4}\sqrt[]{y^8_{}}}{_{}\sqrt[]{5}\sqrt[]{z}} \\ =\frac{4x^2y^4}{\sqrt[]{5}\sqrt[]{z}} \end{gathered}[/tex]
Now, we need to rationalize the denominator. First, we need to multiply the simplified expression by 1
[tex]\begin{gathered} \Rightarrow\sqrt[]{\frac{16x^4y^8}{5z}}=\frac{4x^2y^4_{}}{\sqrt[]{5z}}\cdot1=\frac{4x^2y^4_{}}{\sqrt[]{5z}}\cdot\frac{\sqrt[]{5z}}{\sqrt[]{5z}}=\frac{4x^2y^4\sqrt[]{5z}}{5z} \\ \Rightarrow\sqrt[]{\frac{16x^4y^8}{5z}}=\frac{4x^2y^4\sqrt[]{5z}}{5z} \end{gathered}[/tex]
We can modify how the final expression looks like, but this is the answer.
The answer is (4x^2y^4sqrt(5z))/5z
