Let's examine which of the given expressions are equivalent to [tex]\(\frac{10}{10^{\frac{3}{4}}}\)[/tex].
First, simplify the expression [tex]\(\frac{10}{10^{\frac{3}{4}}}\)[/tex]:
[tex]\[
\frac{10}{10^{\frac{3}{4}}} = 10^{1} \times 10^{-\frac{3}{4}}
\][/tex]
Using the properties of exponents, we can combine the powers of 10:
[tex]\[
10^{1} \times 10^{-\frac{3}{4}} = 10^{1 - \frac{3}{4}} = 10^{\frac{4}{4} - \frac{3}{4}} = 10^{\frac{1}{4}}
\][/tex]
So, the expression [tex]\(\frac{10}{10^{\frac{3}{4}}}\)[/tex] simplifies to [tex]\(10^{\frac{1}{4}}\)[/tex].
Now, let's compare this result with the given expressions:
1. [tex]\(10^{\frac{4}{3}}\)[/tex]
2. [tex]\(10^{\frac{1}{4}}\)[/tex]
3. [tex]\(\sqrt[3]{10^4}\)[/tex]
4. [tex]\(\sqrt[4]{10}\)[/tex]
Start by rewriting the given expressions in terms of exponents for consistency:
1. [tex]\(10^{\frac{4}{3}}\)[/tex] remains as it is.
2. [tex]\(10^{\frac{1}{4}}\)[/tex] is [tex]\(10^{\frac{1}{4}}\)[/tex], which matches our simplified expression [tex]\(10^{\frac{1}{4}}\)[/tex].
3. [tex]\(\sqrt[3]{10^4}\)[/tex] can be written as [tex]\((10^4)^{\frac{1}{3}} = 10^{\frac{4}{3}}\)[/tex].
4. [tex]\(\sqrt[4]{10}\)[/tex] can be written as [tex]\(10^{\frac{1}{4}}\)[/tex], which matches our simplified expression [tex]\(10^{\frac{1}{4}}\)[/tex].
From this, we see that expressions [tex]\(10^{\frac{1}{4}}\)[/tex] and [tex]\(\sqrt[4]{10}\)[/tex], which can be rewritten as [tex]\(10^{\frac{1}{4}}\)[/tex], match the original expression.
Therefore, the expressions equivalent to [tex]\(\frac{10}{10^{\frac{3}{4}}}\)[/tex] are:
[tex]\[
10^{\frac{1}{4}}
\][/tex]
[tex]\[
\sqrt[4]{10}
\][/tex]
In the given list, these correspond to:
- [tex]\(10^{\frac{1}{4}}\)[/tex] (the second expression)
- [tex]\(\sqrt[4]{10}\)[/tex] (the fourth expression)
Thus, the equivalent expressions are:
[tex]\[
2, 4
\][/tex]
