## Laws of Surds

 4.2 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. $$(a+\sqrt{b})(a-\sqrt{b})=a^2-b$$. 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}