A) \[{{6}^{1/3}}\left[ 1+\frac{x}{6}+\frac{2{{x}^{2}}}{{{6}^{2}}}+.... \right]\]
B) \[{{6}^{-1/3}}\left[ 1+\frac{x}{6}+\frac{2{{x}^{2}}}{{{6}^{2}}}+.... \right]\]
C) \[{{6}^{1/3}}\left[ 1-\frac{x}{6}+\frac{2{{x}^{2}}}{{{6}^{2}}}-.... \right]\]
D) \[{{6}^{-1/3}}\left[ 1-\frac{x}{6}+\frac{2{{x}^{2}}}{{{6}^{2}}}-.... \right]\]
Correct Answer: B
Solution :
\[\frac{1}{{{(6-3x)}^{1/3}}}={{(6-3x)}^{-1/3}}={{6}^{-1/3}}{{\left[ 1-\frac{x}{2} \right]}^{-1/3}}\] \[={{6}^{-1/3}}\left[ 1+\left( -\frac{1}{3} \right)\,\left( -\frac{x}{2} \right)x+\frac{\left( -\frac{1}{3} \right)\,\left( -\frac{4}{3} \right)}{2.1}{{\left( -\frac{x}{2} \right)}^{2}}+.... \right]\] \[={{6}^{-1/3}}\left[ 1+\frac{x}{6}+\frac{2{{x}^{2}}}{{{6}^{^{2}}}}+.... \right]\]
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