1. **Stating the problem:** Simplify each expression by applying the laws of exponents in parts (c), (d), and (e). 2. **Part (c): Simplify \((u^{-1} v / v^{-1})^2\)** First, rewrite the expression separating numerator and denominator: $$\left(\frac{u^{-1} v}{v^{-1}}\right)^2 = \frac{(u^{-1} v)^2}{(v^{-1})^2}$$ Apply exponent rules inside numerator and denominator: $$= \frac{u^{-2} v^2}{v^{-2}}$$ Use the quotient rule for exponents on \(v\): $$= u^{-2} v^{2 - (-2)} = u^{-2} v^4$$ Rewrite negative exponent in denominator form: $$= \frac{v^4}{u^2}$$ 3. **Part (d): Simplify \((-2a^3 b^{-1})(5a^{-2} b^2)\)** Multiply coefficients and variables separately: $$= (-2) \cdot 5 \cdot a^{3 + (-2)} \cdot b^{-1 + 2}$$ Calculate exponents: $$= -10 \cdot a^{1} \cdot b^{1} = -10ab$$ 4. **Part (e): Simplify \((x^2 \sqrt{2})^4 (x^2 \sqrt{2})^{-4}\)** Use the product of powers property: $$= (x^2 \sqrt{2})^{4 + (-4)} = (x^2 \sqrt{2})^{0}$$ Any quantity raised to the zero power equals 1: $$= 1$$ **Final answers:** - (c) $$\frac{v^4}{u^2}$$ - (d) $$-10ab$$ - (e) $$1$$