1. **State the problem:** Solve the equation $$2^{x-1} + 2^{x+1} = 40$$ for $x$. 2. **Recall the properties of exponents:** - $2^{x-1} = \frac{2^x}{2}$ - $2^{x+1} = 2 \cdot 2^x$ 3. **Rewrite the equation using these properties:** $$\frac{2^x}{2} + 2 \cdot 2^x = 40$$ 4. **Let $y = 2^x$ to simplify:** $$\frac{y}{2} + 2y = 40$$ 5. **Combine like terms:** $$\frac{y}{2} + 2y = \frac{y}{2} + \frac{4y}{2} = \frac{5y}{2}$$ 6. **Set up the equation:** $$\frac{5y}{2} = 40$$ 7. **Multiply both sides by 2 to clear the denominator:** $$5y = 80$$ 8. **Divide both sides by 5:** $$y = \cancel{\frac{5y}{5}} = \cancel{\frac{80}{5}} = 16$$ 9. **Recall that $y = 2^x$, so:** $$2^x = 16$$ 10. **Express 16 as a power of 2:** $$16 = 2^4$$ 11. **Set exponents equal:** $$x = 4$$ **Final answer:** $$x = 4$$