Using binomial theorem write down the expansions of the following…

Question

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

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Answer

The expansion of $(x + y)^n $ has $n + 1$ terms so the expansion of $\bigg(\sqrt\frac{\text{x}}{\text{a}}-\sqrt\frac{\text{a}}{\text{x}}\bigg)^6$ has $7$ term Using binomial theorem to expand, we get
$\Big(\sqrt\frac{\text{x}}{\text{a}}-\sqrt\frac{\text{a}}{\text{x}}\Big)^6={^6\text{C}}_0\Big(\sqrt\frac{\text{x}}{\text{a}}\Big)^6\Big(\sqrt\frac{\text{a}}{\text{x}}\Big)^0\\-{^6\text{C}}_1\Big(\sqrt\frac{\text{x}}{\text{a}}\Big)^5\Big(\sqrt\frac{\text{a}}{\text{x}}\Big)^1+{^6\text{C}}_2\Big(\sqrt\frac{\text{x}}{\text{a}}\Big)^4\Big(\sqrt\frac{\text{a}}{\text{x}}\Big)^2-{^6\text{C}}_3\Big(\sqrt\frac{\text{x}}{\text{a}}\Big)^3\Big(\sqrt\frac{\text{a}}{\text{x}}\Big)^3\\+{^6\text{C}}_4\Big(\sqrt\frac{\text{x}}{\text{a}}\Big)^2\Big(\sqrt\frac{\text{a}}{\text{x}}\Big)^4-{^6\text{C}}_5\Big(\sqrt\frac{\text{x}}{\text{a}}\Big)\Big(\sqrt\frac{\text{a}}{\text{x}}\Big)^5+{^6\text{C}}_6\Big(\sqrt\frac{\text{x}}{\text{a}}\Big)^0\Big(\sqrt\frac{\text{a}}{\text{x}}\Big)^6$
$=\Big(\frac{\text{x}}{\text{a}}\Big)^{\frac{1}{2}\times5}-6\Big(\frac{\text{x}}{\text{a}}\Big)^{\frac{1}{2}\times5}\Big(\frac{\text{a}}{\text{x}}\Big)^{\frac{1}{2}}+15\Big(\frac{\text{x}}{\text{a}}\Big)^{\frac{1}{2}\times4}\Big(\sqrt\frac{\text{a}}{\text{x}}\Big)^{2\times{\frac{1}{2}}}\\-20\Big(\frac{\text{x}}{\text{a}}\Big)^{3\times{\frac{1}{2}}}\Big(\frac{\text{a}}{\text{x}}\Big)^{3\times\frac{1}{2}}+15\Big(\frac{\text{x}}{\text{a}}\Big)^{2\times\frac{1}{2}}\Big(\frac{\text{a}}{\text{x}}\Big)^{4\times\frac{1}{2}}-6\Big(\frac{\text{x}}{\text{a}}\Big)^\frac{1}{2}\Big(\frac{\text{a}}{\text{x}}\Big)^{5\times\frac{1}{2}}+\Big(\frac{\text{a}}{\text{x}}\Big)^{6\times\frac{1}{2}}$
$=\frac{\text{x}^3}{\text{a}^3}-6\frac{\frac{5\ 1}{\text{x}^2\ 2}}{\frac{5}{\text{a}^2}-\frac{1}{2}}+15\times\frac{\text{x}^2\times\text{a}}{\text{a}^2\times\text{x}}-20\times\frac{\frac{3\ 3}{\text{x}^2\ 2}}{\frac{3}{\text{a}^2}-\frac{3}{2}}\\+15\times\frac{\text{x}}{\text{a}}\times\frac{\text{a}^2}{\text{x}^2}-6\times\frac{\frac{1\ 5}{\text{x}^2\ 2}}{\frac{1}{\text{a}^2}-\frac{ 5}{2}}+\frac{\text{a}^3}{\text{x}^3}$
$=\frac{\text{x}^3}{\text{a}^3}-\frac{6\text{x}^2}{\text{a}^2}+\frac{15\text{x}}{\text{a}}-20+\frac{15\text{a}}{\text{x}}-\frac{6\text{a}^2}{\text{x}^2}+\frac{\text{a}^3}{\text{x}^3}$

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