Sum of Square Roots as Square Root of Sum

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

$\blacksquare$