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,

 

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

 
 

 

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,

 

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