Lesson 1, Topic 3
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Addition & Subtraction of Surds

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Only surds in the same basic form can either be added or subtracted. Sometimes there may be a need to simplify such surds before doing either the addition or the subtraction.

Examples

Simplify the following:

i. \( \scriptsize 2 \sqrt{12}\: + \:3 \sqrt{48} \: – \: \sqrt{75} \)

ii. \( \scriptsize 2 \sqrt{150} \: – \: \sqrt{96} \: – \: 2\sqrt{24} \)

Solution

i. \( \scriptsize 2 \sqrt{12} + 3 \sqrt{48} \: – \: \sqrt{75} \)

= \( \scriptsize 2 \left ( \sqrt{3 \: \times \: 4} \right)\: + \: 3 \left ( \sqrt{3 \: \times \: 16} \right) \: – \: \left ( \sqrt{3 \: \times \: 25} \right) \)

= \( \scriptsize 2 \left ( \sqrt{3} \: \times \: \sqrt{4} \right) \: + \: 3 \left ( \sqrt{3} \: \times \: \sqrt{16} \right) \: – \: \left ( \sqrt{3 \: \times \: 25} \right) \)

= \( \scriptsize 2 \left ( 2 \sqrt{3} \right) \: + \: 3 \left ( 4 \sqrt{3} \right) \: – \: \left (5 \sqrt{3} \right) \)

= \( \scriptsize 4 \sqrt{3} \: + \: 12 \sqrt{3} \: – \: 5 \sqrt{3} \)

= \( \scriptsize 16 \sqrt{3} \: – \: 5 \sqrt{3} \\ \scriptsize = 11 \sqrt {3} \)

ii. \( \scriptsize 2 \sqrt{150} \: – \: \sqrt{96} \: – \: 2\sqrt{24} \)

= \( \scriptsize 2 \left ( \sqrt{6 \: \times \: 25} \right) \: – \: \left ( \sqrt{6 \: \times \: 16} \right) \: – \: 2 \left ( \sqrt{4 \: \times \: 6} \right) \)

= \( \scriptsize 2 \left ( 5 \sqrt{6} \right) \: – \: \left ( 4 \sqrt{6} \right) \: – \: \left (4 \sqrt{6} \right) \)

= \( \scriptsize 10 \sqrt{6}\: – \: 8 \sqrt{6} \\ \scriptsize = 2 \sqrt{6} \)

Exercise:

Evaluate

i. \( \scriptsize \sqrt{32} \: + \: 3 \sqrt{8} \)

ii. \( \scriptsize 3 \sqrt{48} \:-\: \sqrt{75} \: + \: 2 \sqrt{12} \)

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