Difference between revisions of "1986 AIME Problems/Problem 2"

Revision as of 18:17, 24 February 2018

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.

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

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

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

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

@@ Line 8: / Line 8: @@
 :<math>= (13 + 2\sqrt{42} - 5)(5 - (13 - 2\sqrt{42}))</math>
 :<math>= (2\sqrt{42} + 8)(2\sqrt{42} - 8)</math>
-:<math>= (2\sqrt{42})^2 - 8^2 = 104</math>
+:<math>= (2\sqrt{42})^2 - 8^2 =</math> <math>\boxed{104}</math>
 == See also ==

1986 AIME (Problems • Answer Key • Resources)
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