Describe how you would simp expression. \( \left(\frac{20 x^{5} y^{2}}{5 x^{-3} y^{7}}\right)^{-3}, x \neq 0, y \neq 0 \) DONE

To simplify the expression \( \left(\frac{20 x^{5} y^{2}}{5 x^{-3} y^{7}}\right)^{-3} \), start by simplifying the fraction inside the parentheses: 1. Divide the coefficients: \( \frac{20}{5} = 4 \). 2. Simplify the variables using the laws of exponents: - For \( x \): \( x^{5} \div x^{-3} = x^{5 - (-3)} = x^{5 + 3} = x^{8} \). - For \( y \): \( y^{2} \div y^{7} = y^{2 - 7} = y^{-5} \). Now, rewrite the fraction: \[ \frac{20 x^{5} y^{2}}{5 x^{-3} y^{7}} = 4 x^{8} y^{-5} \] Now raise it to the power of -3: \[ \left(4 x^{8} y^{-5}\right)^{-3} = 4^{-3} (x^{8})^{-3} (y^{-5})^{-3} \] Calculate each part: - \( 4^{-3} = \frac{1}{64} \) - \( (x^{8})^{-3} = x^{-24} \) - \( (y^{-5})^{-3} = y^{15} \) Putting it all together gives: \[ \left(\frac{20 x^{5} y^{2}}{5 x^{-3} y^{7}}\right)^{-3} = \frac{y^{15}}{64 x^{24}} \] Thus, the simplified expression is: \[ \frac{y^{15}}{64 x^{24}} \]