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A surd is a root of a non-exponential number, resulting in an irrational number, and cannot be simplified to remove the root. An example of a surd is ${\displaystyle \sqrt{2}}$, because it cannot be simplified further and is irrational. However, a number like ${\displaystyle \sqrt4}$ or ${\displaystyle {\sqrt[{3}]{125}}}$ would not be a surd, because they can be simplified (to 2 and 5, respectively).

Basic rules

For any positive real numbers a and b:

• ${\displaystyle \sqrt{ab}}$ = ${\displaystyle {\sqrt {a}}{\sqrt {b}}}$
• ${\displaystyle {\sqrt {\frac {a}{b}}}}$ = ${\displaystyle {\frac {\sqrt {a}}{\sqrt {b}}}}$
• ${\displaystyle {\sqrt[{a}]{b}}^{a}}$ = b
• ${\displaystyle {\sqrt {a}}+{\sqrt {b}}\neq {\sqrt {a+b}}}$ (in general)