Let's now learn how to simplify given surds.
Remember the rules in surds, as listed below:
1. Addition:
\( \hspace{0.5cm} x\sqrt{a} + y\sqrt{a}\) \(=(x+y)\sqrt{a} \)
2. Subtraction:
\( \hspace{0.5cm} x\sqrt{a} - y\sqrt{a}\) \(=(x-y)\sqrt{a} \)
3. Multiplication:
\( \hspace{0.5cm} \sqrt{x} \times \sqrt{y} = \sqrt{x \times y} \)
\( \hspace{0.5cm} a\sqrt{x} \times b\sqrt{y} = ab\sqrt{xy} \)
4. Division:
\( \hspace{0.5cm} \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} \)
5. Rationalization (using simple surds):
\( \hspace{0.5cm} \frac{a}{\sqrt{b}}\) \(= \frac{a}{\sqrt{b}} \times \frac{\sqrt{b}}{\sqrt{b}} \)
6. Rationalization (using conjugates):
\( \hspace{0.5cm} \frac{c}{a+b\sqrt{n}} = \frac{c}{a+b\sqrt{n}} \times \frac{a-b\sqrt{n}}{a-b\sqrt{n}} \)
Example 21
Simplify: \( \sqrt{50} + \sqrt{18} - \sqrt{32} \).
Solution
\( \sqrt{50} + \sqrt{18} - \sqrt{32} \)
\( \Rightarrow \sqrt{2 \times 25} + \sqrt{2 \times 9} - \sqrt{2 \times 16} \)
\( \Rightarrow \sqrt{2 \times 5^2} + \sqrt{2 \times 3^2} - \sqrt{2 \times 4^2} \)
\( \Rightarrow 5\sqrt{2} + 3\sqrt{2} - 4\sqrt{2} \)
\( \Rightarrow (5 + 3 - 4)\sqrt{2} \)
\( \Rightarrow 4\sqrt{2} \)
Example 22
Rationalize the denominator: \( \frac{5}{\sqrt{7}} \).
Solution
\( \frac{5}{\sqrt{7}} \)
\( \Rightarrow \frac{5}{\sqrt{7}} \times \frac{\sqrt{7}}{\sqrt{7}} \)
\( \Rightarrow \frac{5}{\sqrt{7}} \times \frac{\sqrt{7}}{\sqrt{7}} \)
\( \Rightarrow \frac{5\sqrt{7}}{(\sqrt{7})^2} \)
\( \Rightarrow \frac{5\sqrt{7}}{7} \)
\( \Rightarrow \frac{5}{7} \sqrt{7} \)
Watch the video below for a walk-through of the solution.
Example 23
Simplify: \( \frac{\sqrt{7} + \sqrt{5}}{\sqrt{7} - \sqrt{5}} + \frac{\sqrt{7} - \sqrt{5}}{\sqrt{7} + \sqrt{5}} \).
Solution
\( \frac{\sqrt{7} + \sqrt{5}}{\sqrt{7} - \sqrt{5}} + \frac{\sqrt{7} - \sqrt{5}}{\sqrt{7} + \sqrt{5}} \)
\( \Rightarrow \frac{(\sqrt{7} + \sqrt{5})(\sqrt{7} + \sqrt{5}) + (\sqrt{7} - \sqrt{5})(\sqrt{7} - \sqrt{5}) }{(\sqrt{7})^2 - (\sqrt{5})^2} \)
\( \Rightarrow \frac{( (\sqrt{7})^2 + 2(\sqrt{7} \times \sqrt{5}) + (\sqrt{5})^2 ) + ( (\sqrt{7})^2 - 2(\sqrt{7} \times \sqrt{5}) + (\sqrt{5})^2 )} {(\sqrt{7})^2 - (\sqrt{5})^2}\)
\( \Rightarrow \frac{( 7 + 2(\sqrt{35}) + 5 ) + ( 7 - 2(\sqrt{35}) + 5 )} {7 - 5} \)
\( \Rightarrow \frac{ (12 + 2\sqrt{35}) + (12 - 2\sqrt{35}) } {2} \)
\( \Rightarrow \frac{ 12 + 2\sqrt{35} + 12 - 2\sqrt{35} } {2} \)
\( \Rightarrow \frac{ 12 + 12 + 2\sqrt{35} - 2\sqrt{35} } {2} \)
\( \Rightarrow \frac{ 24 } {2} \)
\( \Rightarrow 12 \)
Example 24
Simplify: \( (\sqrt{5} + 2)^2 - (\sqrt{5} - 2)^2 \).
