Multiplying surd expressions

Multiply every term in one bracket by every term in the other, just as in algebra. A surd multiplied by itself becomes its radicand: \(\sqrt a\sqrt a=a\) for \(a\ge0\).

Example

\((3+\sqrt2)(4-\sqrt2)=12-3\sqrt2+4\sqrt2-2=10+\sqrt2\).

Opposite-sign pairs cancel the cross terms: \((a+\sqrt b)(a-\sqrt b)=a^2-b\). For example, \((5+\sqrt3)(5-\sqrt3)=22\).

\((2+\sqrt3)^2=7+4\sqrt3\), not \(7\): the middle term remains.