Answer :
To determine which of the given options are equivalent to the expression [tex]\(x^{815}\)[/tex], we need to simplify each expression and compare it to [tex]\(x^{815}\)[/tex].
### Option A: [tex]\(\left(x^5\right)^{1 / 8}\)[/tex]
[tex]\[ \left(x^5\right)^{1 / 8} = x^{5 \cdot \frac{1}{8}} = x^{\frac{5}{8}} \][/tex]
Clearly, [tex]\(x^{\frac{5}{8}} \neq x^{815}\)[/tex].
### Option B: [tex]\(\left(x^8\right)^{1/5}\)[/tex]
[tex]\[ \left(x^8\right)^{1/5} = x^{8 \cdot \frac{1}{5}} = x^{\frac{8}{5}} \][/tex]
Clearly, [tex]\(x^{\frac{8}{5}} \neq x^{815}\)[/tex].
### Option C: [tex]\(\sqrt[5]{x^8}\)[/tex]
[tex]\[ \sqrt[5]{x^8} = x^{\frac{8}{5}} \][/tex]
Clearly, [tex]\(x^{\frac{8}{5}} \neq x^{815}\)[/tex].
### Option D: [tex]\((\sqrt[5]{x})^8\)[/tex]
[tex]\[ (\sqrt[5]{x})^8 = (x^{\frac{1}{5}})^8 = x^{\frac{1}{5} \cdot 8} = x^{\frac{8}{5}} \][/tex]
Clearly, [tex]\(x^{\frac{8}{5}} \neq x^{815}\)[/tex].
### Option E: [tex]\((\sqrt[8]{x})^6\)[/tex]
[tex]\[ (\sqrt[8]{x})^6 = (x^{\frac{1}{8}})^6 = x^{\frac{1}{8} \cdot 6} = x^{\frac{6}{8}} = x^{\frac{3}{4}} \][/tex]
Clearly, [tex]\(x^{\frac{3}{4}} \neq x^{815}\)[/tex].
### Option F: [tex]\(\sqrt[8]{x^5}\)[/tex]
[tex]\[ \sqrt[8]{x^5} = x^{\frac{5}{8}} \][/tex]
Clearly, [tex]\(x^{\frac{5}{8}} \neq x^{815}\)[/tex].
After evaluating all options, none of the choices are equivalent to [tex]\(x^{815}\)[/tex].
Therefore, the answer is:
[tex]\[ \boxed{\text{None of the above}} \][/tex]
### Option A: [tex]\(\left(x^5\right)^{1 / 8}\)[/tex]
[tex]\[ \left(x^5\right)^{1 / 8} = x^{5 \cdot \frac{1}{8}} = x^{\frac{5}{8}} \][/tex]
Clearly, [tex]\(x^{\frac{5}{8}} \neq x^{815}\)[/tex].
### Option B: [tex]\(\left(x^8\right)^{1/5}\)[/tex]
[tex]\[ \left(x^8\right)^{1/5} = x^{8 \cdot \frac{1}{5}} = x^{\frac{8}{5}} \][/tex]
Clearly, [tex]\(x^{\frac{8}{5}} \neq x^{815}\)[/tex].
### Option C: [tex]\(\sqrt[5]{x^8}\)[/tex]
[tex]\[ \sqrt[5]{x^8} = x^{\frac{8}{5}} \][/tex]
Clearly, [tex]\(x^{\frac{8}{5}} \neq x^{815}\)[/tex].
### Option D: [tex]\((\sqrt[5]{x})^8\)[/tex]
[tex]\[ (\sqrt[5]{x})^8 = (x^{\frac{1}{5}})^8 = x^{\frac{1}{5} \cdot 8} = x^{\frac{8}{5}} \][/tex]
Clearly, [tex]\(x^{\frac{8}{5}} \neq x^{815}\)[/tex].
### Option E: [tex]\((\sqrt[8]{x})^6\)[/tex]
[tex]\[ (\sqrt[8]{x})^6 = (x^{\frac{1}{8}})^6 = x^{\frac{1}{8} \cdot 6} = x^{\frac{6}{8}} = x^{\frac{3}{4}} \][/tex]
Clearly, [tex]\(x^{\frac{3}{4}} \neq x^{815}\)[/tex].
### Option F: [tex]\(\sqrt[8]{x^5}\)[/tex]
[tex]\[ \sqrt[8]{x^5} = x^{\frac{5}{8}} \][/tex]
Clearly, [tex]\(x^{\frac{5}{8}} \neq x^{815}\)[/tex].
After evaluating all options, none of the choices are equivalent to [tex]\(x^{815}\)[/tex].
Therefore, the answer is:
[tex]\[ \boxed{\text{None of the above}} \][/tex]
