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Rationalization of surds is concerned with the process of removing irrational numbers from the denominator. Consider the surd \( \frac{2}{\sqrt{3}} \)an irrational number dividing a rational number. Here, both the numerator and denominator must be multiplied by \(\scriptsize \sqrt{3} \), the denominator, for it to be removed.

Example

Simplify the following

(a) \( \frac{10}{\sqrt{2}} \)

(b) \( \frac{2 \:+ \:\sqrt{3}}{7 \:-\: \sqrt{5}} \)

Solution

(a) \( \frac{10}{\sqrt{2}} \)

\( \frac{10}{\sqrt{2}} \: \times \: \frac{\sqrt{2}}{\sqrt{2}} \)

= \( \frac{10 \sqrt{2}}{\sqrt{4}} \)

= \( \frac{10 \sqrt{2}}{2 } \)

= \(\scriptsize 5 \sqrt{2} \)

(b) \( \frac{2 \:+ \: \sqrt{3}}{7 \:- \: \sqrt{5}} \)

rationalizing using the conjugate of 7 – √5 which is 7 + √5 we have

\( \frac{2 \:+\: \sqrt{3}}{7 \:- \:\sqrt{5}} \: \times \: \frac{7 \:+ \: \sqrt{5}}{7\: + \: \sqrt{5}} \)

= \( \frac{14 \: + \: 2 \sqrt{5}\: + \: 7 \sqrt{3}\: + \: \sqrt{3} \: \times \: \sqrt{5}}{49 \:+ \: 7 \sqrt{5} \: – \: 7 \sqrt{5} \: – \: 5 } \)

= \( \frac{14\: +\: 2 \sqrt{5} \:+ \:7 \sqrt{3} \:+\: \sqrt{15}} {49 \: – \: 5 } \)

= \( \frac{14 \: + \: 2 \sqrt{5}\: + \:7 \sqrt{3}\: +\: \sqrt{15}} {44 } \)

Exercise

1. Evaluate the following surds 

a. \( \frac{4}{\sqrt{5} \:+\:\sqrt{3}} \)

b. \( \scriptsize \sqrt{32} \:+\: \normalsize \frac{6}{\sqrt{3}} \)

c. \( \frac{3}{2\sqrt{6} \:-\: \sqrt{3}} \)

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