A nested radical

#radicals#identities

Statement

Simplify $\sqrt{3+2\sqrt2}$ .

Hint

Look for integers $a,b$ with $(a+b\sqrt2)^2=3+2\sqrt2$ .

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We compute $(1+\sqrt2)^2=1+2\sqrt2+2=3+2\sqrt2$ .

Since $1+\sqrt2>0$ , the square root of $(1+\sqrt2)^2$ is $1+\sqrt2$ . Hence
$\sqrt{3+2\sqrt2}=1+\sqrt2.$