Miscellaneous Exercise on Chapter 8

Find the coefficient of $x^5$ in the product ${(1+2x)}^6{(1-x)}^7$ using binomial theorem.

If $a$ and $b$ are distinct integers, prove that $a-b$ is a factor of $a^n-b^n,$ whenever $n$ is a positive integer.

Evaluate ${\left(\sqrt{3}+\sqrt{2}\right)}^6-{\left(\sqrt{3}-\sqrt{2}\right)}^6$.
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Evaluate ${\left(a^2+\sqrt{a^2-1}\right)}^4+{\left(a^2-\sqrt{a^2-1}\right)}^4.$

Find an approximations of ${\left(0.99\right)}^5$ using the first three terms of its expansion.

Find $n,$ if the ratio of the fifth term from the beginning to the fifth term from the end in the expansion of ${\left(\sqrt[4]{2}+\frac{1}{\sqrt[4]{3}}\right)}^n$ is $\sqrt{6}:1$.

Expand using Binomial theorem ${\left(1+\frac{x}{2}-\frac{2}{x}\right)}^4,x\ne 0.$

Find the expansion of ${\left(3x^2-2ax+3a^2\right)}^3$ using binomial theorem.