Evaluate the following: (3+ 2)^5-(3- 2)^5 - Vidyadip

Question

Evaluate the following:
$(3+\sqrt2)^5-(3-\sqrt2)^5$

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Answer

$(3+\sqrt2)^5-(3-\sqrt2)^5$
$=2\big[{^5\text{C}}_1(3)^4(\sqrt2)^1+{^5\text{C}}_3(3)^2(\sqrt2)^3+{^5\text{C}}_5(\sqrt3)^5\big]$
$=2\big[5\times81\times\sqrt2+10\times9\times2\sqrt2+4\sqrt2\big]$
$=2\big[405\sqrt2+180\sqrt2+4\sqrt2\big]$
$=2\big[589\sqrt2\big]$
$=1178\sqrt2$

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