Let's find the radical form of the expression [tex]\(4 d^{\frac{3}{8}}\)[/tex].
1. The given expression is [tex]\(4 d^{\frac{3}{8}}\)[/tex].
2. To express [tex]\(d^{\frac{3}{8}}\)[/tex] in radical form, note that the exponent [tex]\(\frac{3}{8}\)[/tex] can be interpreted as taking the 8th root of [tex]\(d\)[/tex] raised to the 3rd power:
[tex]\[
d^{\frac{3}{8}} = (d^3)^{\frac{1}{8}}
\][/tex]
This means we are taking the eighth root of [tex]\(d^3\)[/tex]:
[tex]\[
(d^3)^{\frac{1}{8}} = \sqrt[8]{d^3}
\][/tex]
3. Now, incorporating the factor of 4 in the original expression [tex]\(4 d^{\frac{3}{8}}\)[/tex], we get:
[tex]\[
4 d^{\frac{3}{8}} = 4 \sqrt[8]{d^3}.
\][/tex]
Thus, the correct radical expression for [tex]\(4 d^{\frac{3}{8}}\)[/tex] is:
[tex]\[ 4 \sqrt[8]{d^3} \][/tex]
So, the choice that matches this expression is:
[tex]\[ 4 \sqrt[8]{d^3} \][/tex]
Therefore, the correct answer is:
[tex]\[ 1 \text{ ( } 4 \sqrt[8]{d^3} \text{ )} \][/tex]
