To determine which expression is equivalent to [tex]\(\sqrt{10} \cdot \frac{3}{4} x\)[/tex], we need to simplify [tex]\(\sqrt{10} \cdot \frac{3}{4} x\)[/tex] and then compare it to the given options.
1. Simplifying [tex]\(\sqrt{10} \cdot \frac{3}{4} x\)[/tex]:
We start with the given expression:
[tex]\[
\sqrt{10} \cdot \frac{3}{4} x
\][/tex]
Recall that [tex]\(\sqrt{10}\)[/tex] is the same as [tex]\(10^{1/2}\)[/tex]:
[tex]\[
\sqrt{10} = 10^{1/2}
\][/tex]
Therefore, the expression becomes:
[tex]\[
10^{1/2} \cdot \frac{3}{4} x
\][/tex]
This can be rewritten to show the exponent more clearly:
[tex]\[
10^{(1/2) \cdot \frac{3}{4} x}
\][/tex]
Simplify the exponent:
[tex]\[
10^{(3 x / 4) / 2} = 10^{3x / 8}
\][/tex]
2. Analyzing the given options:
- Option 1: [tex]\((\sqrt[3]{10})^{4 x}\)[/tex]
[tex]\[
(\sqrt[3]{10})^{4x} = (10^{1/3})^{4x} = 10^{(4x/3)}
\][/tex]
- Option 2: [tex]\((\sqrt[4]{10})^{3 x}\)[/tex]
[tex]\[
(\sqrt[4]{10})^{3x} = (10^{1/4})^{3x} = 10^{(3x/4)}
\][/tex]
- Option 3: [tex]\((\sqrt[6]{10})^{4 x}\)[/tex]
[tex]\[
(\sqrt[6]{10})^{4x} = (10^{1/6})^{4x} = 10^{(4x/6)} = 10^{(2x/3)}
\][/tex]
- Option 4: [tex]\((\sqrt[8]{10})^{3 x}\)[/tex]
[tex]\[
(\sqrt[8]{10})^{3x} = (10^{1/8})^{3x} = 10^{(3x/8)}
\][/tex]
3. Comparing the simplified expression to the options:
From the simplification, we found that:
[tex]\[
\sqrt{10} \cdot \frac{3}{4} x = 10^{3x/8}
\][/tex]
Among the options, this matches:
[tex]\[
(\sqrt[8]{10})^{3 x} = 10^{(3x/8)}
\][/tex]
Therefore, the expression equivalent to [tex]\(\sqrt{10} \cdot \frac{3}{4} x\)[/tex] is:
[tex]\[
(\sqrt[8]{10})^{3 x}
\][/tex]
