Laws of Surds

For these real square-root laws, the radicands are non-negative; a denominator radicand must be strictly positive. Thus the product law uses \(a,b\ge0\), and the quotient law uses \(a\ge0,b>0\).

1. \(\sqrt{a} \times \sqrt{b} = \sqrt{a \times b} \)
2. \(\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}} \)
3. \(\sqrt{a} \times \sqrt{a} = \sqrt{a^2} = a\)

4. \(p\sqrt{a} + q\sqrt{a} = (p + q)\sqrt{a}\)
   Example: \(5\sqrt{2} + 2\sqrt{2} = 7\sqrt{2}\)
5. \(p\sqrt{a} - q\sqrt{a} = (p - q)\sqrt{a}\)
   Example: \(9\sqrt{5} - 3\sqrt{5} = 6\sqrt{5} \)