We are given the expression:
[tex]\sqrt[]{\frac{ \sqrt[3]{64}+\sqrt[4]{256}}{\sqrt[]{64}+\sqrt[]{256}}}[/tex]
We will simplify this as shown below:
[tex]\begin{gathered} \sqrt[]{\frac{ \sqrt[3]{64}+\sqrt[4]{256}}{\sqrt[]{64}+\sqrt[]{256}}} \\ \text{Let's consider \& solve the terms one after the order, we have:} \\ \sqrt[3]{64}\Rightarrow\sqrt[3]{4\times4\times4}\Rightarrow4 \\ \sqrt[4]{256}\Rightarrow\sqrt[4]{4\times4\times4\times4}\Rightarrow4 \\ \sqrt[]{64}\Rightarrow\sqrt[]{8\times8}\Rightarrow8 \\ \sqrt[]{256}\Rightarrow\sqrt[]{16\times16}\Rightarrow16 \\ We\text{ will substitute each of these expressions back into the parent expression, we have:} \\ \sqrt[]{\frac{4+4}{8+16}} \\ =\sqrt[]{\frac{8}{24}} \\ =\sqrt[]{\frac{1}{3}} \\ =\frac{\sqrt[]{3}}{\sqrt[]{3}\times\sqrt[]{3}} \\ =\frac{\sqrt[]{3}}{3} \\ \Rightarrow\sqrt[]{\frac{\sqrt[3]{64}+\sqrt[4]{256}}{\sqrt[]{64}+\sqrt[]{256}}}=\frac{\sqrt[]{3}}{3} \\ \\ \therefore\frac{\sqrt[]{3}}{3} \end{gathered}[/tex]
