To determine which of the provided options is the correct radical expression for [tex]\( 4 d^{\frac{3}{8}} \)[/tex], we will begin by expressing [tex]\( d^{\frac{3}{8}} \)[/tex] in radical form and then multiply by 4.
1. Recall that a fractional exponent can be written as a radical:
[tex]\[
d^{\frac{3}{8}} = \sqrt[8]{d^3}
\][/tex]
2. Therefore:
[tex]\[
4 d^{\frac{3}{8}} = 4 \sqrt[8]{d^3}
\][/tex]
Now, we compare this with the given options:
- Option 1: [tex]\( 4 \sqrt[8]{d^3} \)[/tex]
- Option 2: [tex]\( 4 \sqrt[3]{d^8} \)[/tex]
- Option 3: [tex]\( \sqrt[8]{4 d^3} \)[/tex]
- Option 4: [tex]\( \sqrt[3]{4 d^8} \)[/tex]
From step 1 and step 2, it's clear that [tex]\( 4 d^{\frac{3}{8}} \)[/tex] is identical to [tex]\( 4 \sqrt[8]{d^3} \)[/tex]. Therefore, the correct option is:
[tex]\[
\boxed{4 \sqrt[8]{d^3}}
\][/tex]
