To determine which expression is equivalent to [tex]\(\sqrt{10}^{\frac{3}{4} \cdot x}\)[/tex], we can follow these steps:
1. Start by rewriting [tex]\(\sqrt{10}\)[/tex] using exponent notation. The square root of 10 can be expressed as [tex]\(10^{1/2}\)[/tex].
2. Substitute this into the original expression:
[tex]\[
\left( \sqrt{10} \right)^{\frac{3}{4} \cdot x} = \left( 10^{1/2} \right)^{\frac{3}{4} \cdot x}
\][/tex]
3. Apply the power of a power rule [tex]\((a^m)^n = a^{m \cdot n}\)[/tex]:
[tex]\[
\left( 10^{1/2} \right)^{\frac{3}{4} \cdot x} = 10^{(1/2) \cdot (\frac{3}{4} \cdot x)}
\][/tex]
4. Simplify the exponent:
[tex]\[
10^{\frac{1}{2} \cdot \frac{3}{4} \cdot x} = 10^{\frac{3}{8} \cdot x}
\][/tex]
5. Now, convert the expression back to a root form. Recall that [tex]\(a^{1/n} = \sqrt[n]{a}\)[/tex]:
[tex]\[
10^{\frac{3x}{8}} = \left( 10^1 \right)^{\frac{3x}{8}} = \left( \sqrt[8]{10} \right)^{3x}
\][/tex]
So, [tex]\(\sqrt{10}^{\frac{3}{4} \cdot x}\)[/tex] is equivalent to [tex]\(\left( \sqrt[8]{10} \right)^{3x}\)[/tex].
Therefore, the correct answer is:
[tex]\[
(\sqrt[8]{10})^{3x}
\][/tex]
The final choice is (d) [tex]\((\sqrt[8]{10})^{3x}\)[/tex].
