To determine which expression is equivalent to [tex]\(\sqrt{10^{\frac{3}{4}} x}\)[/tex], we need to simplify this expression and compare it to the given options:
Let's start by simplifying [tex]\(\sqrt{10^{\frac{3}{4}} x}\)[/tex].
1. Simplify the expression inside the square root:
[tex]\[
\sqrt{10^{\frac{3}{4}} x}
\][/tex]
2. Use the property of exponents and square roots:
[tex]\[
\sqrt{a} = a^{\frac{1}{2}}
\][/tex]
Apply this property:
[tex]\[
\sqrt{10^{\frac{3}{4}} x} = (10^{\frac{3}{4}} x)^{\frac{1}{2}}
\][/tex]
3. Simplify the expression inside the parentheses:
[tex]\[
(10^{\frac{3}{4}} x)^{\frac{1}{2}} = 10^{\frac{3}{4} \cdot \frac{1}{2}} \cdot x^{\frac{1}{2}}
\][/tex]
Simplify the exponent:
[tex]\[
10^{\frac{3}{4} \cdot \frac{1}{2}} = 10^{\frac{3}{8}}
\][/tex]
So the expression simplifies to:
[tex]\[
10^{\frac{3}{8}} \cdot x^{\frac{1}{2}}
\][/tex]
Now, let's compare each of the given expressions to find the match:
1. Option (a): [tex]\((\sqrt[3]{10})^{4 x}\)[/tex]
Simplify [tex]\(\sqrt[3]{10}\)[/tex]:
[tex]\[
\sqrt[3]{10} = 10^{\frac{1}{3}}
\][/tex]
Raise this to the power [tex]\(4x\)[/tex]:
[tex]\[
(\sqrt[3]{10})^{4 x} = \left(10^{\frac{1}{3}}\right)^{4 x} = 10^{\frac{4 x}{3}}
\][/tex]
This does not match [tex]\(10^{\frac{3}{8}} \cdot x^{\frac{1}{2}}\)[/tex].
2. Option (b): [tex]\((\sqrt[4]{10})^{3 x}\)[/tex]
Simplify [tex]\(\sqrt[4]{10}\)[/tex]:
[tex]\[
\sqrt[4]{10} = 10^{\frac{1}{4}}
\][/tex]
Raise this to the power [tex]\(3 x\)[/tex]:
[tex]\[
(\sqrt[4]{10})^{3 x} = \left(10^{\frac{1}{4}}\right)^{3 x} = 10^{\frac{3 x}{4}}
\][/tex]
This does not match [tex]\(10^{\frac{3}{8}} \cdot x^{\frac{1}{2}}\)[/tex].
3. Option (c): [tex]\((\sqrt[6]{10})^{4 x}\)[/tex]
Simplify [tex]\(\sqrt[6]{10}\)[/tex]:
[tex]\[
\sqrt[6]{10} = 10^{\frac{1}{6}}
\][/tex]
Raise this to the power [tex]\(4 x\)[/tex]:
[tex]\[
(\sqrt[6]{10})^{4 x} = \left(10^{\frac{1}{6}}\right)^{4 x} = 10^{\frac{4 x}{6}} = 10^{\frac{2 x}{3}}
\][/tex]
This does not match [tex]\(10^{\frac{3}{8}} \cdot x^{\frac{1}{2}}\)[/tex].
4. Option (d): [tex]\((\sqrt[8]{10})^{3 x}\)[/tex]
Simplify [tex]\(\sqrt[8]{10}\)[/tex]:
[tex]\[
\sqrt[8]{10} = 10^{\frac{1}{8}}
\][/tex]
Raise this to the power [tex]\(3 x\)[/tex]:
[tex]\[
(\sqrt[8]{10})^{3 x} = \left(10^{\frac{1}{8}}\right)^{3 x} = 10^{\frac{3 x}{8}}
\][/tex]
This matches [tex]\(10^{\frac{3}{8}} \cdot x^{\frac{1}{2}}\)[/tex].
Therefore, the expression [tex]\(\sqrt{10^{\frac{3}{4}} x}\)[/tex] is equivalent to [tex]\((\sqrt[8]{10})^{3 x}\)[/tex].
Thus, the correct answer is:
[tex]\[
(\sqrt[8]{10})^{3 x}
\][/tex]
