1. **State the problem:** Simplify the expression $$\frac{x^2 - 36}{x^2 - 24x + 144} \cdot \left(\frac{x+6}{x-12}\right)^{-2}$$. 2. **Recall important formulas and rules:** - Difference of squares: $$a^2 - b^2 = (a-b)(a+b)$$. - Perfect square trinomial: $$a^2 - 2ab + b^2 = (a-b)^2$$. - Negative exponent rule: $$a^{-n} = \frac{1}{a^n}$$. 3. **Factor the numerator and denominator of the first fraction:** - Numerator: $$x^2 - 36 = (x-6)(x+6)$$ (difference of squares). - Denominator: $$x^2 - 24x + 144 = (x-12)^2$$ (perfect square trinomial). 4. **Rewrite the expression with factored forms:** $$\frac{(x-6)(x+6)}{(x-12)^2} \cdot \left(\frac{x+6}{x-12}\right)^{-2}$$ 5. **Apply the negative exponent rule:** $$\left(\frac{x+6}{x-12}\right)^{-2} = \left(\frac{x-12}{x+6}\right)^2 = \frac{(x-12)^2}{(x+6)^2}$$ 6. **Substitute back:** $$\frac{(x-6)(x+6)}{(x-12)^2} \cdot \frac{(x-12)^2}{(x+6)^2}$$ 7. **Multiply the fractions:** $$\frac{(x-6)(x+6)(x-12)^2}{(x-12)^2 (x+6)^2}$$ 8. **Cancel common factors:** - Cancel $$(x-12)^2$$ from numerator and denominator. - Cancel one $$(x+6)$$ from numerator and denominator. Intermediate step with cancellation: $$\frac{(x-6)\cancel{(x+6)}\cancel{(x-12)^2}}{\cancel{(x-12)^2} (x+6)\cancel{(x+6)}} = \frac{(x-6)}{(x+6)}$$ 9. **Final simplified expression:** $$\boxed{\frac{x-6}{x+6}}$$ This is the simplified form of the original expression, valid for all $x$ except where denominators are zero (i.e., $x \neq 12$ and $x \neq -6$).