Sum of Square Roots as Square Root of Sum
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Theorem
- $\sqrt a + \sqrt b = \sqrt {a + b + \sqrt {4 a b} }$
Proof
\(\ds \sqrt a + \sqrt b\) | \(=\) | \(\ds \sqrt {\paren {\sqrt a + \sqrt b}^2}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \sqrt {\sqrt a^2 + \sqrt b^2 + 2 \sqrt a \sqrt b}\) | Square of Sum | |||||||||||
\(\ds \) | \(=\) | \(\ds \sqrt {a + b + \sqrt {4 a b} }\) | Power of Product |
$\blacksquare$