what is the following quotient?\n\\frac{6 - 3(\\sqrt3{6})}{\\sqrt3{9}}\n\\bigcirc 2(\\sqrt3{3})-\\sqrt3{18}\n…

what is the following quotient?\n\\frac{6 - 3(\\sqrt3{6})}{\\sqrt3{9}}\n\\bigcirc 2(\\sqrt3{3})-\\sqrt3{18}\n\\bigcirc 2(\\sqrt3{3})-3(\\sqrt3{2})\n\\bigcirc 3(\\sqrt3{3})-\\sqrt3{18}\n\\bigcirc 3(\\sqrt3{3})-3(\\sqrt3{2})

what is the following quotient?\n\\frac{6 - 3(\\sqrt3{6})}{\\sqrt3{9}}\n\\bigcirc 2(\\sqrt3{3})-\\sqrt3{18}\n\\bigcirc 2(\\sqrt3{3})-3(\\sqrt3{2})\n\\bigcirc 3(\\sqrt3{3})-\\sqrt3{18}\n\\bigcirc 3(\\sqrt3{3})-3(\\sqrt3{2})

Answer

Explanation:

Step1: Split the fraction

Split (\frac{6 - 3\sqrt[3]{6}}{\sqrt[3]{9}}) into (\frac{6}{\sqrt[3]{9}}-\frac{3\sqrt[3]{6}}{\sqrt[3]{9}}).

Step2: Simplify each term

For (\frac{6}{\sqrt[3]{9}}), rewrite (6) as (2\times3) and use (\frac{a}{b^{\frac{m}{n}}}=a\times b^{-\frac{m}{n}}). (6 = 2\times3), (\sqrt[3]{9}=9^{\frac{1}{3}}=(3^{2})^{\frac{1}{3}} = 3^{\frac{2}{3}}). Then (\frac{6}{\sqrt[3]{9}}=\frac{2\times3}{3^{\frac{2}{3}}}=2\times3^{1-\frac{2}{3}}=2\sqrt[3]{3}). For (\frac{3\sqrt[3]{6}}{\sqrt[3]{9}}), use (\frac{\sqrt[3]{a}}{\sqrt[3]{b}}=\sqrt[3]{\frac{a}{b}}). (\frac{3\sqrt[3]{6}}{\sqrt[3]{9}}=3\sqrt[3]{\frac{6}{9}}=3\sqrt[3]{\frac{2}{3}}). Rewrite (\sqrt[3]{\frac{2}{3}}) as (\frac{\sqrt[3]{2}}{\sqrt[3]{3}}), and rationalize the denominator: (3\times\frac{\sqrt[3]{2}}{\sqrt[3]{3}}\times\frac{\sqrt[3]{3^{2}}}{\sqrt[3]{3^{2}}}=3\times\frac{\sqrt[3]{2\times3^{2}}}{3}=\sqrt[3]{18}).

Answer:

(2\sqrt[3]{3}-\sqrt[3]{18}) (corresponding to the first option (2(\sqrt[3]{3})-\sqrt[3]{18}))