1. **State the problem:** Solve the equation $$4^{x^2} = 8^x$$ for $x$. 2. **Rewrite the bases as powers of 2:** Since $4 = 2^2$ and $8 = 2^3$, rewrite the equation as: $$ (2^2)^{x^2} = (2^3)^x $$ 3. **Apply the power of a power rule:** $$ 2^{2x^2} = 2^{3x} $$ 4. **Set the exponents equal:** Since the bases are the same and nonzero, the exponents must be equal: $$ 2x^2 = 3x $$ 5. **Solve the quadratic equation:** Bring all terms to one side: $$ 2x^2 - 3x = 0 $$ Factor out $x$: $$ x(2x - 3) = 0 $$ 6. **Find the roots:** Set each factor equal to zero: - $x = 0$ - $2x - 3 = 0 \Rightarrow 2x = 3 \Rightarrow x = \frac{3}{2}$ 7. **Final answer:** $$ x = 0 \text{ or } x = \frac{3}{2} $$