Rules For Basic Surd Operations - SS3 Mathematics Lesson Note
The rules for basic operations with surd for any positive number \(a\) and \(b\) are:
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\(\sqrt{a} - \sqrt{b} \neq \sqrt{a - b}\)
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\(\sqrt{a} + \sqrt{b} \neq \sqrt{a + b}\), but
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\(\sqrt{a} \times \sqrt{b} = \sqrt{ab}\)
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\(\sqrt{a} \div \sqrt{b} = \sqrt{\frac{a}{b}}\)
Example 4 Evaluate \(\sqrt{54} \times \sqrt{24} \times 2\sqrt{20}\)
Solution
\[\sqrt{54} \times \sqrt{24} \times 2\sqrt{20} = \ \sqrt{9 \times 6} \times \sqrt{4 \times 6} \times 2\sqrt{4 \times 5}\]
\[3\sqrt{6} \times 2\sqrt{6} \times 2 \times 2\sqrt{5} = \ 3\sqrt{6} \times 2\sqrt{6} \times 4\sqrt{5}\]
\[3 \times 2\sqrt{6^{2}} \times 4\sqrt{5} = \ 6 \times 6 \times 4\sqrt{5} = 144\sqrt{5}\]
Example 5 Evaluate \(\frac{5\sqrt{3}}{2\sqrt{50}}\)
Solution
\[\frac{5\sqrt{3}}{2\sqrt{50}} = \ \frac{5\sqrt{3}}{2\sqrt{25 \times 2}} = \frac{5\sqrt{3}}{2 \times 5\sqrt{2}} = \ \frac{\sqrt{3}}{2\sqrt{2}} = \frac{\sqrt{3}}{2\sqrt{2}}.\frac{\sqrt{2}}{\sqrt{2}} = \ \frac{\sqrt{6}}{4}\]
Example 6 Simplify \((5\sqrt{3} - 2\sqrt{7})(\sqrt{27} + 3\sqrt{7})\)
Solution
\[\left( 5\sqrt{3} - 2\sqrt{7} \right)\left( \sqrt{27} + 3\sqrt{7} \right) = \ 5\sqrt{3}\left( \sqrt{27} + 3\sqrt{7} \right) - 2\sqrt{7}\left( \sqrt{9 \times 3} + 3\sqrt{7} \right)\]
\[5\sqrt{27 \times 3} + 15\sqrt{21} - 6\sqrt{21} - 6\left( 7^{2} \right) = 5\sqrt{81} + 15\sqrt{21} - 6\sqrt{21} - 42\]
\[5 \times 9 + 9\sqrt{21} - 42 = 45 + 9\sqrt{21} - 42 = 3 + 9\sqrt{21}\]