1. **State the problem:** Simplify fully the expression $$\left(\frac{9t^4 w^9}{18t^6 w^{10}}\right)^{-2}$$. 2. **Recall the rules:** - When dividing powers with the same base, subtract exponents: $$a^m / a^n = a^{m-n}$$. - When raising a power to another power, multiply exponents: $$(a^m)^n = a^{mn}$$. - Negative exponents mean reciprocal: $$a^{-n} = \frac{1}{a^n}$$. 3. **Simplify inside the parentheses first:** - Simplify the coefficient: $$\frac{9}{18} = \frac{1}{2}$$. - Simplify the $t$ terms: $$t^{4} / t^{6} = t^{4-6} = t^{-2}$$. - Simplify the $w$ terms: $$w^{9} / w^{10} = w^{9-10} = w^{-1}$$. So inside the parentheses we have: $$\frac{1}{2} t^{-2} w^{-1}$$ 4. **Apply the exponent $-2$ to the entire expression:** $$\left(\frac{1}{2} t^{-2} w^{-1}\right)^{-2} = \left(\frac{1}{2}\right)^{-2} (t^{-2})^{-2} (w^{-1})^{-2}$$ 5. **Calculate each part:** - $$\left(\frac{1}{2}\right)^{-2} = 2^{2} = 4$$ - $$(t^{-2})^{-2} = t^{(-2) \times (-2)} = t^{4}$$ - $$(w^{-1})^{-2} = w^{(-1) \times (-2)} = w^{2}$$ 6. **Combine all parts:** $$4 t^{4} w^{2}$$ **Final answer:** $$4 t^{4} w^{2}$$