Obtain the value of (3+1)^6+(3-1)^6 using binomial expansion meth… - Vidyadip

Question

Obtain the value of $(\sqrt{3}+1)^6+(\sqrt{3}-1)^6$ using binomial expansion method.

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Answer

$(\sqrt{3}+1)^6+(\sqrt{3}-1)^6$
$=\left(6 C_0(\sqrt{3})^6 \times 1^0 +6 C_1(\sqrt{3})^5 \times 1^1 +6 C_2(\sqrt{3})^4 \times 1^2 6 C_3(\sqrt{3})^3 \times 1^3 6 C_4(\sqrt{3})^2 \times 1^4 6 C_5(\sqrt{3})^1 \times 1^5 6 C_6(\sqrt{3})^0 \times 1^6 \right)$
$+\left( 6 C_0(\sqrt{3})^6 \times 1^0 \\ -6 C_1(\sqrt{3})^5 \times 1^1 +6 C_2(\sqrt{3})^4 \times 1^2 -6 C_3(\sqrt{3})^3 \times 1^3 +6 C_4(\sqrt{3})^2 \times 1^4 -6 C_5(\sqrt{3})^1 \times 1^5 +6 C_6(\sqrt{3})^0 \times 1^6 \right)$
$=(\sqrt{3})^6+6(\sqrt{3})^5+15(\sqrt{3})^4+20(\sqrt{3})^3+15(\sqrt{3})^2+6(\sqrt{3})+1$$+(\sqrt{3})^6-6(\sqrt{3})^5+15(\sqrt{3})^4-20(\sqrt{3})^3+15(\sqrt{3})^2-6(\sqrt{3})+1$
$=2(3)^3+30(3)^2+30(3)+2 $
$=2 \times 27+30 \times 9+90+2$
$=54+270+92$
$=416$

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