1. **State the problem:** Evaluate the expressions: 1) $\left(\sqrt[3]{64}\right)^2$ 2) $(-1000)^{\frac{1}{3}}$ 2. **Recall the formulas and rules:** - The cube root of a number $a$ is $a^{\frac{1}{3}}$. - When raising a power to another power, multiply the exponents: $\left(a^m\right)^n = a^{mn}$. - Cube roots of negative numbers are real and equal to the negative of the cube root of the positive number. 3. **Evaluate the first expression:** $\left(\sqrt[3]{64}\right)^2 = \left(64^{\frac{1}{3}}\right)^2 = 64^{\frac{1}{3} \times 2} = 64^{\frac{2}{3}}$ Since $64 = 4^3$, then: $$64^{\frac{2}{3}} = \left(4^3\right)^{\frac{2}{3}} = 4^{3 \times \frac{2}{3}} = 4^2 = 16$$ 4. **Evaluate the second expression:** $(-1000)^{\frac{1}{3}} = \sqrt[3]{-1000}$ Since $1000 = 10^3$, then: $$\sqrt[3]{-1000} = -\sqrt[3]{1000} = -10$$ **Final answers:** 1) $16$ 2) $-10$