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

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== Solution ==
 
== Solution ==
{{solution}}
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Simplify by repeated application of the [[difference of squares]].
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:<math>\left((\sqrt{6} + \sqrt{7})^2 - \sqrt{5}^2\right)\left(\sqrt{5}^2 - (\sqrt{6} - \sqrt{7})^2\right)</math>
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:<math>= (13 + 2\sqrt{42} - 5)(5 - (13 - 2\sqrt{42}))</math>
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:<math>= (2\sqrt{42} + 8)(2\sqrt{42} - 8)</math>
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:<math>= (2\sqrt{42})^2 - 8^2 = 104</math>
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== See also ==
 
== See also ==
* [[1986 AIME Problems]]
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{{AIME box|year=1986|num-b=1|num-a=3}}
  
{{AIME box|year=1986|num-b=1|num-a=3}}
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[[Category:Intermediate Algebra Problems]]

Revision as of 20:02, 23 March 2007

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 (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