Laws of Surds


 Laws of Surds

\(\blacksquare\) Rational numbers are the numbers that can be expressed in fractional form \(\dfrac{a}{b}\) where \(a\) and \(b\) are integers and \(b \ne 0\).
\(\blacksquare\) Irrational numbers are the numbers that cannot be expressed in fractional form.
Below are some examples:
Rational number Irrational number
\(-3=-\dfrac{3}{1}\) \(\pi=3.14159265...\)
\(\dfrac{1}{3}\) \(e=2.71828182...\)(Euler number)
\(1.75=\dfrac{7}{4}\) (terminating decimal) \(\varphi=1.61803398\) (golden ratio)
\(0.555...=\dfrac{111}{200}\)(recurring decimal) \(\sqrt{3}=1.732050808...\)
\(\sqrt{25}=5\) \(\sqrt[3]{9}=2.080083823\)

From the examples, root of a number can be either rational or irrational.

An irrational number in the form of root is called surd.

\(\sqrt[3]{9}\) is read as "surd \(9\) order \(3\)".

Laws of surds:


\(\begin{aligned} \bullet \quad\sqrt{a} \times \sqrt{b} &=\sqrt{ab} \\\\ \bullet \quad \sqrt{a} \div \sqrt{b}&=\sqrt{\dfrac{a}{b}} \end{aligned}\)

for \(a \gt 0\) and \(b \gt 0\).


Conjugate surd of \(a+\sqrt{b}\) is \(a-\sqrt{b}\), similarly \(a-\sqrt{b}\) is \(a+\sqrt{b}\).

A rational number is obtained when the conjugate pair is multiplied.



To simplify an expression involving surd as denominator, rationalising the denominator by multiplying the numerator and denominator with conjugate surd.

For example,


\(\begin{aligned} &\bullet \quad\dfrac{1}{m\sqrt{a}} \times\dfrac{m\sqrt{a}}{m\sqrt{a}} \\\\ &\bullet \dfrac{1}{m\sqrt{a}+n\sqrt{b}}\times\dfrac{m\sqrt{a}-n\sqrt{b}}{m\sqrt{a}-n\sqrt{b}} \\\\ &\bullet \dfrac{1}{m\sqrt{a}-n\sqrt{b}}\times\dfrac{m\sqrt{a}+n\sqrt{b}}{m\sqrt{a}+n\sqrt{b}} \end{aligned}\)