Converting a radical to a fractional exponent

Question

I want to understand how to convert a radical to a fractional exponent. Given the following equation:

$\sqrt[3]{(x)^6\cdot x^9}=\sqrt[3]{x^{24}\cdot x^9}=\sqrt[3]{x^{33}}=x^{\frac{33}3}=x^{11}$

How does: $\sqrt[3]{(x)^6\cdot x^9} = \sqrt[3]{x^{24}\cdot x^9}\;\;$??

Answer 1

Power rules of exponent :

1. $(a^m)^n=a^{mn}$ $\to \sqrt[3]{(x^4)^6\cdot x^9} = \sqrt[3]{x^{24}\cdot x^9}$
2. $a^ma^n=a^{m+n}$ $\to \sqrt[3]{x^{24}\cdot x^9}$$=\sqrt[3]{x^{33}}$
3. $\sqrt[n]{a}=a^{\frac{1}{n}},\sqrt[n]{a^m}=a^{\frac{m}{n}} \to \sqrt[3]{x^{33}}$$=x^{\frac{33}{3}}=x^{11}\;$

Or,

$\sqrt[3]{(x^4)^6\cdot x^9} = \sqrt[3]{x^{24}\cdot x^9}$(2)$=\sqrt[3]{x^{33}}$$=x^{\frac{33}{3}}=x^{11}\;$