1. The problem is to simplify the expression $(2m^2)^{-1} / m^2$. 2. Recall the rule for negative exponents: $a^{-n} = \frac{1}{a^n}$. 3. Simplify $(2m^2)^{-1}$ using the negative exponent rule: $$ (2m^2)^{-1} = \frac{1}{2m^2} $$ 4. Now divide by $m^2$: $$ \frac{1}{2m^2} \div m^2 = \frac{1}{2m^2} \times \frac{1}{m^2} = \frac{1}{2m^{2+2}} = \frac{1}{2m^4} $$ 5. Expressing with negative exponents: $$ \frac{1}{2m^4} = \frac{1}{2} m^{-4} = \frac{1}{2} m^{-4} $$ 6. The original expression simplifies to: $$ \frac{1}{2} m^{-4} $$ 7. The user's answer was $-2 m^{-4}$ which is incorrect because the coefficient should be $\frac{1}{2}$ and positive. Final answer: $$ \frac{1}{2} m^{-4} $$