1. **State the problem:** Solve the equation $16^x = 8^{2x+1}$ for $x$. 2. **Recall the formula and rules:** We can express both sides with the same base to compare exponents. Note that $16 = 2^4$ and $8 = 2^3$. 3. **Rewrite the equation using base 2:** $$16^x = (2^4)^x = 2^{4x}$$ $$8^{2x+1} = (2^3)^{2x+1} = 2^{3(2x+1)} = 2^{6x+3}$$ 4. **Set the exponents equal since bases are the same:** $$4x = 6x + 3$$ 5. **Solve for $x$:** $$4x - 6x = 3$$ $$-2x = 3$$ $$x = -\frac{3}{2}$$ 6. **Final answer:** $$x = -\frac{3}{2}$$