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1986 AIME Problems/Problem 2

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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 = 104

See also

1986 AIME (Problems • Resources)
Preceded by
Problem 1 		Followed by
Problem 3
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
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