1986 AIME Problems/Problem 2

Problem

Evaluate the product $(\sqrt 5+\sqrt6+\sqrt7)(-\sqrt 5+\sqrt6+\sqrt7)(\sqrt 5-\sqrt6+\sqrt7)(\sqrt 5+\sqrt6-\sqrt7)$ .

Simplify by repeated application of the difference of squares.

$\left((\sqrt{6} + \sqrt{7})^2 - \sqrt{5}^2\right)\left(\sqrt{5}^2 - (\sqrt{6} - \sqrt{7})^2\right)$

$= (13 + 2\sqrt{42} - 5)(5 - (13 - 2\sqrt{42}))$

$= (2\sqrt{42} + 8)(2\sqrt{42} - 8)$

$= (2\sqrt{42})^2 - 8^2 =$ $\boxed{104}$

1986 AIME (Problems • Answer Key • Resources)
Preceded by Problem 1	Followed by Problem 3
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