Select the correct answer. Simplify. ((3x^2y^3)/(z^3))^3 A. (27x^6y^9)/(z^3) B. (3x^6y^9)/(z^9) C. (3x^5y^6)/(z^6) D. (27x^6y^9)/(z^9) c A ) B

To simplify the expression \((\frac{3x^2y^3}{z^3})^3\), we need to apply the power of a quotient rule and the power of a power rule.<br /><br />The power of a quotient rule states that \((\frac{a}{b})^n = \frac{a^n}{b^n}\).<br /><br />The power of a power rule states that \((a^m)^n = a^{m \cdot n}\).<br /><br />Let's apply these rules step by step:<br /><br />1. Apply the power of a quotient rule:<br />\[<br />\left(\frac{3x^2y^3}{z^3}\right)^3 = \frac{(3x^2y^3)^3}{(z^3)^3}<br />\]<br /><br />2. Apply the power of a power rule to both the numerator and the denominator:<br />\[<br />\frac{(3^3)(x^2)^3(y^3)^3}{(z^3)^3}<br />\]<br /><br />3. Simplify each term:<br />\[<br />3^3 = 27<br />\]<br />\[<br />(x^2)^3 = x^{2 \cdot 3} = x^6<br />\]<br />\[<br />(y^3)^3 = y^{3 \cdot 3} = y^9<br />\]<br />\[<br />(z^3)^3 = z^{3 \cdot 3} = z^9<br />\]<br /><br />Putting it all together, we get:<br />\[<br />\frac{27x^6y^9}{z^9}<br />\]<br /><br />Therefore, the correct answer is:<br />D. \(\frac{27x^6y^9}{z^9}\)