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Lesson 1, Topic 6
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# Surds Rationalization

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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 – âˆš5 which is 7 + âˆš5 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 surdsÂ

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

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

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

error: