It is not possible to find the square roots of many numbers.
For example, √3 cannot be written exactly as a fraction or a decimal. This is because it is an irrational number.
Therefore, it is often better to write the number as a square root sign √3.
A surd is therefore – √3.
When multiplying surds, it is squaring the square root.
Therefore, √a × √a = a
Which can also be written as a√a
Also the following is also possible, √a × √b = √ab
Points to note:
- If “a” and “b” are irrational numbers their product can be a rational number
When dividing surds, the following formula is used:
√a ÷ √b = √ a/b
Points to note:
- Similar to multiplication, if “a” and “b” are irrational numbers their product can be a rational number.
To simplify surds it is written in the form a√b.
For example, simplify √75 by writing it in the form a√b.
Start by finding the largest square number that divides into 75.
This is 25. We can use this to write:
√75 = √(25 × 3)
= √25 × √3
Adding and subtracting surds
Adding and subtracting surds is possible if the number under the square root sign is the same.
For example, simplify √27 + √75.
Start by writing √27 and √75 in their simplest forms.
√27 = √(9 x 3)
= √9 x √3
√75 = √(25 x 3)
= √25 x √3
√27 + √75 = 3√3 + 5√3 = 8√3
Rationalizing the denominator
Rationalising the denominator is when a fraction has a surd as the denominator and is written so that the denominator is a rational number.
Here, the numerator and denominator is multiplied by √2 to make the denominator into a whole number. The denominator is then normally re-writen as a rational number.