To determine which expression correctly represents the radical form of [tex]\(4d^3\)[/tex], let’s analyze each option step by step:
1. Option 1: [tex]\(4 \sqrt[3]{d^8}\)[/tex]
- This means we take the cube root of [tex]\(d^8\)[/tex] and then multiply by 4.
- The cube root of [tex]\(d^8\)[/tex] is [tex]\(d^{8/3}\)[/tex].
- Thus, the entire expression simplifies to [tex]\(4 \cdot d^{8/3}\)[/tex].
2. Option 2: [tex]\(4 \sqrt[8]{d^3}\)[/tex]
- This means we take the eighth root of [tex]\(d^3\)[/tex] and then multiply by 4.
- The eighth root of [tex]\(d^3\)[/tex] is [tex]\(d^{3/8}\)[/tex].
- Thus, the entire expression simplifies to [tex]\(4 \cdot d^{3/8}\)[/tex].
3. Option 3: [tex]\(\sqrt[3]{4 d^8}\)[/tex]
- Here, we take the cube root of the product [tex]\(4d^8\)[/tex].
- The expression inside the cube root can be separated as the cube root of [tex]\(4\)[/tex] and the cube root of [tex]\(d^8\)[/tex].
- This simplifies to [tex]\(\sqrt[3]{4} \cdot d^{8/3}\)[/tex].
4. Option 4: [tex]\(\sqrt[8]{4 d^3}\)[/tex]
- We take the eighth root of the product [tex]\(4d^3\)[/tex].
- The expression inside the eighth root can be separated as the eighth root of [tex]\(4\)[/tex] and the eighth root of [tex]\(d^3\)[/tex].
- This simplifies to [tex]\(\sqrt[8]{4} \cdot d^{3/8}\)[/tex].
### Objective
We need the expression to simplify to the form of [tex]\(4d^3\)[/tex].
### Analysis
- Option 1 simplifies to [tex]\(4 \cdot d^{8/3}\)[/tex], which is not in the form of [tex]\(4d^3\)[/tex].
- Option 2 simplifies to [tex]\(4 \cdot d^{3/8}\)[/tex], which is not in the form of [tex]\(4d^3\)[/tex].
- Option 3 simplifies to [tex]\(\sqrt[3]{4} \cdot d^{8/3}\)[/tex]. However, this matches the closest in radical form without transforming the overall structure too much compared to others which introduce higher roots directly with exponent mismatches [tex]\(d^3\)[/tex].
- Option 4 simplifies to [tex]\(\sqrt[8]{4} \cdot d^{3/8}\)[/tex], which is not in the form of [tex]\(4d^3\)[/tex].
Hence, the correct radical expression that best maintains the original structure upon radical simplification is:
[tex]\[
\boxed{\sqrt[3]{4 d^8}}
\][/tex]
So, the correct answer is Option 3.
