Solve (1+3)^4+(1-3)^4 by using binomial theorem. - Vidyadip

Question

Solve $(1+\sqrt{3})^4+(1-\sqrt{3})^4$ by using binomial theorem.

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Answer

By binomial theorem :
$ \begin{aligned} (1+\sqrt{3})^4=1+{ }^4 C_1(\sqrt{3})^1+{ }^4 C_2 & (\sqrt{3})^2+{ }^4 C_3(\sqrt{3})^3 +{ }^4 C_4(\sqrt{3})^4 \ldots\ldots (1) \end{aligned} $
$\begin{array}{r}\text {and}\quad (1-\sqrt{3})^4=1-{ }^4 C _1(\sqrt{3})^1+{ }^4 C _2(\sqrt{3})^2-{ }^4 C _3 (\sqrt{3})^3+{ }^4 C _4(\sqrt{3})^4\ldots\ldots (2)\end{array}$
Adding equations (1) and (2),
$\begin{aligned}(1+\sqrt{3})^4+(1-\sqrt{3})^4 & =2\left[1+{ }^4 C _2(\sqrt{3})^2+{ }^4 C _4(\sqrt{3})^4\right] \\ & =2[1+6 \times 3+1 \times 9] \\ & =2(1+18+9)=2 \times 28 \\ & =56 \end{aligned}$

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