To convert the expression [tex]\(4 d^{\frac{3}{8}}\)[/tex] into its radical form, we need to understand the exponent notation and how it translates to radicals.
1. Understand the exponent: The exponent [tex]\(\frac{3}{8}\)[/tex] in the expression [tex]\(d^{\frac{3}{8}}\)[/tex] indicates a root and a power:
- The denominator (8) represents the root.
- The numerator (3) represents the power.
2. Rewrite the expression [tex]\(d^{\frac{3}{8}}\)[/tex] using radical notation:
- The general form for converting a fractional exponent [tex]\(a^{\frac{m}{n}}\)[/tex] to a radical is [tex]\(a^{\frac{m}{n}} = \sqrt[n]{a^m}\)[/tex].
3. Apply the conversion:
- In this case, [tex]\(d^{\frac{3}{8}}\)[/tex] becomes [tex]\(\sqrt[8]{d^3}\)[/tex].
4. Incorporate the coefficient '4':
- The given expression [tex]\(4 d^{\frac{3}{8}}\)[/tex] then becomes [tex]\(4 \sqrt[8]{d^3}\)[/tex].
Now, we compare this result with the provided options:
1. [tex]\(4 \sqrt[8]{d^3}\)[/tex]
2. [tex]\(4 \sqrt[3]{d^8}\)[/tex]
3. [tex]\(\sqrt[8]{4 d^3}\)[/tex]
4. [tex]\(\sqrt[3]{4 d^8}\)[/tex]
Clearly, the option that matches our converted expression is:
[tex]\[4 \sqrt[8]{d^3}\][/tex]
Therefore, the correct answer is the first option: [tex]\(4 \sqrt[8]{d^3}\)[/tex].
