AoPSWiki

Our Precalculus course starts on Dec. 4. Master trig, complex numbers, and vectors and matrices in 2 and 3 dimensions. Click here to enroll today!

Personal tools

Search

Toolbox

1986 AIME Problems/Problem 2

From AoPSWiki

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

Retrieved from "http://www.artofproblemsolving.com/Wiki/index.php/1986_AIME_Problems/Problem_2"

Category: Intermediate Algebra Problems

Do you have what it takes to be the next brilliant trader, researcher, or developer at Jane Street Capital? Find out in the Careers in Mathematics Forum.

© Copyright 2008 AoPS Incorporated. All Rights Reserved. • Foundation • Privacy • Contact Us