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Question
Find \[\left( a + b \right)^4 - \left( a - b \right)^4\] . Hence, evaluate \[\left( \sqrt{3} + \sqrt{2} \right)^4 - \left( \sqrt{3} - \sqrt{2} \right)^4\] .
Solution
The expression \[(a + b )^4 - (a - b )^4\] can be written as
\[(a + b )^4 - (a - b )^4 = 2[ ^{4}{}{C}_1 a^3 b^1 + ^{4}{}{C}_3 a^1 b^3 ] \]
\[ = 2[4 a^3 b + 4a b^3 ]\]
\[ = 8( a^3 b + a b^3 )\]
\[\text{ Putting a } = \sqrt{3} \text{ and } b = \sqrt{2}, \text{ we get } : \]
\[ (\sqrt{3} + \sqrt{2} )^4 - (\sqrt{3} - \sqrt{2} )^4 = 8[(\sqrt{3} )^3 \times \sqrt{2} + \sqrt{3} \times (\sqrt{2} )^3 ]\]
\[ = 8(3\sqrt{6} + 2\sqrt{6})\]
\[ = 40\sqrt{6}\]
\[\therefore (\sqrt{3} + \sqrt{2} )^4 - (\sqrt{3} - \sqrt{2} )^4 = 40\sqrt{6}\]
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