To determine the expression equivalent to [tex]\((\sqrt[3]{125})^x\)[/tex], let's break down the problem step by step.
1. Understanding the Radical Expression:
- The expression [tex]\(\sqrt[3]{125}\)[/tex] represents the cube root of 125.
2. Rewriting the Cube Root in Exponential Form:
- We know that a cube root can be expressed as an exponent of [tex]\(\frac{1}{3}\)[/tex]. Therefore, [tex]\(\sqrt[3]{125}\)[/tex] can be written as [tex]\(125^{\frac{1}{3}}\)[/tex].
3. Raising to the Power [tex]\(x\)[/tex]:
- We need to raise this expression to the power [tex]\(x\)[/tex], so [tex]\((\sqrt[3]{125})^x\)[/tex] can be written as [tex]\((125^{\frac{1}{3}})^x\)[/tex].
4. Applying the Power Rule:
- According to the power of a power property in exponents, [tex]\((a^m)^n = a^{m \cdot n}\)[/tex]. Here, [tex]\(a\)[/tex] is 125, [tex]\(m\)[/tex] is [tex]\(\frac{1}{3}\)[/tex], and [tex]\(n\)[/tex] is [tex]\(x\)[/tex].
- Therefore, [tex]\((125^{\frac{1}{3}})^x\)[/tex] simplifies to [tex]\(125^{\frac{1}{3} \cdot x}\)[/tex], or [tex]\(125^{\frac{1}{3} x}\)[/tex].
So the expression [tex]\((\sqrt[3]{125})^x\)[/tex] is equivalent to [tex]\(125^{\frac{1}{3} x}\)[/tex].
The correct answer is:
[tex]\[
125^{\frac{1}{3} x}
\][/tex]
