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]
