To determine the correct radical expression for [tex]\(\frac{3}{8}\)[/tex] among the given options, let's analyze each option carefully:
1. [tex]\(4 \sqrt[8]{d^3}\)[/tex]:
- This expression signifies taking the eighth root of [tex]\(d^3\)[/tex] and then multiplying the result by 4. It does not simplify to match [tex]\(\frac{3}{8}\)[/tex].
2. [tex]\(4 \sqrt[3]{d^8}\)[/tex]:
- This means taking the cube root of [tex]\(d^8\)[/tex] and then multiplying the result by 4. It does not simplify to match [tex]\(\frac{3}{8}\)[/tex].
3. [tex]\(\sqrt[8]{40^3}\)[/tex]:
- This expression represents the eighth root of [tex]\(40^3\)[/tex]. Breaking it down, it means you are taking the radical of a large number, which does not match [tex]\(\frac{3}{8}\)[/tex].
4. [tex]\(\sqrt[3]{4 d^8}\)[/tex]:
- This means taking the cube root of [tex]\(4d^8\)[/tex]. Although it involves the correct numbers, it doesn’t simplify to [tex]\(\frac{3}{8}\)[/tex].
From this analysis, none of the given radical expressions directly simplify to [tex]\(\frac{3}{8}\)[/tex]. However, upon careful reconsideration and if there might have been an intention to match certain variable relationships with given constants, the simple fraction [tex]\(\frac{3}{8}\)[/tex] does not map directly to any of the complex radical forms without more context or transformation. Thus:
The true approach for such exams should ensure proper understanding of algebraic manipulations or transformations for clarity. In this case, none of the provided radical expressions simplifies directly back to [tex]\(\frac{3}{8}\)[/tex]. Thus, no exact match would be the proper answer.
