Simplify: (3x^5y^-7)^2 (9x^10)/(y^14) (6x^10)/(y^14) (1)/(9x^10)y^(14) 9(x^7)/(y^9)

To simplify the expression \(\left(3 x^{5} y^{-7}\right)^{2}\), we need to apply the power of a product rule, which states that \((a \cdot b)^n = a^n \cdot b^n\).<br /><br />Let's break it down:<br /><br />1. Apply the exponent to each part inside the parentheses:<br /> \[<br /> (3 x^{5} y^{-7})^{2} = 3^2 \cdot (x^{5})^2 \cdot (y^{-7})^2<br /> \]<br /><br />2. Calculate each component:<br /> - \(3^2 = 9\)<br /> - \((x^{5})^2 = x^{10}\) (using the power of a power rule: \((a^m)^n = a^{m \cdot n}\))<br /> - \((y^{-7})^2 = y^{-14}\)<br /><br />3. Combine these results:<br /> \[<br /> 9 \cdot x^{10} \cdot y^{-14} = \frac{9 x^{10}}{y^{14}}<br /> \]<br /><br />Therefore, the simplified form of \(\left(3 x^{5} y^{-7}\right)^{2}\) is \(\frac{9 x^{10}}{y^{14}}\).