Let's solve this problem assuming the original expression is:
[tex](\sqrt[6]{x^5})^7[/tex]So, we need to use some properties of exponentials:
[tex]\begin{gathered} \sqrt[n]{y}^{}=y^{\frac{1}{n}} \\ \\ (y^a)^b=y^{a\cdot b} \end{gathered}[/tex]So, let's use:
[tex]\begin{gathered} y=x^5 \\ \\ n=6 \end{gathered}[/tex]We have:
[tex]\sqrt[6]{x^5}=(x^5)^{\frac{1}{6}}=x^{5\cdot\frac{1}{6}}=x^{\frac{5}{6}}[/tex]Now, using the second property again, we obtain:
[tex](\sqrt[6]{x^5})^7=(x^{\frac{5}{6}})^7=x^{\frac{5}{6}\cdot7}=x^{\frac{35}{6}}[/tex]Therefore, the expression with a rational exponent is:
[tex]x^{\frac{35}{6}}[/tex]