Question

mplest radical form.
$(125x)^{\frac{1}{3}}$

Explanation:

Step1: Apply the power of a product rule

The power of a product rule states that \((ab)^n = a^n b^n\). So, we can apply this to \((125x)^{\frac{1}{3}}\) to get \(125^{\frac{1}{3}} \cdot x^{\frac{1}{3}}\).

Step2: Simplify \(125^{\frac{1}{3}}\)

We know that \(125 = 5^3\), so \(125^{\frac{1}{3}}=(5^3)^{\frac{1}{3}}\). Using the power of a power rule \((a^m)^n = a^{mn}\), we have \((5^3)^{\frac{1}{3}} = 5^{3\times\frac{1}{3}} = 5^1 = 5\).

Step3: Rewrite \(x^{\frac{1}{3}}\) in radical form

Recall that \(a^{\frac{1}{n}}=\sqrt[n]{a}\), so \(x^{\frac{1}{3}}=\sqrt[3]{x}\).

Step4: Combine the results

Putting it all together, \(125^{\frac{1}{3}} \cdot x^{\frac{1}{3}} = 5\sqrt[3]{x}\).

Answer:

\(5\sqrt[3]{x}\)