To determine which expression is equivalent to [tex]\(\sqrt[3]{8}^{\frac{1}{4} x}\)[/tex], we need to simplify the given expression step-by-step.
1. First, let's rewrite [tex]\(\sqrt[3]{8}\)[/tex]:
[tex]\[
\sqrt[3]{8} = 8^{\frac{1}{3}}
\][/tex]
2. Now substitute this result into the original expression:
[tex]\[
\sqrt[3]{8}^{\frac{1}{4} x} = \left(8^{\frac{1}{3}}\right)^{\frac{1}{4} x}
\][/tex]
3. Use the property of exponents [tex]\((a^m)^n = a^{mn}\)[/tex] to simplify the expression:
[tex]\[
\left(8^{\frac{1}{3}}\right)^{\frac{1}{4} x} = 8^{(\frac{1}{3} \cdot \frac{1}{4} x)} = 8^{\frac{x}{12}}
\][/tex]
Now, we need to compare this simplified form [tex]\(8^{\frac{x}{12}}\)[/tex] to the options given:
Option 1: [tex]\(8^{\frac{3}{4} x}\)[/tex]
[tex]\[
8^{\frac{3}{4} x} \neq 8^{\frac{x}{12}}
\][/tex]
Option 2: [tex]\(\sqrt[7]{8}^x\)[/tex]
[tex]\[
\sqrt[7]{8}^x = (8^{\frac{1}{7}})^x = 8^{\frac{x}{7}} \neq 8^{\frac{x}{12}}
\][/tex]
Option 3: [tex]\(\sqrt[12]{8} x\)[/tex]
[tex]\[
\sqrt[12]{8} x = (8^{\frac{1}{12}}) x \neq 8^{\frac{x}{12}}
\][/tex]
Option 4: [tex]\(8^{\frac{3}{4 x}}\)[/tex]
[tex]\[
8^{\frac{3}{4 x}} \neq 8^{\frac{x}{12}}
\][/tex]
None of the given options are equivalent to [tex]\(8^{\frac{x}{12}}\)[/tex].
Thus, without any errors, the equivalent expression to [tex]\(\sqrt[3]{8}^{\frac{1}{4} x}\)[/tex] is:
[tex]\[8^{\frac{x}{12}}\][/tex]
There appears to be an oversight or missing option in the list provided.
