Question

(64x^4)^{\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 \((64x^4)^{\frac{1}{3}}\) as follows:
\((64x^4)^{\frac{1}{3}} = 64^{\frac{1}{3}} \cdot (x^4)^{\frac{1}{3}}\)

Step2: Simplify \(64^{\frac{1}{3}}\)

We know that \(4^3 = 64\), so \(64^{\frac{1}{3}} = 4\) (because the cube root of 64 is 4).

Step3: Simplify \((x^4)^{\frac{1}{3}}\)

Using the power of a power rule \((a^m)^n = a^{mn}\), we have:
\((x^4)^{\frac{1}{3}} = x^{4 \cdot \frac{1}{3}} = x^{\frac{4}{3}}\)

Step4: Combine the results

Now, we multiply the results from Step 2 and Step 3:
\(64^{\frac{1}{3}} \cdot (x^4)^{\frac{1}{3}} = 4 \cdot x^{\frac{4}{3}}\)

Answer:

\(4x^{\frac{4}{3}}\)