Simplify Cube Root Ee5705

1. **State the problem:** Simplify the cube root $$\sqrt[3]{-8x^{12}y^{9}}$$ assuming all variables represent positive real numbers. 2. **Recall the cube root properties:** - The cube root of a product is the product of the cube roots: $$\sqrt[3]{abc} = \sqrt[3]{a} \cdot \sqrt[3]{b} \cdot \sqrt[3]{c}$$. - Cube roots of perfect cubes simplify nicely: $$\sqrt[3]{a^3} = a$$. - Since variables are positive, $$\sqrt[3]{x^{3k}} = x^k$$. - Cube root of a negative number is negative: $$\sqrt[3]{-a} = -\sqrt[3]{a}$$. 3. **Break down the expression:** $$\sqrt[3]{-8x^{12}y^{9}} = \sqrt[3]{-8} \cdot \sqrt[3]{x^{12}} \cdot \sqrt[3]{y^{9}}$$ 4. **Simplify each part:** - $$\sqrt[3]{-8} = -2$$ because $$-8 = (-2)^3$$. - $$\sqrt[3]{x^{12}} = x^{\frac{12}{3}} = x^4$$. - $$\sqrt[3]{y^{9}} = y^{\frac{9}{3}} = y^3$$. 5. **Combine the simplified parts:** $$-2 \cdot x^4 \cdot y^3 = -2x^4y^3$$ 6. **Final answer:** $$\sqrt[3]{-8x^{12}y^{9}} = -2x^4y^3$$ Since the cube root of a negative number is defined for real numbers, choice B is incorrect. **Answer:** Choice A: $$\sqrt[3]{-8x^{12}y^{9}} = -2x^4y^3$$