Answered

Which of the following choices are equivalent to the expression [tex]x^{\frac{8}{5}}[/tex]? Select all that apply.

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]



Answer :

GinnyAnswer
To determine which of the given expressions are equivalent to the expression \((x^8)^{1/5}\), we need to simplify each expression and see if they match.

Let's simplify each option step by step:

Expression A: \((x^8)^{1/5}\)

[tex]\[ (x^8)^{1/5} = x^{(8 \cdot 1/5)} = x^{8/5} \][/tex]

Expression B: \(\sqrt[8]{x^5}\)

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

Expression C: \(\sqrt[5]{x^8}\)

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

Expression D: \((x^5)^{1/8}\)

[tex]\[ (x^5)^{1/8} = x^{(5 \cdot 1/8)} = x^{5/8} \][/tex]

Expression E: \((\sqrt[8]{x})^5\)

[tex]\[ (\sqrt[8]{x})^5 = (x^{1/8})^5 = x^{(1/8 \cdot 5)} = x^{5/8} \][/tex]

Expression F: \((\sqrt[5]{x})^8\)

[tex]\[ (\sqrt[5]{x})^8 = (x^{1/5})^8 = x^{(1/5 \cdot 8)} = x^{8/5} \][/tex]

To see which of these expressions are equivalent to \((x^8)^{1/5} = x^{8/5}\):

- Expression A: \((x^8)^{1/5} = x^{8/5}\)
- Expression B: \(\sqrt[8]{x^5} = x^{5/8}\) (not equivalent)
- Expression C: \(\sqrt[5]{x^8} = x^{8/5}\)
- Expression D: \((x^5)^{1/8} = x^{5/8}\) (not equivalent)
- Expression E: \((\sqrt[8]{x})^5 = x^{5/8}\) (not equivalent)
- Expression F: \((\sqrt[5]{x})^8 = x^{8/5}\)

Thus, the expressions that are equivalent to \((x^8)^{1/5}\) are:

[tex]\[ \boxed{\text{A, C, F}} \][/tex]

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