To determine which expressions are equivalent to [tex]\(\frac{10}{10^{\frac{3}{4}}}\)[/tex], we will simplify [tex]\(\frac{10}{10^{\frac{3}{4}}}\)[/tex] and then check each of the given expressions to see if they match the simplified result.
### Simplifying [tex]\(\frac{10}{10^{\frac{3}{4}}}\)[/tex]
First, rewrite:
[tex]\[
\frac{10}{10^{\frac{3}{4}}}
\][/tex]
This can be expressed as:
[tex]\[
10 \cdot 10^{-\frac{3}{4}}
\][/tex]
Recall the exponent rule [tex]\(a^m \cdot a^n = a^{m+n}\)[/tex]. Thus:
[tex]\[
10^{1} \cdot 10^{-\frac{3}{4}} = 10^{1 - \frac{3}{4}}
\][/tex]
Simplify the exponent:
[tex]\[
10^{1 - \frac{3}{4}} = 10^{\frac{4}{4} - \frac{3}{4}} = 10^{\frac{1}{4}}
\][/tex]
Therefore, [tex]\(\frac{10}{10^{\frac{3}{4}}} = 10^{\frac{1}{4}}\)[/tex].
### Checking the given expressions:
1. [tex]\(10^{\frac{4}{3}}\)[/tex]:
[tex]\[
10^{\frac{4}{3}} \approx 21.544
\][/tex]
This is not the same as [tex]\(10^{\frac{1}{4}}\)[/tex].
2. [tex]\(10^{\frac{1}{4}}\)[/tex]:
[tex]\[
10^{\frac{1}{4}} \approx 1.778
\][/tex]
This matches our simplified result [tex]\(10^{\frac{1}{4}}\)[/tex].
3. [tex]\(\sqrt[3]{10^4}\)[/tex]:
[tex]\[
\sqrt[3]{10^4} = (10^4)^{\frac{1}{3}} = 10^{\frac{4}{3}} \approx 21.544
\][/tex]
This is not the same as [tex]\(10^{\frac{1}{4}}\)[/tex].
4. [tex]\(\sqrt[4]{10}\)[/tex]:
[tex]\[
\sqrt[4]{10} = 10^{\frac{1}{4}} \approx 1.778
\][/tex]
This matches our simplified result [tex]\(10^{\frac{1}{4}}\)[/tex].
### Conclusion
The expressions that are equivalent to [tex]\(\frac{10}{10^{\frac{3}{4}}}\)[/tex] are:
- [tex]\(10^{\frac{1}{4}}\)[/tex]
- [tex]\(\sqrt[4]{10}\)[/tex]
