SHS 1: E-Maths
Topic: Surds
\(*\) Definition
\(*\) Operations on Surds
\(*\) Rationalization
\(*\) Equations with Surds
Surds belong to the set of irrational numbers.
A surd is an irrational number that cannot be simplified to remove the square root \((\sqrt{})\), or other roots, eg. \((\sqrt[3]{})\), etc.
Surds are typically represented using the radical symbol \((\sqrt{})\). They are irrational numbers left in root form to ensure accuracy in mathematical calculations.
The most common surds are the square roots of non-perfect squares, eg. \(\sqrt{2}\), cube root of non-perfect cubes, eg. \(\sqrt[3]{5}\), etc.
You say:
1. \(\sqrt{a}\) \(\Rightarrow\) root \(a\) or square root of \(a\).
2. \(\sqrt[3]{a}\) \(\Rightarrow\) cubic root of \(a\) or third root of \(a\).
3. \(\sqrt[4]{a}\) \(\Rightarrow\) fourth root of \(a\).
4. \(\sqrt[5]{a}\) \(\Rightarrow\) fifth root of \(a\).
And so on.
Note:
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.
Surds are used when exact values are needed in calculations.
There are basically two (2) kinds of surds:
1. Simple surds
2. Compound surds
Simple surds
Simple surds are surds that have only one term under the radical symbol.
Eg. \(\sqrt{2}\), \(\sqrt[3]{5}\), \(\sqrt{3}\), \(\sqrt[4]{10}\), etc.
Compound surds
A compound surd is an expression that contains two or more surds connected by addition or subtraction.
Eg. \(\sqrt{2} + \sqrt{3}\), \(\sqrt{5} - \sqrt{3}\), etc.
Quick Test
Test yourself on what you have learnt so far. Click on the link below when you are ready.
To perform any operation on surds, you will need to first simplify the given surd before performing the operation. To do that, express the number under the radical symbol as a product of a perfect square (square number), example 4, 9, 16 etc., and a non-perfect square, example 2, 3, 5, 6, etc.
For instance,
\(\sqrt{72} = \sqrt{36 \times 2}\)
Next, write the square root of the square number outside of the radical symbol as shown below:
\(\sqrt{72} = \sqrt{36 \times 2}\)
\(\Rightarrow \sqrt{72} = \sqrt{6^2 \times 2}\)
\(\Rightarrow \sqrt{72} = 6\sqrt{2}\)
Example 1
\(\sqrt{27}\)
Solution
\(\sqrt{27}\)
\(\sqrt{27} = \sqrt{9 \times 3}\)
\(\Rightarrow \sqrt{27} = \sqrt{3^2 \times 3}\)
\(\Rightarrow \sqrt{27} = 3\sqrt{3}\)
Watch a walk through of the solution.
Example 2
\(\sqrt{125}\)
Solution
\(\sqrt{125}\)
\(\sqrt{125} = \sqrt{25 \times 5}\)
\(\Rightarrow \sqrt{125} = \sqrt{5^2 \times 5}\)
\(\Rightarrow \sqrt{125} = 5\sqrt{5}\)
Watch a walk through of the solution.
Example 3
\(\sqrt{160}\)
Solution
\(\sqrt{160}\)
\(\sqrt{160} = \sqrt{16 \times 10}\)
\(\Rightarrow \sqrt{160} = \sqrt{4^2 \times 10}\)
\(\Rightarrow \sqrt{160} = 4\sqrt{10}\)
Watch a walk through of the solution.
Example 4
\(\sqrt{52}\)
Solution
\(\sqrt{52}\)
\(\sqrt{52} = \sqrt{4 \times 13}\)
\(\Rightarrow \sqrt{52} = \sqrt{2^2 \times 13}\)
\(\Rightarrow \sqrt{52} = 2\sqrt{13}\)
Watch a walk through of the solution.
Example 5
\(\sqrt{63}\)
Solution
\(\sqrt{63}\)
\(\sqrt{63} = \sqrt{9 \times 7}\)
\(\Rightarrow \sqrt{63} = \sqrt{3^2 \times 7}\)
\(\Rightarrow \sqrt{63} = 3\sqrt{7}\)
Watch a walk through of the solution.
