Simplify each expression (a) \( \left(\frac{2}{y}\right)^{3}\left(\frac{3}{y}\right)^{2} \) (b) \( \left(\frac{b^{-4}}{a^{-4}}\right)\left(\frac{b}{a}\right)^{4} \) \( \square \)

Let's simplify each of the given expressions step by step. --- ### Problem (a) \[ \left(\frac{2}{y}\right)^{3}\left(\frac{3}{y}\right)^{2} \] **Step 1:** Apply the power to both the numerator and the denominator for each fraction. \[ \left(\frac{2}{y}\right)^{3} = \frac{2^{3}}{y^{3}} = \frac{8}{y^{3}} \] \[ \left(\frac{3}{y}\right)^{2} = \frac{3^{2}}{y^{2}} = \frac{9}{y^{2}} \] **Step 2:** Multiply the two results together. \[ \frac{8}{y^{3}} \times \frac{9}{y^{2}} = \frac{8 \times 9}{y^{3} \times y^{2}} = \frac{72}{y^{5}} \] **Final Answer:** \[ \left(\frac{2}{y}\right)^{3}\left(\frac{3}{y}\right)^{2} = \frac{72}{y^{5}} \] --- ### Problem (b) \[ \left(\frac{b^{-4}}{a^{-4}}\right)\left(\frac{b}{a}\right)^{4} \] **Step 1:** Simplify the first fraction by using the property \( \frac{x^{-n}}{y^{-n}} = \left(\frac{y}{x}\right)^n \). \[ \frac{b^{-4}}{a^{-4}} = \left(\frac{a}{b}\right)^{4} \] **Step 2:** Rewrite the entire expression with the simplification from Step 1. \[ \left(\frac{a}{b}\right)^{4} \times \left(\frac{b}{a}\right)^{4} \] **Step 3:** Use the property of exponents \( x^{n} \times x^{-n} = x^{0} = 1 \). \[ \left(\frac{a}{b} \times \frac{b}{a}\right)^{4} = \left(1\right)^{4} = 1 \] **Final Answer:** \[ \left(\frac{b^{-4}}{a^{-4}}\right)\left(\frac{b}{a}\right)^{4} = 1 \] --- If you have any further questions or need additional explanations, feel free to ask!