To determine which expression is equivalent to [tex]\((\sqrt[3]{125})^x\)[/tex], let’s break down the given expression step by step.
1. Understanding the Cube Root:
The cube root of 125, written as [tex]\(\sqrt[3]{125}\)[/tex], is the number that, when raised to the power of 3, gives 125. Since [tex]\(125 = 5^3\)[/tex], it follows that:
[tex]\[
\sqrt[3]{125} = 5
\][/tex]
Therefore, [tex]\((\sqrt[3]{125})^x\)[/tex] simplifies to:
[tex]\[
5^x
\][/tex]
2. Expressing in Base 125:
Next, we want to express [tex]\(5^x\)[/tex] in terms of [tex]\(125\)[/tex]. Recall that [tex]\(125 = 5^3\)[/tex]. Thus:
[tex]\[
125 = 5^3
\][/tex]
3. Raising to the Power [tex]\(x\)[/tex]:
If we raise [tex]\(125\)[/tex] raised to an exponent [tex]\(\frac{1}{3}\)[/tex], we get:
[tex]\[
125^{\frac{1}{3}} = (5^3)^{\frac{1}{3}}
\][/tex]
Using the property of exponents [tex]\((a^m)^n = a^{m \cdot n}\)[/tex], this simplifies to:
[tex]\[
(5^3)^{\frac{1}{3}} = 5^{3 \cdot \frac{1}{3}} = 5^1 = 5
\][/tex]
Notice [tex]\((\sqrt[3]{125})^x\)[/tex] is just rewriting [tex]\((125^{\frac{1}{3}})^x\)[/tex]:
4. Final Expression:
We can now simplify:
[tex]\[
\left(125^{\frac{1}{3}}\right)^x = 125^{\frac{1}{3} \cdot x} = 125^{\frac{x}{3}}
\][/tex]
Thus, the expression [tex]\((\sqrt[3]{125})^x\)[/tex] is equivalent to [tex]\(125^{\frac{x}{3}}\)[/tex].
So, the correct equivalent expression is:
[tex]\[
125^{\frac{1}{3} x}
\][/tex]
The answer is:
[tex]\[
\boxed{125^{\frac{1}{3} x}}
\][/tex]
