1. **State the problem:** Simplify the expression $$\left(\frac{3}{2} u^{-2}\right)^2$$. 2. **Recall the exponent rules:** When raising a product to a power, apply the power to each factor: $$\left(ab\right)^n = a^n b^n$$. 3. **Apply the power to each factor:** $$\left(\frac{3}{2}\right)^2 \cdot \left(u^{-2}\right)^2$$ 4. **Simplify each part:** $$\left(\frac{3}{2}\right)^2 = \frac{3^2}{2^2} = \frac{9}{4}$$ $$\left(u^{-2}\right)^2 = u^{-2 \times 2} = u^{-4}$$ 5. **Combine the results:** $$\frac{9}{4} u^{-4}$$ 6. **Rewrite with positive exponents:** $$\frac{9}{4} \cdot \frac{1}{u^4} = \frac{9}{4 u^4}$$ **Final answer:** $$\frac{9}{4 u^4}$$