Evaluate the following: ( 3+ 2)^6-( 3- 2)^6 - Vidyadip

Question

Evaluate the following:

$(\sqrt3+\sqrt2)^6-(\sqrt3-\sqrt2)^6$

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Answer

$(\sqrt3+\sqrt2)^6-(\sqrt3-\sqrt2)^6$

$=2\big[{^6\text{C}}_1(\sqrt3)^5(\sqrt2)+{^6\text{C}}_3(\sqrt3)^3(\sqrt2)^3+{^6\text{C}}_5(\sqrt3)(\sqrt2)^5\big]$

$=2\big[6\times\sqrt6\times9+20\times3\sqrt3\times2\sqrt2+6\times\sqrt3\times4\sqrt2\big]$

$=2\big[54\sqrt6+120\sqrt6+24\sqrt6\big]$

$=2\big[198\sqrt6\big]$

$=396\sqrt6$

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