Using binomial theorem write down the expansions of the following… - Vidyadip

Question

Using binomial theorem write down the expansions of the following:
$(^3\sqrt{\text{x}}-^3\sqrt{\text{a}})^6$

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Answer

$(\sqrt[3]{\text{x}}-\sqrt[3]{\text{a}})^6$
$=\Big(\frac{6}{0}\Big)\big(\sqrt[3]{\text{x}}\big)^6\big(-\sqrt[3]{\text{a}}\big)^0+\Big(\frac{6}{1}\Big)\big(\sqrt[3]{\text{x}}\big)^5\big({-}\sqrt[3]{\text{a}}\big)^1+\Big(\frac{6}{2}\Big)\big(-\sqrt[3]{\text{x}}\big)^4\big(\sqrt[3]{\text{a}}\big)^2\\\Big(\frac{6}{3}\Big)\big(\sqrt[3]{\text{x}}\big)^3\big(-\sqrt[3]{\text{a}}\big)^0+\Big(\frac{6}{4}\Big)\big(\sqrt[3]{\text{x}}\big)^2\big(-\sqrt[3]{\text{a}}\big)^4+\Big(\frac{6}{5}\Big)\big(\sqrt[3]{\text{x}}\big)^1\big(-\sqrt[3]{\text{a}}\big)^5\\\Big(\frac{6}{6}\Big)\big(\sqrt[3]{\text{x}}\big)^0\big(-\sqrt[3]{\text{a}}\big)^6$
$=\text{x}^2-6\text{x}^\frac{5}{3}\text{a}^\frac{2}{3}+15\text{x}^\frac{4}{3}\text{a}^\frac{2}{3}-20\text{ax}+15\text{x}^\frac{2}{3}\text{a}^\frac{4}{3}-6\text{x}^\frac{1}{3}\text{a}^\frac{5}{3}+\text{a}^2$

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