Exercise 1
Simplify the following surds.
a. \(\sqrt{150}\)
b. \(\sqrt{90}\)
c. \(\sqrt{18}\)
d. \(\sqrt{50}\)
e. \(\sqrt{44}\)
Answers
a. \(5\sqrt{6}\)
b. \(3\sqrt{10}\)
c. \(3\sqrt{2}\)
d. \(5\sqrt{2}\)
e. \(2\sqrt{11}\)
Watch a 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.
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 6
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 7
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 8
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 9
Simplify: \(\frac{1}{2}\sqrt{3} - \frac{1}{3}\sqrt{3}\)
Solution
\(\frac{1}{2}\sqrt{3} - \frac{1}{3}\sqrt{3}\)
\(\Rightarrow (\frac{1}{2} - \frac{1}{3})\sqrt{3}\)
\(\Rightarrow (\frac{3(1) - 2(1)}{6})\sqrt{3}\)
\(\Rightarrow (\frac{3 - 2}{6})\sqrt{3}\)
\(\Rightarrow \frac{1}{6}\sqrt{3}\)
\(\Rightarrow \frac{\sqrt{3}}{6}\)
Watch a walk through of the solution.
Example 10
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.
Exercise 2
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. \(3\sqrt{7}\)
Quick Test
Test yourself on what you have learnt so far. Click on the link below when you are ready.
Multiplication
Follow the rule below when finding the product of two or more surds.
\[\sqrt{x} \times \sqrt{y} = \sqrt{x \times y}\]
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.
Example 11
Simplify:
\((\sqrt{3})^3 (\sqrt{7})^4\)
Solution
\((\sqrt{3})^3 (\sqrt{7})^4\)
\(\Rightarrow (\sqrt{3} \times \sqrt{3} \times \sqrt{3})\)\((\sqrt{7} \times \sqrt{7} \times \sqrt{7} \times \sqrt{7})\)
\(\Rightarrow (\sqrt{3\times3\times3})\)\((\sqrt{7\times7\times7\times7})\)
\(\Rightarrow (\sqrt{3^2\times3})\)\((\sqrt{7^2\times7^2})\)
\(\Rightarrow (3\sqrt{3})\)\((7\times7)\)
\(\Rightarrow 3\sqrt{3}\times 49\)
\(\Rightarrow 147\sqrt{3}\)
Example 12
Simplify:
\((\sqrt{5})^3 (\sqrt{2})^3\)
Solution
\((\sqrt{5})^3 (\sqrt{2})^3\)
\(\Rightarrow (\sqrt{5} \times \sqrt{5} \times \sqrt{5})\)\((\sqrt{2} \times \sqrt{2} \times \sqrt{2})\)
\(\Rightarrow (\sqrt{5\times5\times5})\)\((\sqrt{2\times2\times2})\)
\(\Rightarrow (\sqrt{5^2\times5})\)\((\sqrt{2^2\times2})\)
\(\Rightarrow (5\sqrt{5})(2\sqrt{2})\)
\(\Rightarrow 5\times2\times\sqrt{5}\times\sqrt{2}\)
\(\Rightarrow 10\times\sqrt{5\times2}\)
\(\Rightarrow 10\sqrt{10}\)
Example 13
Solution
Example 14
Solution
Example 15
Solution
Exercise 3
Solution
Quick Test
Test yourself on what you have learnt so far. Click on the link below when you are ready.
Multiplication of Surds of the form \(a + b\sqrt{n}\)
We often use the distributive property when it comes to multiplication of surds of the form \(a + b\sqrt{n}\).
Hence:
\(\sqrt{x}(a + b\sqrt{n})\)
\(\Rightarrow (\sqrt{x} \times a) + (\sqrt{x} \times b\sqrt{n})\)
\(\Rightarrow a\sqrt{x} + b\sqrt{x \times n}\)
\(\Rightarrow a\sqrt{x} + b\sqrt{xn}\)
Also;
\((x + y\sqrt{n})(a + b\sqrt{n})\)
\(\Rightarrow\) \(x(a + b\sqrt{n}) + \) \(y\sqrt{n}(a + b\sqrt{n})\)
Exercise 4
Answers
1.
2.
3.
4.
5.
Exercise 5
Answers
1.
2.
3.
4.
5.
Exercise 6
Answers
1.
2.
3.
4.
5.
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.
To advertise on our website kindly call on 0208711375 or 0249969740.