To convert the expression [tex]\(4 d^{\frac{3}{8}}\)[/tex] into its radical form, follow these steps:
1. Understand the Exponential Form:
The expression [tex]\(d^{\frac{3}{8}}\)[/tex] means that [tex]\(d\)[/tex] is raised to the power of [tex]\(\frac{3}{8}\)[/tex].
2. Rewrite the Exponential Expression:
The exponent [tex]\(\frac{3}{8}\)[/tex] can be interpreted as a radical by considering the denominator and the numerator separately.
- The denominator (8) indicates the root.
- The numerator (3) indicates the power.
3. Apply Radical Form:
To convert [tex]\(d^{\frac{3}{8}}\)[/tex] into radical form, we interpret this as the eighth root of [tex]\(d\)[/tex] cubed:
[tex]\[
d^{\frac{3}{8}} = \sqrt[8]{d^3}
\][/tex]
4. Combine with the Coefficient:
The given expression is [tex]\(4 d^{\frac{3}{8}}\)[/tex]. Thus, combining the coefficient (4) with the radical form of [tex]\(d^{\frac{3}{8}}\)[/tex], we get:
[tex]\[
4 d^{\frac{3}{8}} = 4 \sqrt[8]{d^3}
\][/tex]
Therefore, the correct radical expression for [tex]\(4 d^{\frac{3}{8}}\)[/tex] is:
[tex]\[
\boxed{4 \sqrt[8]{d^3}}
\][/tex]
So, the answer is:
[tex]\[
4 \sqrt[8]{d^3}
\][/tex]
