Question

question
express in simplest radical form.
$(3125x^3)^{\frac{6}{5}}$
answer attempt 1 out of 2

Explanation:

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

\((3125a^{3})^{\frac{6}{5}} = 3125^{\frac{6}{5}} \cdot (a^{3})^{\frac{6}{5}}\)

Step2: Simplify \(3125^{\frac{6}{5}}\). Note that \(3125 = 5^5\), so \(3125^{\frac{6}{5}}=(5^5)^{\frac{6}{5}}\)

Using the rule \((a^m)^n = a^{mn}\), we get \((5^5)^{\frac{6}{5}} = 5^{5\times\frac{6}{5}} = 5^6 = 15625\)

Step3: Simplify \((a^{3})^{\frac{6}{5}}\)

Using the rule \((a^m)^n = a^{mn}\), we have \((a^{3})^{\frac{6}{5}} = a^{3\times\frac{6}{5}} = a^{\frac{18}{5}}\)

Step4: Rewrite \(a^{\frac{18}{5}}\) as a radical. Recall that \(a^{\frac{m}{n}}=\sqrt[n]{a^m}\), so \(a^{\frac{18}{5}}=\sqrt[5]{a^{18}}\)

Simplify \(\sqrt[5]{a^{18}}\): \(a^{18}=a^{15 + 3}=a^{15}\cdot a^{3}\), so \(\sqrt[5]{a^{18}}=\sqrt[5]{a^{15}\cdot a^{3}}=\sqrt[5]{a^{15}}\cdot\sqrt[5]{a^{3}} = a^{3}\sqrt[5]{a^{3}}\)

Step5: Combine the results from Step2 and Step4

\(3125^{\frac{6}{5}} \cdot (a^{3})^{\frac{6}{5}} = 15625 \cdot a^{3}\sqrt[5]{a^{3}}\)

Answer:

\(15625a^{3}\sqrt[5]{a^{3}}\)