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

Revision as of 19:02, 23 March 2007

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.

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

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

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

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

@@ Line 3: / Line 3: @@
 == Solution ==
-{{solution}}
+Simplify by repeated application of the [[difference of squares]].
+:<math>\left((\sqrt{6} + \sqrt{7})^2 - \sqrt{5}^2\right)\left(\sqrt{5}^2 - (\sqrt{6} - \sqrt{7})^2\right)</math>
+:<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>
 == See also ==
-* [[1986 AIME Problems]]
+{{AIME box|year=1986|num-b=1|num-a=3}}
-{{AIME box|year=1986|num-b=1|num-a=3}}
+[[Category:Intermediate Algebra Problems]]

1986 AIME (Problems • Answer Key • Resources)
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