$\sqrt x $ means the positive square root of $x$.

A number that can be expressed in the form $\displaystyle \frac{a}{b}$, where $a$ and $b$ are integers, is said to be a rational number e.g. $\displaystyle \frac{3}{4}$, $\displaystyle \frac{5}{9}$, $\displaystyle \frac{2}{3}$.

An irrational number is one which cannot be expressed in this form e.g. $\sqrt { 19}, \sqrt {13}, \sqrt {27}$.

Numbers such as $\sqrt {3}, \sqrt {5}, \sqrt {7}$ are said to be in surd form.

We use surd form when we want to be exact.

Simplifying Surds

$\sqrt {9\hbox{x}16} \quad =\sqrt {144} \quad =\quad 12$

Also $\sqrt {9\hbox{x}16} =\sqrt {9}\times \sqrt {16}=3\hbox{x}4=12$

So $\sqrt {a\times b} =\sqrt a \times \sqrt b $

Similarly $ \sqrt {\displaystyle \frac{a}{b}} = \displaystyle \frac{\sqrt {a}}{\sqrt {b}}$eg$\displaystyle \sqrt {{9\over 16}}={\sqrt {9}\over \sqrt {16}}={3\over 4}$

But $ \sqrt {a+b} \ne \sqrt {a} + \sqrt {b}$

$\sqrt {a-b} \ne \sqrt {a} - \sqrt {b}$

Simplifying products involving surds

$\begin{array}{rcl}& & (x+\sqrt {3})(x+\sqrt {2})\\ & & = x.x+x\sqrt {2} + x\sqrt {3}+\sqrt {3}\sqrt {2}\\ & & = x^{2}+x(\sqrt {2}+\sqrt {3}) + \sqrt {6}\\ \end{array}$

Pairs of values such as $2 +\sqrt {3},2-\sqrt {3}$ are called pairs of conjugates. To simplify an expression such as $\displaystyle \frac{2 + \sqrt {5}}{3 - \sqrt {5}}$, multiply the numerator and denominator by the conjugate of the denominator, i.e. $3 + \sqrt {5}$

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