In our previous lesson on identifying surds, we learnt that in indices:
1. \(\sqrt{a} = a^\frac{1}{2}\)
2. \(\sqrt[3]{a} = a^\frac{1}{3}\)
3. \(\sqrt[4]{a} = a^\frac{1}{4}\)
4. \(\sqrt[5]{a} =a^\frac{1}{5}\)
And so on.
In this lesson, we will learn about the various rules in surds, that is, how to perform basic operations such as addition, subtraction, multiplication and division on surds.
Consider the algebraic expression below:
\(2x + 6x\)
Since the two terms have the same variable (\(x\)) in common, we say that \(2x\) and \(6x\) are like-terms.
\(\therefore 2x + 6x\)
\(\Rightarrow (2 + 6)x\)
\(\Rightarrow 8x\)
Similarly, when two or more surds have the same simple or compound surd in common, we can refer to the surds as like-terms, treating the simple or compound surd like a variable.
When the roots are the same:
Addition
1. \(x\sqrt{a} + y\sqrt{a}\) \(=(x+y)\sqrt{a}\)
Subtraction
2. \(x\sqrt{a} - y\sqrt{a}\) \(=(x-y)\sqrt{a}\)
Try the examples below then click "show solution" to compare the solution with your answer.
Example 1
Simplify: \(5\sqrt{2} + 7\sqrt{2}\)
Solution
\(5\sqrt{2} + 7\sqrt{2}\)
\(\Rightarrow (5 + 7)\sqrt{2}\)
\(\Rightarrow 12\sqrt{2}\)
Watch a walk through of the solution.
Example 2
Simplify: \(5\sqrt{2} - 7\sqrt{2}\)
Solution
\(5\sqrt{2} - 7\sqrt{2}\)
\(\Rightarrow (5 - 7)\sqrt{2}\)
\(\Rightarrow -2\sqrt{2}\)
Watch a walk through of the solution.
Example 3
Simplify: \(5\sqrt{5} - 7\sqrt{2} + 11\sqrt{5} - 3\sqrt{2}\)
Here, you may group like terms in your calculation.
Solution
\(5\sqrt{5} - 7\sqrt{2} + 11\sqrt{5} - 3\sqrt{2}\)
Grouping like terms:
\(- 7\sqrt{2} - 3\sqrt{2} + 5\sqrt{5} + 11\sqrt{5}\)
\(\Rightarrow (- 7 - 3)\sqrt{2} + (5 + 11)\sqrt{5} \)
\(\Rightarrow - 10\sqrt{2} + 16\sqrt{5} \)
\(\Rightarrow 16\sqrt{5} - 10\sqrt{2} \)
Watch a walk through of the solution.
Example 4
Simplify: \(\frac{1}{2}\sqrt{3} - \frac{1}{3}\sqrt{3}\)
Solution
\(\frac{1}{2}\sqrt{3} - \frac{1}{3}\sqrt{3}\)
\(\Rightarrow (\frac{1}{2} - \frac{1}{3})\sqrt{3}\)
\(\Rightarrow (\frac{3(1) - 2(1)}{6})\sqrt{3}\)
\(\Rightarrow (\frac{3 - 2}{6})\sqrt{3}\)
\(\Rightarrow \frac{1}{6}\sqrt{3}\)
\(\Rightarrow \frac{\sqrt{3}}{6}\)
Example 5
Simplify: \(\sqrt{2} + \sqrt{5}\)
Solution
\(\sqrt{2} + \sqrt{5}\)
\(= \sqrt{2} + \sqrt{5}\)
Since the roots are not the same, you cannot add or subtract them.
Try Work
Simplify the following:
1. \(8\sqrt{6} - 5\sqrt{6}\)
2. \(10\sqrt{5} + \sqrt{5}\)
3. \(\frac{1}{5}\sqrt{11} - 5\sqrt{11}\)
4. \(8\sqrt{3} - 5\sqrt{5} + 6\sqrt{3}\)
5. \(4\sqrt{7} - \sqrt{7}\)
Answers
1. \(3\sqrt{6}\)
2. \(11\sqrt{5}\)
3. \(\frac{-24}{5}\sqrt{11}\)
4. \(14\sqrt{3} - 5\sqrt{5}\)
5. \(3\sqrt{7}\)
Multiplication
Follow the rule below when finding the product of two or more surds.
