Question

question
express in simplest radical form.
$(32x)^{\frac{4}{5}}$

Explanation:

Step1: Recall exponent rule \( (ab)^n = a^n b^n \)

\( (32x)^{\frac{4}{5}} = 32^{\frac{4}{5}} \cdot x^{\frac{4}{5}} \)

Step2: Simplify \( 32^{\frac{4}{5}} \). Note \( 32 = 2^5 \), so \( (2^5)^{\frac{4}{5}} = 2^{5 \cdot \frac{4}{5}} = 2^4 = 16 \)

Step3: Rewrite \( x^{\frac{4}{5}} \) as radical. \( x^{\frac{m}{n}} = \sqrt[n]{x^m} \), so \( x^{\frac{4}{5}} = \sqrt[5]{x^4} \)

Step4: Multiply results from Step2 and Step3

\( 16 \cdot \sqrt[5]{x^4} = 16\sqrt[5]{x^4} \)

Answer:

\( 16\sqrt[5]{x^4} \)