Rewrite the radical expression as an expression with rational exponents.

[tex]\[
\sqrt[3]{x^8}
\][/tex]

A. [tex]\(x^{\frac{3}{8}}\)[/tex]
B. [tex]\(x^{\frac{8}{3}}\)[/tex]
C. [tex]\(8 x^3\)[/tex]
D. [tex]\(3 x^8\)[/tex]



Answer :

Certainly! Let's rewrite the radical expression [tex]\(\sqrt[3]{x^8}\)[/tex] using rational exponents.

1. Understand the Radical Expression:
- The expression [tex]\(\sqrt[3]{x^8}\)[/tex] denotes the cube root of [tex]\(x^8\)[/tex].

2. Applying the Rule for Radicals:
- A general rule for converting radicals to rational exponents is [tex]\( \sqrt[n]{a^m} = a^{\frac{m}{n}} \)[/tex].
- In our case, [tex]\( n = 3 \)[/tex] (the cube root) and [tex]\( m = 8 \)[/tex] (the exponent inside the radical).

3. Rewrite the Expression:
- Applying the rule, we rewrite [tex]\(\sqrt[3]{x^8}\)[/tex] as [tex]\(x^{\frac{8}{3}}\)[/tex].

Therefore, the correct expression with rational exponents for [tex]\(\sqrt[3]{x^8}\)[/tex] is:

[tex]\[ x^{\frac{8}{3}} \][/tex]

Given the options:
- [tex]\(x^{\frac{3}{8}}\)[/tex]
- [tex]\(x^{\frac{8}{3}}\)[/tex]
- [tex]\(8 x^3\)[/tex]
- [tex]\(3 x^8\)[/tex]

The correct answer is:

[tex]\[ \boxed{x^{\frac{8}{3}}} \][/tex]