3. \(\sqrt{x} \times \sqrt{y} = \sqrt{x \times y}\)
It implies that:
\(\sqrt{x} \times \sqrt{x} = \sqrt{x^2} = x\)
\(\therefore \sqrt{x^2} = x\)
Remember:
\(\sqrt{x^2} \ \Rightarrow\) the square root of perfect squares.
Division
The square root of fractions can also be expressed in this form,
4. \(\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}\)
We will need to rationalize the surd. We will learn about Rationalization in the next section. It simply means expressing the surd such that the denominator is no more an irrational number but a rational number, hence the term, "rationalization".
Try the examples below and click on "show solution" to compare your answer to the solutions.
Example 6
Simplify \(\sqrt{3} \times \sqrt{4}\)
Solution
\(\sqrt{3} \times \sqrt{4}\)
\(\Rightarrow \sqrt{3} \times \sqrt{2^2}\)
\(\Rightarrow \sqrt{3} \times 2\)
\(\Rightarrow 2\sqrt{3}\)
Watch a video walk through of the solution.
Example 7
Sometimes you need to express the number given under the radical symbol as a product of prime factors.
\(72 = 2 \times 2 \times 2 \times 3 \times 3\)
Simplify: \(\sqrt{72}\)
Solution
\(\sqrt{72}\)
\(\Rightarrow \sqrt{2 \times 2 \times 2 \times 3 \times 3}\)
\(\Rightarrow \sqrt{2 \times 2^2 \times 3^2}\)
\(\Rightarrow \sqrt{2} \times \sqrt{2^2} \times \sqrt{3^2}\)
\(\Rightarrow \sqrt{2} \times 2 \times 3\)
\(\Rightarrow \sqrt{2} \times 6\)
\(\Rightarrow 6\sqrt{2}\)
Watch a video walk through of the solution.
Example 8
Simplify: \(\frac{\sqrt{12}}{16}\)
Solution
\(\frac{\sqrt{12}}{16}\)
\(\Rightarrow \frac{\sqrt{4 \times 3}}{16}\)
\(\Rightarrow \frac{\sqrt{2^2 \times 3}}{16}\)
\(\Rightarrow \frac{2\sqrt{3}}{16}\)
\(\Rightarrow \frac{\sqrt{3}}{8}\)
Example 9
Simplify: \((\sqrt{5})^2\)
Solution
\((\sqrt{5})^2\)
\(\Rightarrow 5\)
Watch a video walk-through of the solution.
Example 10
Simplify: \(\frac{3\sqrt{45}}{9}\)
Solution
\(\frac{3\sqrt{45}}{9}\)
\(\Rightarrow \frac{\sqrt{45}}{3}\)
\(\Rightarrow \frac{\sqrt{9 \times 5}}{3}\)
\(\Rightarrow \frac{\sqrt{3^2 \times 5}}{3}\)
\(\Rightarrow \frac{3\sqrt{5}}{3}\)
\(\Rightarrow \sqrt{5}\)
Watch a video walk-through of the solution.
Try Work
Simplify the following:
1. \(\sqrt{108}\)
2. \(6\sqrt{27} \times \frac{1}{2}\sqrt{128}\)
3. \(\frac{49\sqrt{7}}{21}\)
4. \(\sqrt{3}(\sqrt{4} - \sqrt{2})\)
5. \(\sqrt{7} + \sqrt{3} \times \sqrt{6} - \sqrt{28}\)
Answers
1. \(6\sqrt{3}\)
2. \(72\sqrt{6}\)
3. \(\frac{7\sqrt{7}}{3}\)
4. \(2\sqrt{3} - \sqrt{6}\)
5. \(3\sqrt{2} - \sqrt{7}\)
Fractions are always supposed to be written in such a way that their denominator is always a rational number (with the exception of zero (0)). As a result of this, when a fraction has it's denominator being an irrational number (a surd in this case), we need to find a way to rewrite the fraction such that the denominator will become a rational number (except 0).
It is the process of changing this surd so it has a rational denominator that we call rationalization.
When the denominator is a simple surd such as \(\sqrt{b}\), you will need to express 1 as a fraction, in terms of \(\sqrt{b}\), that is \((\frac{\sqrt{b}}{\sqrt{b}})\) and multiply it by the original surd given.
