1. **Problem:** Write the expression $(2^3)^4$ as a power with base 2. 2. **Formula:** When raising a power to another power, multiply the exponents: $\left(a^m\right)^n = a^{m \cdot n}$. 3. **Solution:** $$ (2^3)^4 = 2^{3 \cdot 4} = 2^{12} $$ 4. **Explanation:** We multiply the exponents 3 and 4 to get 12, so the expression simplifies to $2^{12}$. 1. **Problem:** Write the expression $2\sqrt{2}^3$ as a power with base 2. 2. **Formula:** Recall that $\sqrt{2} = 2^{1/2}$, so $\sqrt{2}^3 = \left(2^{1/2}\right)^3 = 2^{3/2}$. 3. **Solution:** $$ 2 \sqrt{2}^3 = 2 \cdot 2^{3/2} = 2^{1} \cdot 2^{3/2} = 2^{1 + \frac{3}{2}} = 2^{\frac{2}{2} + \frac{3}{2}} = 2^{\frac{5}{2}} $$ 4. **Explanation:** We express the square root as a fractional exponent and then add exponents when multiplying powers with the same base. Final answers: - a) $2^{12}$ - b) $2^{\frac{5}{2}}$