1. **State the problem:** Solve the equation $$2^{x+7} - 2^{x+6} = 1$$ for $x$. 2. **Recall the properties of exponents:** - $a^{m+n} = a^m \cdot a^n$ - We can factor expressions with common bases and exponents. 3. **Rewrite the terms:** $$2^{x+7} = 2^x \cdot 2^7 = 2^x \cdot 128$$ $$2^{x+6} = 2^x \cdot 2^6 = 2^x \cdot 64$$ 4. **Substitute back into the equation:** $$2^x \cdot 128 - 2^x \cdot 64 = 1$$ 5. **Factor out $2^x$:** $$2^x (128 - 64) = 1$$ $$2^x \cdot 64 = 1$$ 6. **Divide both sides by 64:** $$2^x = \frac{1}{64}$$ 7. **Express $\frac{1}{64}$ as a power of 2:** Since $64 = 2^6$, then $$\frac{1}{64} = 2^{-6}$$ 8. **Set the exponents equal:** $$2^x = 2^{-6} \implies x = -6$$ **Final answer:** $$x = -6$$