Remember that \(\frac{\sqrt{b}}{\sqrt{b}} = 1\), and every number multiplied by 1 will give the number itself.
A summary of the explannation is the calculation below:
5. \(\frac{a}{\sqrt{b}}\) \(= \frac{a}{\sqrt{b}} \times \frac{\sqrt{b}}{\sqrt{b}}\)
\(\hspace{1.5cm} \Rightarrow \frac{a\sqrt{b}}{\sqrt{b}^2}\)
\(\hspace{1.5cm} \Rightarrow \frac{a\sqrt{b}}{b}\)
Example 11
Simplify: \(\frac{1}{\sqrt{7}}\)
Solution
\(\frac{1}{\sqrt{7}}\)
rationalizing the surd:
\(\Rightarrow \frac{1}{\sqrt{7}} \times \frac{\sqrt{7}}{\sqrt{7}}\)
\(\Rightarrow \frac{\sqrt{1 \times 7}}{\sqrt{7 \times 7}}\)
\(\Rightarrow \frac{\sqrt{7}}{\sqrt{7^2}}\)
\(\Rightarrow \frac{\sqrt{7}}{7}\)
Watch the walk-through of the solution.
Example 12
Simplify: \(\frac{2\sqrt{3}}{\sqrt{6}}\)
Solution
\(\frac{2\sqrt{3}}{\sqrt{6}}\)
\(\Rightarrow \frac{2\sqrt{3}}{\sqrt{6}} \times \frac{\sqrt{6}}{\sqrt{6}}\)
\(\Rightarrow \frac{2\sqrt{3 \times 6}}{\sqrt{6 \times 6}}\)
\(\Rightarrow \frac{2\sqrt{18}}{\sqrt{6^2}}\)
\(\Rightarrow \frac{2\sqrt{2 \times 9}}{6}\)
\(\Rightarrow \frac{2\sqrt{2 \times 3^2}}{6}\)
\(\Rightarrow \frac{(2 \times 3)\sqrt{2}}{6}\)
\(\Rightarrow \frac{6\sqrt{2}}{6}\)
\(\Rightarrow \sqrt{2}\)
Watch a walk-through of the solution.
Example 13
Simplify: \(\frac{3\sqrt{7}}{4\sqrt{5}}\)
Solution
\(\frac{3\sqrt{7}}{4\sqrt{5}}\)
\(\Rightarrow \frac{3\sqrt{7}}{4\sqrt{5}}\)
\(\Rightarrow \frac{3\sqrt{7}}{4\sqrt{5}} \times \frac{\sqrt{5}}{\sqrt{5}}\)
\(\Rightarrow \frac{3\sqrt{7 \times 5}}{4\sqrt{5^2}}\)
\(\Rightarrow \frac{3\sqrt{35}}{4 \times 5}\)
\(\Rightarrow \frac{3\sqrt{35}}{20}\)
\(\Rightarrow \frac{3}{20}\sqrt{5}\)
A walk-through of the solution.
Example 14
Simplify: \(\frac{13\sqrt{12}}{4\sqrt{18}}\)
Solution
\(\frac{13\sqrt{12}}{4\sqrt{18}}\)
\(\Rightarrow \frac{13\sqrt{3 \times 4}}{4\sqrt{2 \times 9}}\)
\(\Rightarrow \frac{(13 \times 2)\sqrt{3}}{(4 \times 3)\sqrt{2}}\)
\(\Rightarrow \frac{13\sqrt{3}}{(2 \times 3)\sqrt{2}}\)
Rationalizing the surd:
\(\Rightarrow \frac{13\sqrt{3}}{6\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}}\)
\(\Rightarrow \frac{13\sqrt{3 \times 2}}{6\sqrt{2 \times 2}}\)
\(\Rightarrow \frac{13\sqrt{6}}{6\sqrt{2^2}}\)
\(\Rightarrow \frac{13\sqrt{6}}{6\times2}\)
\(\Rightarrow \frac{13}{12}\sqrt{6}\)
A walk-through of the solution.
