1. Surds
Surds are numbers that have been left in square root form and are utilized when precise calculations are necessary. They are figures that, if expressed in decimal form, would go on forever.
1.1 Rational Numbers
A number that can be written as an integer (whole number) or a simple fraction is called a rational number. Rational numbers can be terminating decimals or recurring decimals.
For example: 2, 100, -3, 2⁄11.
1.2 Irrational Numbers
Irrational numbers in decimal form are infinite, with no recurring or repeating pattern. E.g. π is an example of an irrational number,
When a root (square root, cube root or higher) gives an irrational number, it is called a surd.
For example: √9 = 3, which is an integer. The square root of 9 is not a surd.
√5 = 2.23606, which is an infinitely long decimal with no recurring or repeating pattern, i.e. an irrational number. The square root of 5 is a surd.
1.3 What do you mean by surds?
Surds are irrational numbers that are left as square roots. An irrational number cannot be expressed as a fraction, and it would be infinitely long in decimal form with no recurring pattern.
Surds can be a square root, cube root, or other root and are used when detailed accuracy is required in a calculation.
The examples of surds are √2, √3, √5, etc., as these values cannot be further simplified. If we further simply them, we get decimal values, such as:
√2 = 1.4142135…
√3 = 1.7320508…
√5 = 2.2360679…
1.4 How to simplify surds?
In order to simplify a surd, follow these steps:
- Find a square number that is a factor of the number under the root.
- Rewrite the surd as a product of this square number and another number, then evaluate the root of the square number.
- Repeat if the number under the root still has square factors.
Example: Simplify √24
Solution : 
Example: Simplify 3 x 4√54
Solution:
1.5 Adding and subtracting surds
In order to add/subtract surds, we follow these steps:
- Check whether the terms are 'like surds'.
- If they aren't like surds, simplify each surd as far as possible.
- Combine the like surd terms by adding/subtracting.
Example - Simplify : √45 - 2√5
Solution:
1.6 Multiplying and Dividing surds
Multiplying surds with the same number inside the square root
We know that: (√3)² = √3 x √3 = √9 = 3
In order to multiply/divide surds, we follow these steps:
- Simplify the surds if possible.
- Use surd laws to fully simplify the numerator and denominator of the fraction.
- Divide the numerator by the denominator.
Example - Simplify :
Solution:
