We need to write as an exponential in fractional form the expression:
[tex]\text{ The eighth root of fifty-seven to the sixth degree}[/tex]The "eighth root" is represented by the symbol:
[tex]\sqrt[8]{\ldots}[/tex]And the exponential "fifty-seven to the sixth degree" is:
[tex]57^6[/tex]So, the whole expression is written as:
[tex]\sqrt[8]{57^6}[/tex]Now, we need to use the following properties of exponentials:
[tex]\begin{gathered} \sqrt[n]{x}=x^{\frac{1}{n}} \\ \\ (y^a)^b=y^{a\cdot b} \end{gathered}[/tex]Then, using those properties, we obtain:
[tex]\sqrt[8]{57^6}=(57^6)^{\frac{1}{8}}=57^{6\cdot\frac{1}{8}}[/tex]Notice that:
[tex]6\cdot\frac{1}{8}=\frac{6}{8}=\frac{6\div2}{8\div2}=\frac{3}{4}[/tex]Thus, we obtain:
[tex]\sqrt[8]{57^6}=57^{\frac{3}{4}}[/tex]