Example 15
Simplify: \(\frac{9\sqrt{5}}{6\sqrt{6}}\)
Solution
\(\frac{9\sqrt{5}}{6\sqrt{6}}\)
\(\Rightarrow \frac{3 \times 3\sqrt{5}}{2 \times 3 \sqrt{6}}\)
\(\Rightarrow \frac{3\sqrt{5}}{2\sqrt{6}}\)
Rationalizing the surds:
\(\Rightarrow \frac{3\sqrt{5}}{2\sqrt{6}} \times \frac{\sqrt{6}}{\sqrt{6}}\)
\(\Rightarrow \frac{3\sqrt{5 \times 6}}{2\sqrt{6 \times 6}}\)
\(\Rightarrow \frac{3\sqrt{30}}{2\sqrt{6^2}}\)
\(\Rightarrow \frac{3\sqrt{30}}{2 \times 6}\)
\(\Rightarrow \frac{3\sqrt{30}}{2 \times 2 \times 3}\)
\(\Rightarrow \frac{\sqrt{30}}{2 \times 2}\)
\(\Rightarrow \frac{\sqrt{30}}{4}\)
\(\Rightarrow \frac{1}{4}\sqrt{30}\)
Watch a video walk-through of the solution.
Try Work
Simplify the following:
1. \(\frac{9\sqrt{5x^3}}{3\sqrt{5}}\)
2. \(\frac{7}{\sqrt{7}}\)
3. \(\frac{9\sqrt{8}}{5\sqrt{6}}\)
4. \(\frac{21}{7\sqrt{3}}\)
5. \(\frac{3x}{\sqrt{7}}\)
Answers
1. \(3x\sqrt{x}\)
2. \(\sqrt{7}\)
3. \(\frac{6\sqrt{3}}{5}\)
4. \(\sqrt{3}\)
5. \(\frac{3x \sqrt{7} }{7}\)
Watch a video walk-through of the solution.
Quick Test
Test yourself on what you have learnt so far. Click on the link below when you are ready.
Kindly contact the administrator on 0208711375 for the link to the test.
Given a compound surd \((a+\sqrt{b})\), its conjugate is said to be the compound surd \((a-\sqrt{b})\).
The reverse is also true, as the conjugate of \((a-\sqrt{b})\) is also \((a+\sqrt{b})\).
Notice how the operation symbols change in the conjugate surds respectively?
When rationalizing a surd whose denominator is a compound surd, we express 1 in terms of the conjugate of the denominator and multiply it by the given surd as shown below:
When the addition operator is present:
6. \(\frac{c}{a+b\sqrt{n}} = \frac{c}{a+b\sqrt{n}} \times \frac{a-b\sqrt{n}}{a-b\sqrt{n}}\)
When the subtraction operator is present:
7. \(\frac{c}{a-b\sqrt{n}} = \frac{c}{a-b\sqrt{n}} \times \frac{a+b\sqrt{n}}{a+b\sqrt{n}}\)
Notice how we get a difference of two squares as the denominators in each case?
Difference of 2 squares:
\(a^2 - b^2\) \(= (a + b)(a - b)\)
Remember:
Rationalization is necessary to remove all irrational numbers from the denominator side of a fraction.
Let's try these examples:
Example 16
Simplify the expression:
\(\frac{5}{\sqrt{3} - 2}\)
Solution
\(\frac{5}{\sqrt{3} - 2}\)
Rationalizing the denominator:
\(\Rightarrow \frac{5}{\sqrt{3} - 2} \times \frac{\sqrt{3} + 2}{\sqrt{3} + 2}\)
\(\Rightarrow \frac{5(\sqrt{3} + 2)}{(\sqrt{3} - 2)(\sqrt{3} + 2)}\)
\(\Rightarrow \frac{5\sqrt{3} + 10}{(\sqrt{3})^2 - (2)^2}\)
\(\Rightarrow \frac{5\sqrt{3} + 10}{3 - 4}\)
\(\Rightarrow \frac{5\sqrt{3} + 10}{- 1}\)
\(\Rightarrow - (5\sqrt{3} + 10)\)
