Question: 9. Simplify: $$\sqrt[6]{9x^{8}y^{12}}$$
(A) $$x^{6}y^{12} \sqrt[6]{9x^{2}}$$
(B) $$xy^{23} \sqrt[3]{3x}$$
(C) $$x^{6}y^{12} \sqrt[3]{3x}$$
(D) $$xy^{23} \sqrt[6]{9x^{2}}$$
1. **State the problem:** Simplify the expression $$\sqrt[6]{9x^{8}y^{12}}$$.
2. **Recall the rule for radicals:** $$\sqrt[n]{a^{m}} = a^{\frac{m}{n}}$$.
3. **Rewrite the expression using fractional exponents:**
$$\sqrt[6]{9x^{8}y^{12}} = (9)^{\frac{1}{6}} \cdot x^{\frac{8}{6}} \cdot y^{\frac{12}{6}}$$
4. **Simplify the exponents:**
$$x^{\frac{8}{6}} = x^{\frac{4}{3}}, \quad y^{\frac{12}{6}} = y^{2}$$
5. **Rewrite the constant 9 as a power:**
$$9 = 3^{2}$$
6. **Apply the fractional exponent to 9:**
$$9^{\frac{1}{6}} = (3^{2})^{\frac{1}{6}} = 3^{\frac{2}{6}} = 3^{\frac{1}{3}} = \sqrt[3]{3}$$
7. **Rewrite the expression:**
$$\sqrt[6]{9x^{8}y^{12}} = \sqrt[3]{3} \cdot x^{\frac{4}{3}} \cdot y^{2}$$
8. **Separate the integer and fractional parts of the exponent for x:**
$$x^{\frac{4}{3}} = x^{1 + \frac{1}{3}} = x \cdot x^{\frac{1}{3}} = x \sqrt[3]{x}$$
9. **Combine all parts:**
$$\sqrt[6]{9x^{8}y^{12}} = y^{2} \cdot x \cdot \sqrt[3]{3} \cdot \sqrt[3]{x} = x y^{2} \sqrt[3]{3x}$$
10. **Final simplified form:**
$$x y^{2} \sqrt[3]{3x}$$
11. **Match with the options:** This corresponds to option (B).