To determine which expression is equivalent to [tex]\(\sqrt{10^{\frac{3}{4}} x}\)[/tex], let's go through the simplification step by step:
1. Start with the original expression:
[tex]\[
\sqrt{10^{\frac{3}{4}} x}
\][/tex]
2. Recall that the square root of a product can be expressed as the product of the square roots:
[tex]\[
\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}
\][/tex]
So we can write:
[tex]\[
\sqrt{10^{\frac{3}{4}} x} = \sqrt{10^{\frac{3}{4}}} \cdot \sqrt{x}
\][/tex]
3. Now, simplify [tex]\(\sqrt{10^{\frac{3}{4}}}\)[/tex]. The property of square roots and exponents tells us:
[tex]\[
\sqrt{a^b} = a^{b/2}
\][/tex]
Therefore:
[tex]\[
\sqrt{10^{\frac{3}{4}}} = 10^{\frac{3}{4 \cdot 2}} = 10^{\frac{3}{8}}
\][/tex]
4. Substitute back into our expression:
[tex]\[
\sqrt{10^{\frac{3}{4}} x} = 10^{\frac{3}{8}} \cdot \sqrt{x}
\][/tex]
5. Next, we look to match our simplified expression with one of the given choices. Let's transform each of the given choices and compare:
- [tex]\((\sqrt[3]{10})^{4 x}\)[/tex]:
[tex]\[
(\sqrt[3]{10})^{4 x} = (10^{\frac{1}{3}})^{4 x} = 10^{\frac{4 x}{3}}
\][/tex]
- [tex]\((\sqrt[4]{10})^{3 x}\)[/tex]:
[tex]\[
(\sqrt[4]{10})^{3 x} = (10^{\frac{1}{4}})^{3 x} = 10^{\frac{3 x}{4}}
\][/tex]
- [tex]\((\sqrt[6]{10})^{4 x}\)[/tex]:
[tex]\[
(\sqrt[6]{10})^{4 x} = (10^{\frac{1}{6}})^{4 x} = 10^{\frac{4 x}{6}} = 10^{\frac{2 x}{3}}
\][/tex]
- [tex]\((\sqrt[8]{10})^{3 x}\)[/tex]:
[tex]\[
(\sqrt[8]{10})^{3 x} = (10^{\frac{1}{8}})^{3 x} = 10^{\frac{3 x}{8}}
\][/tex]
6. Comparing these, we see that the expression [tex]\(10^{\frac{3 x}{8}}\)[/tex] matches our simplified form.
Therefore, the equivalent expression to [tex]\(\sqrt{10^{\frac{3}{4}} x}\)[/tex] is:
[tex]\[
(\sqrt[8]{10})^{3 x}
\][/tex]
