1. **State the problem:** Evaluate the cube root of -81, i.e., find $\sqrt[3]{-81}$. 2. **Recall the formula and rules:** The cube root of a number $a$ is a number $b$ such that $b^3 = a$. For negative numbers, cube roots are defined because an odd power of a negative number is negative. 3. **Evaluate the cube root:** We want to find $b$ such that $b^3 = -81$. 4. **Express 81 as a power:** $81 = 3^4$. 5. **Rewrite the cube root:** $$\sqrt[3]{-81} = \sqrt[3]{-1 \times 3^4} = \sqrt[3]{-1} \times \sqrt[3]{3^4}$$ 6. **Evaluate each part:** $$\sqrt[3]{-1} = -1$$ $$\sqrt[3]{3^4} = 3^{\frac{4}{3}}$$ 7. **Combine:** $$\sqrt[3]{-81} = -1 \times 3^{\frac{4}{3}} = -3^{\frac{4}{3}}$$ 8. **Simplify the exponent:** $$3^{\frac{4}{3}} = 3^{1 + \frac{1}{3}} = 3^1 \times 3^{\frac{1}{3}} = 3 \times \sqrt[3]{3}$$ 9. **Final answer:** $$\sqrt[3]{-81} = -3 \times \sqrt[3]{3}$$ This is the exact form. Numerically, $\sqrt[3]{3} \approx 1.442$, so $$\sqrt[3]{-81} \approx -3 \times 1.442 = -4.326$$