1. **State the problem:** Solve the equation $$2^{-115x - 15} = (2^9)^{-13x + 14}$$ for $x$. 2. **Use the property of exponents:** Recall that $(a^m)^n = a^{mn}$. 3. **Rewrite the right side:** $$ (2^9)^{-13x + 14} = 2^{9(-13x + 14)} = 2^{-117x + 126} $$ 4. **Set the exponents equal:** Since the bases are the same and nonzero, the exponents must be equal: $$ -115x - 15 = -117x + 126 $$ 5. **Solve for $x$:** $$ -115x - 15 = -117x + 126 $$ Add $117x$ to both sides: $$ 2x - 15 = 126 $$ Add $15$ to both sides: $$ 2x = 141 $$ Divide both sides by $2$: $$ x = \frac{141}{2} = 70.5 $$ **Final answer:** $$ x = 70.5 $$