Rationalize the Denominator. Note that a surd is an irrational number. In this section, we will learn how to rationalize a surd, that is, the process of making a surd into a rational number.
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Definition of Rationalisation of surds
The process of converting a surd into a rational number is called the rationalisation of surds. This is done by multiplying the given surd with a proper auxiliary surd. In this case, the auxiliary surd is called the surd-rationalizing factor of the given surd.
How to Rationalize the Denominator? In the next two sections, we will learn it.
How to Rationalise a surd
How to rationalise surds? See the below process on rationalizing surds.
(i) To rationalize a simple surd we just need to multiply the given surd with its surd-factor. Let us rationalize the simple surd $5\sqrt{2}.$ So we have to multiply it by $\sqrt{2}.$
Note that $5 \sqrt{2} \times \sqrt{2}$ $=5 \times 2$ $=10$ is a rational number. Here $\sqrt{2}$ is called the surd-rationalizing factor of $5\sqrt{2}.$
Rationalize the denominator
Benefits of Rationalisation of surds: Why do we rationlize the denominator? Lets understand this with the help of an example. We know that $\sqrt{2} \approx 1.414.$ Using this fact, we cannot easily calculate the value of $\frac{1}{\sqrt{2}}.$ But rationalization the denominator of $\frac{1}{\sqrt{2}}$ makes the calculation easy as follows:
Multiplying the numerator and the denominator of $\dfrac{1}{\sqrt{2}}$ by $\sqrt{2},$ we have
$\dfrac{1}{\sqrt{2}}$ $=\dfrac{1 \times \sqrt{2}}{\sqrt{2} \times \sqrt{2}}$
$=\dfrac{\sqrt{2}}{2}$
∴ $\dfrac{1}{\sqrt{2}}$ $=\dfrac{\sqrt{2}}{2}$ $=\dfrac{1.414}{2}$ $=0.707$
Problem 1: Rationalize the denominator of $\dfrac{2}{\sqrt{3}}$
Solution:
Multiplying both the numerator and denominator by $\sqrt{3},$ we get
$\dfrac{2}{\sqrt{3}}$ $=\dfrac{2 \times \sqrt{3}}{\sqrt{3} \times \sqrt{3}}$
$=\dfrac{2\sqrt{3}}{3}$ $[\because \sqrt{a} \times \sqrt{a}=a]$
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