1. **Problem:** Simplify and solve the equation $$2^{x+1} = 5 \cdot 2^{x-2}$$. 2. **Formula and rules:** Recall the properties of exponents: - $$a^{m+n} = a^m \cdot a^n$$ - $$a^{m-n} = \frac{a^m}{a^n}$$ - To solve equations with the same base, set the exponents equal if the bases are equal. 3. **Step-by-step solution:** - Rewrite both sides using exponent rules: $$2^{x+1} = 5 \cdot 2^{x-2}$$ - Divide both sides by $$2^{x-2}$$ to isolate terms: $$\frac{2^{x+1}}{2^{x-2}} = 5$$ - Simplify the left side using $$a^{m}/a^{n} = a^{m-n}$$: $$2^{(x+1)-(x-2)} = 5$$ $$2^{3} = 5$$ - Calculate $$2^3$$: $$8 = 5$$ 4. **Interpretation:** Since $$8 \neq 5$$, the equation has no solution. **Final answer:** No solution exists for $$x$$ in the equation $$2^{x+1} = 5 \cdot 2^{x-2}$$.