1. **State the problem:** Solve the equation $$4^{2x} = 8^{1-x}$$ for $x$. 2. **Rewrite bases as powers of 2:** Since $4 = 2^2$ and $8 = 2^3$, rewrite the equation as: $$ (2^2)^{2x} = (2^3)^{1-x} $$ 3. **Apply power of a power rule:** $$ 2^{4x} = 2^{3(1-x)} $$ 4. **Set exponents equal:** Since the bases are the same and nonzero, the exponents must be equal: $$ 4x = 3(1-x) $$ 5. **Solve for $x$:** $$ 4x = 3 - 3x $$ Add $3x$ to both sides: $$ 4x + 3x = 3 $$ $$ 7x = 3 $$ Divide both sides by 7: $$ x = \frac{3}{7} $$ **Final answer:** $$ x = \frac{3}{7} $$