\(\Rightarrow -5\sqrt{3} - 10\)
Example 17
Simplify the product:
\((\sqrt{5} + \sqrt{3})(\sqrt{5} - \sqrt{3})\)
Solution
\((\sqrt{5} + \sqrt{3})(\sqrt{5} - \sqrt{3})\)
Using difference of 2 squares:
\(\Rightarrow \sqrt{5}^2 - \sqrt{3}^2\)
\(\Rightarrow 5 - 3\)
\(\Rightarrow 2\)
Example 18
Rationalize the denominator of:
\(\frac{\sqrt{7} + 3}{\sqrt{7} - 2}\)
Solution
\(\frac{\sqrt{7} + 3}{\sqrt{7} - 2}\)
Rationalizing the denominator:
\(\Rightarrow \frac{\sqrt{7} + 3}{\sqrt{7} - 2} \times \frac{\sqrt{7} + 2}{\sqrt{7} + 2}\)
\(\Rightarrow \frac{(\sqrt{7} + 3)(\sqrt{7} + 2)}{(\sqrt{7} - 2)(\sqrt{7} + 2)}\)
\(\Rightarrow \frac{\sqrt{7}(\sqrt{7} + 2) + 3(\sqrt{7} + 2)}{(\sqrt{7})^2 - 2^2}\)
\(\Rightarrow \frac{(\sqrt{7})^2 + 2\sqrt{7} + 3\sqrt{7} + 6}{7 - 4}\)
\(\Rightarrow \frac{7 + (2 + 3)\sqrt{7} + 6}{3}\)
\(\Rightarrow \frac{7 + 6 + 5\sqrt{7} }{3}\)
\(\Rightarrow \frac{13 + 5\sqrt{7}}{3}\)
Example 19
Find the value of:
\((\sqrt{6} - \sqrt{2})^2\)
Solution
\((\sqrt{6} - \sqrt{2})^2\)
\(\Rightarrow (\sqrt{6} - \sqrt{2})(\sqrt{6} - \sqrt{2})\)
\(\Rightarrow \sqrt{6}(\sqrt{6} - \sqrt{2})\)\( - \sqrt{2}(\sqrt{6} - \sqrt{2})\)
\(\Rightarrow (\sqrt{6})^2 - \sqrt{12}\)\( - \sqrt{12} + (\sqrt{2})^2\)
\(\Rightarrow 6 \)\( - 2\sqrt{12} + 2\)
\(\Rightarrow 6 + 2 \)\( - 2\sqrt{3 \times 4} \)
\(\Rightarrow 8 \)\( - 2\sqrt{3 \times 2^2} \)
\(\Rightarrow 8 \)\( - (2 \times 2)\sqrt{3} \)
\(\Rightarrow 8 \)\( - 4\sqrt{3} \)
Example 20
If \(x = \frac{1}{\sqrt{2} + \sqrt{5}}\), find \(x\) in simplified form.
Solution
\(x = \frac{1}{\sqrt{2} + \sqrt{5}}\)
Rationalizing the denominator:
\(x = \frac{1}{\sqrt{2} + \sqrt{5}} \times \frac{\sqrt{2} - \sqrt{5}}{\sqrt{2} - \sqrt{5}}\)
\(x = \frac{\sqrt{2} - \sqrt{5}}{(\sqrt{2})^2 - (\sqrt{5})^2}\)
\(x = \frac{\sqrt{2} - \sqrt{5}}{2 - 5}\)
\(x = \frac{\sqrt{2} - \sqrt{5}}{- 3}\)
\(x = \frac{-(\sqrt{2} - \sqrt{5})}{3}\)
\(x = \frac{\sqrt{5} - \sqrt{2}}{3}\)
\(\therefore x\) is \(\frac{\sqrt{5} - \sqrt{2}}{3}\)
Try Work
Simplify the following expressions:
1. \(\frac{\sqrt{2} + \sqrt{3}}{\sqrt{2} - \sqrt{3}}\)
2. \((3\sqrt{2} - \sqrt{5})(3\sqrt{2} + \sqrt{5})\)
3. \(\frac{\sqrt{11} - 4}{2\sqrt{11} + 3}\)
4. \((\sqrt{7} + \sqrt{2})\)\((\sqrt{7} - \sqrt{2})\) \(+ (\sqrt{3} - \sqrt{5})\)\((\sqrt{3} + \sqrt{5})\)
5. If \(y = \frac{5 + \sqrt{3}}{\sqrt{3} - 1}\), find \(y\) in its simplest form.
Answers
1. \(-5 - 2\sqrt{6}\)
2. \(13\)
3. \(\frac{34 - 11\sqrt{11}}{35}\)
4. \(3\)
5. \(y = 3\sqrt{3} + 4\)
Quick Test
Test yourself on what you have learnt so far. Click on the link below when you are ready.
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