1986 AIME Problems/Problem 2

Revision as of 19:17, 24 February 2018 by Bluebacon008 (talk | contribs) (Solution)

Problem

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

Solution

Simplify by repeated application of the difference of squares.

$\left((\sqrt{6} + \sqrt{7})^2 - \sqrt{5}^2\right)\left(\sqrt{5}^2 - (\sqrt{6} - \sqrt{7})^2\right)$
$= (13 + 2\sqrt{42} - 5)(5 - (13 - 2\sqrt{42}))$
$= (2\sqrt{42} + 8)(2\sqrt{42} - 8)$
$= (2\sqrt{42})^2 - 8^2 =$ $\boxed{104}$

See also

1986 AIME (ProblemsAnswer KeyResources)
Preceded by
Problem 1
Followed by
Problem 3
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
All AIME Problems and Solutions

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