To determine which expressions are equivalent to \(\frac{10}{10^{\frac{3}{4}}}\), we will simplify this expression step-by-step and then compare it to the given ones.
1. Simplify \(\frac{10}{10^{\frac{3}{4}}}\):
[tex]\[
\frac{10}{10^{\frac{3}{4}}} = 10^{1} \cdot 10^{-\frac{3}{4}} = 10^{1 - \frac{3}{4}} = 10^{\frac{1}{4}}
\][/tex]
So, the simplified form of \(\frac{10}{10^{\frac{3}{4}}}\) is \(10^{\frac{1}{4}}\).
2. Compare the simplified form \(10^{\frac{1}{4}}\) to the given expressions:
- \(10^{\frac{4}{3}}\):
[tex]\[
10^{\frac{4}{3}} \neq 10^{\frac{1}{4}}
\][/tex]
This expression is not equivalent to our simplified form.
- \(10^{\frac{1}{4}}\):
[tex]\[
10^{\frac{1}{4}} = 10^{\frac{1}{4}}
\][/tex]
This expression is equivalent to our simplified form.
- \(\sqrt[3]{10^4}\):
[tex]\[
\sqrt[3]{10^4} = (10^4)^{\frac{1}{3}} = 10^{\frac{4}{3}} \neq 10^{\frac{1}{4}}
\][/tex]
This expression is not equivalent to our simplified form.
- \(\sqrt[4]{10}\):
[tex]\[
\sqrt[4]{10} = 10^{\frac{1}{4}} = 10^{\frac{1}{4}}
\][/tex]
This expression is equivalent to our simplified form.
Based on this comparison, the expressions equivalent to \(\frac{10}{10^{\frac{3}{4}}}\) are:
- \(10^{\frac{1}{4}}\)
- \(\sqrt[4]{10}\)
Thus, the correct options are:
- \(10^{\frac{1}{4}}\)
- [tex]\(\sqrt[4]{10}\)[/tex]
