Answer :
We are given the expression
[tex]$$
x^{\frac{5}{8}},
$$[/tex]
and we need to determine which of the following choices are equivalent to it.
1. The first expression is directly
[tex]$$
x^{\frac{5}{8}},
$$[/tex]
which is exactly the same as the given expression.
2. The second expression is
[tex]$$
\left(x^8\right)^{\frac{1}{5}}.
$$[/tex]
By using the power rule, where [tex]$\left(x^a\right)^b = x^{a \cdot b}$[/tex], we have
[tex]$$
\left(x^8\right)^{\frac{1}{5}} = x^{8 \cdot \frac{1}{5}} = x^{\frac{8}{5}}.
$$[/tex]
Since [tex]$\frac{8}{5} \neq \frac{5}{8}$[/tex], this option is not equivalent to the given expression.
3. The third expression is
[tex]$$
\sqrt[8]{x^5}.
$$[/tex]
The radical expression can be rewritten in exponential form as
[tex]$$
x^{\frac{5}{8}},
$$[/tex]
which is equivalent to the given expression.
4. The fourth expression is
[tex]$$
\sqrt[5]{x^8}.
$$[/tex]
Similarly, rewriting it in exponential form we have
[tex]$$
x^{\frac{8}{5}},
$$[/tex]
which is not equivalent to [tex]$x^{\frac{5}{8}}$[/tex].
Thus, we find that only two of the given expressions (the first and the third) are equivalent to [tex]$x^{\frac{5}{8}}$[/tex].
The final answer is therefore [tex]$\boxed{2}$[/tex].
[tex]$$
x^{\frac{5}{8}},
$$[/tex]
and we need to determine which of the following choices are equivalent to it.
1. The first expression is directly
[tex]$$
x^{\frac{5}{8}},
$$[/tex]
which is exactly the same as the given expression.
2. The second expression is
[tex]$$
\left(x^8\right)^{\frac{1}{5}}.
$$[/tex]
By using the power rule, where [tex]$\left(x^a\right)^b = x^{a \cdot b}$[/tex], we have
[tex]$$
\left(x^8\right)^{\frac{1}{5}} = x^{8 \cdot \frac{1}{5}} = x^{\frac{8}{5}}.
$$[/tex]
Since [tex]$\frac{8}{5} \neq \frac{5}{8}$[/tex], this option is not equivalent to the given expression.
3. The third expression is
[tex]$$
\sqrt[8]{x^5}.
$$[/tex]
The radical expression can be rewritten in exponential form as
[tex]$$
x^{\frac{5}{8}},
$$[/tex]
which is equivalent to the given expression.
4. The fourth expression is
[tex]$$
\sqrt[5]{x^8}.
$$[/tex]
Similarly, rewriting it in exponential form we have
[tex]$$
x^{\frac{8}{5}},
$$[/tex]
which is not equivalent to [tex]$x^{\frac{5}{8}}$[/tex].
Thus, we find that only two of the given expressions (the first and the third) are equivalent to [tex]$x^{\frac{5}{8}}$[/tex].
The final answer is therefore [tex]$\boxed{2}$[/tex].