Let's analyze the given expression step-by-step:
We are given the expression [tex]\(\sqrt[3]{8}^{\frac{1}{4} x}\)[/tex].
1. First, we will simplify [tex]\(\sqrt[3]{8}\)[/tex]:
- [tex]\(\sqrt[3]{8}\)[/tex] represents the cube root of 8.
- We know that [tex]\(2^3 = 8\)[/tex], thus [tex]\(\sqrt[3]{8} = 2\)[/tex].
2. Now, we substitute this back into the original expression:
- Therefore, the expression becomes [tex]\(2^{\frac{1}{4} x}\)[/tex].
Next, we need to match this expression [tex]\(2^{\frac{1}{4} x}\)[/tex] with the given choices:
1. [tex]\(8^{\frac{3}{4} x}\)[/tex]:
- This choice does not match since the base is [tex]\(8\)[/tex] rather than [tex]\(2\)[/tex].
2. [tex]\(\sqrt[7]{8}^x\)[/tex]:
- This is the seventh root of [tex]\(8\)[/tex] raised to [tex]\(x\)[/tex]. This also does not match since the base operation is different and doesn't simplify to give a base [tex]\(2\)[/tex].
3. [tex]\(\sqrt[12]{8} x\)[/tex]:
- This expression implies the 12th root of [tex]\(8\)[/tex] multiplied by [tex]\(x\)[/tex], which again does not align with [tex]\(2^{\frac{1}{4} x}\)[/tex].
4. [tex]\(8^{\frac{3}{4 x}}\)[/tex]:
- This expression also does not match as it involves a different exponent and base.
Given these observations, none of the given choices match the simplified form [tex]\(2^{\frac{1}{4} x}\)[/tex]. Therefore, the correct answer is:
None of the options given could simplify to match [tex]\(2^{\frac{1}{4} x}\)[/tex].
