Answer :
To determine which of the given choices are equivalent to the expression \( x^{3/5} \), we need to simplify and compare each choice to \( x^{3/5} \).
### Given Expression:
[tex]\[ x^{3/5} \][/tex]
### Choices:
#### A. \( \left(x^8\right)^{1/5} \)
Simplify using the power of a power rule, \((a^m)^n = a^{m \cdot n}\):
[tex]\[ \left(x^8\right)^{1/5} = x^{8 \cdot (1/5)} = x^{8/5} \][/tex]
This is not equivalent to \( x^{3/5} \).
#### B. \( \sqrt[8]{x^5} \)
Express in fractional exponents:
[tex]\[ \sqrt[8]{x^5} = x^{5/8} \][/tex]
This is not equivalent to \( x^{3/5} \).
#### C. \( \sqrt[5]{x^8} \)
Express in fractional exponents:
[tex]\[ \sqrt[5]{x^8} = x^{8/5} \][/tex]
This is not equivalent to \( x^{3/5} \).
#### D. \( \left(x^5\right)^{1/8} \)
Simplify using the power of a power rule:
[tex]\[ \left(x^5\right)^{1/8} = x^{5 \cdot (1/8)} = x^{5/8} \][/tex]
This is not equivalent to \( x^{3/5} \).
#### E. \( \left(\sqrt[8]{x}\right)^5 \)
Express in fractional exponents and simplify:
[tex]\[ \left(\sqrt[8]{x}\right)^5 = \left(x^{1/8}\right)^5 = x^{(1/8) \cdot 5} = x^{5/8} \][/tex]
This is not equivalent to \( x^{3/5} \).
#### F. \( \left(\sqrt[5]{x}\right)^8 \)
Express in fractional exponents and simplify:
[tex]\[ \left(\sqrt[5]{x}\right)^8 = \left(x^{1/5}\right)^8 = x^{(1/5) \cdot 8} = x^{8/5} \][/tex]
This is not equivalent to \( x^{3/5} \).
### Conclusion:
None of the given choices \(A, B, C, D, E,\) and \(F\) are equivalent to \( x^{3/5} \).
So, there are no equivalent expressions from the choices provided.
### Given Expression:
[tex]\[ x^{3/5} \][/tex]
### Choices:
#### A. \( \left(x^8\right)^{1/5} \)
Simplify using the power of a power rule, \((a^m)^n = a^{m \cdot n}\):
[tex]\[ \left(x^8\right)^{1/5} = x^{8 \cdot (1/5)} = x^{8/5} \][/tex]
This is not equivalent to \( x^{3/5} \).
#### B. \( \sqrt[8]{x^5} \)
Express in fractional exponents:
[tex]\[ \sqrt[8]{x^5} = x^{5/8} \][/tex]
This is not equivalent to \( x^{3/5} \).
#### C. \( \sqrt[5]{x^8} \)
Express in fractional exponents:
[tex]\[ \sqrt[5]{x^8} = x^{8/5} \][/tex]
This is not equivalent to \( x^{3/5} \).
#### D. \( \left(x^5\right)^{1/8} \)
Simplify using the power of a power rule:
[tex]\[ \left(x^5\right)^{1/8} = x^{5 \cdot (1/8)} = x^{5/8} \][/tex]
This is not equivalent to \( x^{3/5} \).
#### E. \( \left(\sqrt[8]{x}\right)^5 \)
Express in fractional exponents and simplify:
[tex]\[ \left(\sqrt[8]{x}\right)^5 = \left(x^{1/8}\right)^5 = x^{(1/8) \cdot 5} = x^{5/8} \][/tex]
This is not equivalent to \( x^{3/5} \).
#### F. \( \left(\sqrt[5]{x}\right)^8 \)
Express in fractional exponents and simplify:
[tex]\[ \left(\sqrt[5]{x}\right)^8 = \left(x^{1/5}\right)^8 = x^{(1/5) \cdot 8} = x^{8/5} \][/tex]
This is not equivalent to \( x^{3/5} \).
### Conclusion:
None of the given choices \(A, B, C, D, E,\) and \(F\) are equivalent to \( x^{3/5} \).
So, there are no equivalent expressions from the choices provided.
