To find which of the given choices are equivalent to [tex]\( x^{5/4} \)[/tex], let's rewrite each option using properties of exponents and roots.
### Given Expression:
[tex]\[ x^{5/4} \][/tex]
### Choices:
#### A. [tex]\((\sqrt[3]{x})^4\)[/tex]
First, rewrite the cube root:
[tex]\[ \sqrt[3]{x} = x^{1/3} \][/tex]
Now raise this to the fourth power:
[tex]\[ (\sqrt[3]{x})^4 = (x^{1/3})^4 = x^{(1/3) \cdot 4} = x^{4/3} \][/tex]
This is not equivalent to [tex]\( x^{5/4} \)[/tex].
#### B. [tex]\((\sqrt[4]{x})^5\)[/tex]
First, rewrite the fourth root:
[tex]\[ \sqrt[4]{x} = x^{1/4} \][/tex]
Now raise this to the fifth power:
[tex]\[ (\sqrt[4]{x})^5 = (x^{1/4})^5 = x^{(1/4) \cdot 5} = x^{5/4} \][/tex]
This is equivalent to [tex]\( x^{5/4} \)[/tex].
#### C. [tex]\(\left(x^4\right)^{1/5}\)[/tex]
Raise [tex]\( x^4 \)[/tex] to the power of [tex]\( 1/5 \)[/tex]:
[tex]\[ \left(x^4\right)^{1/5} = x^{4 \cdot (1/5)} = x^{4/5} \][/tex]
This is not equivalent to [tex]\( x^{5/4} \)[/tex].
#### D. [tex]\(\sqrt[5]{x^4}\)[/tex]
Rewrite the fifth root:
[tex]\[ \sqrt[5]{x^4} = (x^4)^{1/5} \][/tex]
[tex]\[ = x^{4 \cdot (1/5)} = x^{4/5} \][/tex]
This is not equivalent to [tex]\( x^{5/4} \)[/tex].
#### E. [tex]\(\left(x^5\right)^{1/4}\)[/tex]
Raise [tex]\( x^5 \)[/tex] to the power of [tex]\( 1/4 \)[/tex]:
[tex]\[ \left(x^5\right)^{1/4} = x^{5 \cdot (1/4)} = x^{5/4} \][/tex]
This is equivalent to [tex]\( x^{5/4} \)[/tex].
#### F. [tex]\(\sqrt[4]{x^5}\)[/tex]
Rewrite the fourth root:
[tex]\[ \sqrt[4]{x^5} = (x^5)^{1/4} \][/tex]
[tex]\[ = x^{5 \cdot (1/4)} = x^{5/4} \][/tex]
This is equivalent to [tex]\( x^{5/4} \)[/tex].
### Conclusion:
The choices that are equivalent to [tex]\( x^{5/4} \)[/tex] are:
[tex]\[ B. (\sqrt[4]{x})^5 \][/tex]
[tex]\[ E. \left(x^5\right)^{1/4} \][/tex]
[tex]\[ F. \sqrt[4]{x^5} \][/tex]
So, the equivalent choices are B, E, and F.
