Which is equivalent to \( (\sqrt[3]{125})^{x} \) ? \( 125^{\frac{1}{3} x} \) \( 125^{\frac{1}{3 x}} \) \( 125^{3 x} \) \( 1255^{\left[\frac{1}{3}\right)^{x}} \)

The expression \( (\sqrt[3]{125})^{x} \) can be simplified step by step to find its equivalent form among the provided options. 1. **Simplify the Cube Root:** \[ \sqrt[3]{125} = 5 \] This is because \( 125 = 5^3 \). 2. **Raise to the Power of \( x \):** \[ (\sqrt[3]{125})^{x} = 5^{x} \] Now, let's evaluate each of the given options to see which one matches \( 5^{x} \): 1. **Option 1: \( 125^{\frac{1}{3} x} \)** \[ 125^{\frac{1}{3} x} = (5^3)^{\frac{1}{3} x} = 5^{3 \cdot \frac{1}{3} x} = 5^{x} \] This matches exactly with \( (\sqrt[3]{125})^{x} \). 2. **Option 2: \( 125^{\frac{1}{3 x}} \)** \[ 125^{\frac{1}{3 x}} = (5^3)^{\frac{1}{3 x}} = 5^{\frac{3}{3 x}} = 5^{\frac{1}{x}} \] This does not match \( 5^{x} \). 3. **Option 3: \( 125^{3 x} \)** \[ 125^{3 x} = (5^3)^{3 x} = 5^{9 x} \] This does not match \( 5^{x} \). 4. **Option 4: \( 1255^{\left(\frac{1}{3}\right)^{x}} \)** Assuming it's a typographical error and should be \( 125^{\left(\frac{1}{3}\right)^{x}} \): \[ 125^{\left(\frac{1}{3}\right)^{x}} = (5^3)^{\left(\frac{1}{3}\right)^{x}} = 5^{3 \left(\frac{1}{3}\right)^{x}} \] This does not match \( 5^{x} \). **Conclusion:** The equivalent expression to \( (\sqrt[3]{125})^{x} \) is: **\( 125^{\frac{1}{3} x} \)** **Answer:** \( 125^{\frac{1}{3} x} \)