Discrete Mathematics with Applications 4th Edition

Published by Cengage Learning
ISBN 10: 0-49539-132-8
ISBN 13: 978-0-49539-132-6

Chapter 4 - Elementary Number Theory and Methods of Proof - Exercise Set 4.1 - Page 163: 61

Answer

Claim: Suppose that integers m and n are perfect squares. Then m + n + 2√(mn) is also a perfect square. Proof: Suppose both m and n are particular but arbitrarily chosen perfect squares. By definition of perfect square, let m = r² and n = s². By substitution and algebra, it follows that: m + n + 2√(mn) = r² + s² + 2√(r² * s²) = r² + s² + 2 * √r² * √s² = r² + s² + 2rs = (r+s)(r+s) = (r+s)² . Let t = r+s, then (r+s)² = t². By definition of a perfect square, it follows that t² = m + n + 2√(mn) is a perfect square.

Work Step by Step

A perfect square is define as y = x². Define m and n as perfect squares. Substitute m and n into the equation m + n + 2√(mn) and use algebra to determine m + n + 2√(mn) = (r+s)². Let t = (r + s), then m + n + 2√(mn) = t². Then conclude that t² is a perfect square.
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