To determine which expressions are equivalent to [tex]\(\frac{10}{10^{\frac{3}{4}}}\)[/tex], let's analyze each option step-by-step.
First, let's simplify the given expression:
[tex]\[
\frac{10}{10^{\frac{3}{4}}}
\][/tex]
We can rewrite the expression as:
[tex]\[
10 \times 10^{-\frac{3}{4}} = 10^{1 - \frac{3}{4}} = 10^{\frac{4}{4} - \frac{3}{4}} = 10^{\frac{1}{4}}
\][/tex]
Thus, [tex]\(\frac{10}{10^{\frac{3}{4}}} = 10^{\frac{1}{4}}\)[/tex].
Now, let's compare this with each of the given options:
1. [tex]\(10^{\frac{4}{3}}\)[/tex]
This is not equivalent to [tex]\(10^{\frac{1}{4}}\)[/tex].
2. [tex]\(10^{\frac{1}{4}}\)[/tex]
This is equivalent to [tex]\(10^{\frac{1}{4}}\)[/tex].
3. [tex]\(\sqrt[3]{10^4}\)[/tex]
We can rewrite [tex]\(\sqrt[3]{10^4}\)[/tex] as:
[tex]\[
(10^4)^{\frac{1}{3}} = 10^{\frac{4}{3}}
\][/tex]
This is not equivalent to [tex]\(10^{\frac{1}{4}}\)[/tex].
4. [tex]\(\sqrt[4]{10}\)[/tex]
We can rewrite [tex]\(\sqrt[4]{10}\)[/tex] as:
[tex]\[
10^{\frac{1}{4}}
\][/tex]
This is equivalent to [tex]\(10^{\frac{1}{4}}\)[/tex].
Therefore, the equivalent expressions to [tex]\(\frac{10}{10^{\frac{3}{4}}}\)[/tex] are:
[tex]\[
10^{\frac{1}{4}} \quad \text{and} \quad \sqrt[4]{10}.
\][/tex]
