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Algebra (Olympiad): Product Expression

JEE Maths question with a full step-by-step solution.

Question
The product of the expression
(5+6+7)(−5+6+7)(5−6+7)(5+6−7)(\sqrt5+\sqrt6+\sqrt7)(-\sqrt5+\sqrt6+\sqrt7)(\sqrt5-\sqrt6+\sqrt7)(\sqrt5+\sqrt6-\sqrt7)
Solution
Answer: 104
Step 1: Group the four factors into two conjugate pairs:
[(6+7)+5][(6+7)−5]×[5+(7−6)][5−(7−6)].\big[(\sqrt6+\sqrt7)+\sqrt5\big]\big[(\sqrt6+\sqrt7)-\sqrt5\big] \times \big[\sqrt5+(\sqrt7-\sqrt6)\big]\big[\sqrt5-(\sqrt7-\sqrt6)\big].
 Step 2: Apply (X+Y)(X−Y)=X2−Y2(X+Y)(X-Y) = X^2 - Y^2 to each pair:
=[(6+7)2−5][5−(7−6)2].= \big[(\sqrt6+\sqrt7)^2 - 5\big]\big[5 - (\sqrt7-\sqrt6)^2\big].
 Step 3: Expand. Since (6+7)2=13+242(\sqrt6+\sqrt7)^2 = 13 + 2\sqrt{42} and (7−6)2=13−242(\sqrt7-\sqrt6)^2 = 13 - 2\sqrt{42},
=[(13+242)−5][5−(13−242)]=(8+242)(−8+242).= \big[(13 + 2\sqrt{42}) - 5\big]\big[5 - (13 - 2\sqrt{42})\big] = (8 + 2\sqrt{42})(-8 + 2\sqrt{42}).
 Step 4: This is a difference of squares:
=(242)2−82=4(42)−64=168−64=104.= (2\sqrt{42})^2 - 8^2 = 4(42) - 64 = 168 - 64 = 104.
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