Sure, let's rewrite the given radical expression [tex]\(\sqrt[3]{x^8}\)[/tex] as an expression with rational exponents step by step.
1. Understand the Radical Expression:
The given expression is the cube root of [tex]\(x^8\)[/tex], which we denote as [tex]\(\sqrt[3]{x^8}\)[/tex].
2. Rewrite the Radical as an Exponent:
The cube root of a number can be expressed as raising that number to the power of [tex]\(\frac{1}{3}\)[/tex]. Thus, [tex]\(\sqrt[3]{x^8}\)[/tex] can be written as:
[tex]\[
(x^8)^{\frac{1}{3}}
\][/tex]
3. Apply the Power of a Power Property:
To simplify [tex]\((x^8)^{\frac{1}{3}}\)[/tex], we use the power of a power property for exponents, which states [tex]\((a^m)^n = a^{m \cdot n}\)[/tex]. Here, [tex]\(a = x\)[/tex], [tex]\(m = 8\)[/tex], and [tex]\(n = \frac{1}{3}\)[/tex]. So, we get:
[tex]\[
x^{8 \cdot \frac{1}{3}}
\][/tex]
4. Simplify the Exponent:
Multiply the exponents [tex]\(8\)[/tex] and [tex]\(\frac{1}{3}\)[/tex]:
[tex]\[
8 \cdot \frac{1}{3} = \frac{8}{3}
\][/tex]
5. Write the Final Expression:
Therefore, the expression [tex]\((x^8)^{\frac{1}{3}}\)[/tex] simplifies to:
[tex]\[
x^{\frac{8}{3}}
\][/tex]
So, the correct option is [tex]\(x^{\frac{8}{3}}\)[/tex].
