Questions: (27 x^(-6) y^9 / z^9)^(1 / 3) = A. y^3 / (x^2 z^3) B. 3 y^3 x^2 z^3 C. 3 x^3 / (y^2 z^3) D. y^3 / (3 x^2 z^3) E. 3 y^3 / (x^2 z^3)

Solution Steps

The given expression is: \[ \left(\frac{27 x^{-6} y^{9}}{z^{9}}\right)^{1 / 3} \] First, recognize that \(27\) is a perfect cube, since \(27 = 3^3\). Rewrite the expression as: \[ \left(\frac{3^3 x^{-6} y^{9}}{z^{9}}\right)^{1 / 3} \]

Using the property \((a^m)^n = a^{m \cdot n}\), apply the exponent \(1/3\) to each term inside the parentheses: \[ 3^{3 \cdot \frac{1}{3}} \cdot x^{-6 \cdot \frac{1}{3}} \cdot y^{9 \cdot \frac{1}{3}} \cdot z^{-9 \cdot \frac{1}{3}} \] Simplify the exponents: \[ 3^{1} \cdot x^{-2} \cdot y^{3} \cdot z^{-3} \] This simplifies to: \[ 3 \cdot x^{-2} \cdot y^{3} \cdot z^{-3} \]

To make the expression more readable, rewrite \(x^{-2}\) and \(z^{-3}\) as \(\frac{1}{x^2}\) and \(\frac{1}{z^3}\), respectively: \[ 3 \cdot \frac{1}{x^2} \cdot y^{3} \cdot \frac{1}{z^3} \] Combine the terms: \[ \frac{3 y^{3}}{x^{2} z^{3}} \]

Final Answer