Hello!
So, this is quite the complex question, and here are the following steps:
What is the quotient of [tex] \frac{6-3\sqrt[3]{6}}{\sqrt[3]{9}} [/tex]?
[tex] \frac{(6 - 3\sqrt[3]{6})\sqrt[3]{9^{2}}}{9} [/tex] (rationalize the denominator)
[tex] \frac{3(2 -\sqrt[3]{6})\sqrt[3]{9^{2}}}{9} [/tex] (factor 3 from the expression)
[tex] \frac{(2-\sqrt[3]{6})\sqrt[3]{9^{2}}}{3} [/tex] (reduce the fraction with 3)
[tex] \frac{2\sqrt[3]{9^{2}}-\sqrt[3]{9^{2}}}{3} [/tex] (distributive property)
[tex] \frac{2\sqrt[3]{81}-\sqrt[3]{486}}{3} [/tex] (simplify 3 · 9²)
[tex] \frac{6\sqrt[3]{3}-3\sqrt[3]{18}}{3} [/tex] (simplify the radical)
[tex] \frac{3(2\sqrt[3]{3}-3\sqrt[3]{18})}{3} [/tex] (factor 3 from the expression)
[tex] 2\sqrt[3]{3}-\sqrt[3]{18} [/tex] (reduce the fraction)
The answer, is simply, choice A, [tex] 2\sqrt[3]{3} -\sqrt[3]{18} [/tex] ≈ 0.263758.
