Westonci.ca connects you with experts who provide insightful answers to your questions. Join us today and start learning! Join our platform to connect with experts ready to provide precise answers to your questions in various areas. Discover detailed answers to your questions from a wide network of experts on our comprehensive Q&A platform.

Which of the following choices are equivalent to the expression below? Check all that apply.

[tex] x^{3 / 5} [/tex]

A. [tex] \left(x^8\right)^{1 / 5} [/tex]

B. [tex] \sqrt[8]{x^5} [/tex]

C. [tex] \sqrt[5]{x^8} [/tex]

D. [tex] \left(x^5\right)^{1 / 8} [/tex]

E. [tex] (\sqrt[8]{x})^5 [/tex]

F. [tex] (\sqrt[5]{x})^8 [/tex]

Sagot :

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.