Answer :
To determine which of the choices are equivalent to the expression [tex]\( x^{5/4} \)[/tex], let's break down each option and compare them to the given expression:
1. Option A: [tex]\(\sqrt[5]{x^4}\)[/tex]
The expression [tex]\(\sqrt[5]{x^4}\)[/tex] can be rewritten using exponentiation as:
[tex]\[
(x^4)^{1/5} = x^{4 \cdot (1/5)} = x^{4/5}
\][/tex]
Comparing [tex]\( x^{4/5} \)[/tex] with [tex]\( x^{5/4} \)[/tex], they are not equivalent.
2. Option B: [tex]\(\sqrt[4]{x^5}\)[/tex]
The expression [tex]\(\sqrt[4]{x^5}\)[/tex] can be rewritten as:
[tex]\[
(x^5)^{1/4} = x^{5 \cdot (1/4)} = x^{5/4}
\][/tex]
This matches exactly with the original expression [tex]\( x^{5/4} \)[/tex], so it is equivalent.
3. Option C: [tex]\((\sqrt[5]{x})^4\)[/tex]
The expression [tex]\((\sqrt[5]{x})^4\)[/tex] is:
[tex]\[
(x^{1/5})^4 = x^{1/5 \cdot 4} = x^{4/5}
\][/tex]
Comparing [tex]\( x^{4/5} \)[/tex] with [tex]\( x^{5/4} \)[/tex], they are not equivalent.
Therefore, the only equivalent expression to [tex]\( x^{5/4} \)[/tex] is option B: [tex]\(\sqrt[4]{x^5}\)[/tex].
1. Option A: [tex]\(\sqrt[5]{x^4}\)[/tex]
The expression [tex]\(\sqrt[5]{x^4}\)[/tex] can be rewritten using exponentiation as:
[tex]\[
(x^4)^{1/5} = x^{4 \cdot (1/5)} = x^{4/5}
\][/tex]
Comparing [tex]\( x^{4/5} \)[/tex] with [tex]\( x^{5/4} \)[/tex], they are not equivalent.
2. Option B: [tex]\(\sqrt[4]{x^5}\)[/tex]
The expression [tex]\(\sqrt[4]{x^5}\)[/tex] can be rewritten as:
[tex]\[
(x^5)^{1/4} = x^{5 \cdot (1/4)} = x^{5/4}
\][/tex]
This matches exactly with the original expression [tex]\( x^{5/4} \)[/tex], so it is equivalent.
3. Option C: [tex]\((\sqrt[5]{x})^4\)[/tex]
The expression [tex]\((\sqrt[5]{x})^4\)[/tex] is:
[tex]\[
(x^{1/5})^4 = x^{1/5 \cdot 4} = x^{4/5}
\][/tex]
Comparing [tex]\( x^{4/5} \)[/tex] with [tex]\( x^{5/4} \)[/tex], they are not equivalent.
Therefore, the only equivalent expression to [tex]\( x^{5/4} \)[/tex] is option B: [tex]\(\sqrt[4]{x^5}\)[/tex].