Solution
\( (\sqrt{5} + 2)^2 - (\sqrt{5} - 2)^2 \)
The above expression is difference of 2 squares:
\( \Rightarrow ( (\sqrt{5} + 2) + (\sqrt{5} - 2) ) ( (\sqrt{5} + 2) - (\sqrt{5} - 2) ) \)
\( \Rightarrow ( \sqrt{5} + 2 + \sqrt{5} - 2 ) ( \sqrt{5} + 2 - \sqrt{5} + 2 ) \)
\( \Rightarrow ( 2 \sqrt{5} ) ( 4 ) \)
\( \Rightarrow ( 2 \times 4) \sqrt{5} \)
\( \Rightarrow 8 \sqrt{5} \)
Example 25
Rationalize the denominator: \( \frac{2\sqrt{6} + 3}{\sqrt{3} - \sqrt{2}} \).
Solution
Example 26
Simplify: \( \sqrt{8} \times \sqrt{12} \).
Solution
Example 27
Express in the form \( a + b\sqrt{c} \): \( (2 + \sqrt{3})(3 - \sqrt{3}) \).
Solution
Example 28
Solve for \( x \): \( x = \sqrt{45} - \sqrt{20} \).
Solution
\( x \): \( x = \sqrt{45} - \sqrt{20} \)
\( \Rightarrow x = \sqrt{45} - \sqrt{20} \)
\( \Rightarrow x = \sqrt{5 \times 9} - \sqrt{5 \times 4} \)
\( \Rightarrow x = \sqrt{5 \times 3^2} - \sqrt{5 \times 2^2} \)
\( \Rightarrow x = 3\sqrt{5} - 2\sqrt{5} \)
\( \Rightarrow x = (3 - 2)\sqrt{5} \)
\( \Rightarrow x = \sqrt{5} \)
\( \therefore x\) is \(\sqrt{5}\)
Example 29
If \( a = \sqrt{2} + \sqrt{3} \), find the value of \( \frac{1}{a} \) in the form \( m + n\sqrt{p} \), where \( m, n, p \) are integers.
Solution
Example 30
Simplify: \( \frac{\sqrt{3} + 1}{\sqrt{3} - 1} \).
Solution
Try Work
Solve the following:
1. Simplify: \( 3\sqrt{27} - 2\sqrt{12} + \sqrt{75} \).
2. Rationalize the denominator: \( \frac{4}{\sqrt{5} + 3} \).
3. Multiply and simplify: \( (\sqrt{2} + \sqrt{3})(\sqrt{2} - \sqrt{3}) \).
4. Simplify: \( \frac{\sqrt{18} + \sqrt{2}}{\sqrt{8}} \).
5. If \( a = \sqrt{3} - 1 \), find \( a^2 \).
6. Solve for \( x \): \( \sqrt{x + 3} = 2\sqrt{3} \).
7. Simplify: \( \sqrt{\frac{50}{2}} + \sqrt{\frac{72}{4}} \).
8. Express in the form \( a + b\sqrt{c} \): \( \frac{\sqrt{6} + \sqrt{2}}{\sqrt{3}} \).
9. Simplify: \( \sqrt{12} + \sqrt{27} - \sqrt{48} \).
10. Rationalize and simplify: \( \frac{1}{\sqrt{2} + \sqrt{3} + \sqrt{5}} \).
Answers
1. \( 10\sqrt{3} \)
2. \(-\sqrt{5} + 3\)
3. \(-1\)
4. \(2\)
5. \( a = 4 - 2\sqrt{3}\)
6. \(x = 9\)
7. \(5 + 3\sqrt{2}\)
8. \(\sqrt{2} + \frac{\sqrt{6}}{3}\)
9. \(\sqrt{3}\)
10. \(\frac{\sqrt{3}}{6} + \frac{\sqrt{2}}{4} - \frac{\sqrt{30}}{12}\)
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