1. Surds
Objective: Use surds to write the value of a number having an irrational square root such as, for example, √20 in surd form
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2. Goes on forever and has no recurring pattern
Sometimes, the square root of a number is a whole number All of these square roots, for example, are whole numbers:
√9 = 3, √25 = 5, √81 = 9, √625 = 25
But often this is not the case and the answer is a decimal that goes on forever without recurring.
The square root of 2 is an example of this
√2 = …
Goes on forever and has no recurring pattern

3. The Greeks discovered this in their work with Pythagoras’ theorem.1
They could see that a right angle triangle like this one had a hypotenuse with an exact size but could not calculate it exactly! 1
h2 =
h2 = 1 + 1
h2 = 2
h = √2
h = …

4. We call numbers like the √2 irrational numbers because we know that they exist but cannot find their exact values.
The square root of any prime number, for example, is an irrational number.
In real life, we rarely need exact answers so we can find an approximation for an irrational number of a suitable accuracy to fulfil our needs.

5. This is called surd form
But pure mathematicians want to give exact answers, and they would write an irrational square roots exactly like this
√50 = …
√50 = 5√2
This is called surd form

6. In the GCSE, there is often a question where you are asked to write a square root in surd form. Here is how to do it…

7. Square numbers can be found by
To write a number in surd form, we need to know the square numbers. Here are the first fourteen square numbers:
4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225
Square numbers can be found by
2 x 2 = 4
3 x 3 = 9
4 x 4 = 16
5 x 5 = 25
etc.

8. Here is an example of how we would use a square to write √18 in surd form…
Use this square number to write √18 as…
Find the largest square number that will divide exactly into 18
√18 = √(9 x 2)
 √18 = 3√2
18 ÷ 4 = 4 r 2
18 ÷ 9 = 2
Take out the square root of 9 like this…
This is √18 in surd form
9 is the largest

9. Here is another example of how to write √75 in surd form…
Use 25 to write √75 as…
Find the largest square number that will divide exactly into 75
√75 = √(25 x 3)
 √75 = 5√3
75 ÷ 4 = 18 r 3
75 ÷ 9 = 8 r 3
75 ÷ 16 = 4 r 11
72 ÷ 25 = 3
Take out the square root of 25 like this…
This is √75 in surd form

10. Write in surd form 1. √8 6.
√32 2.
√12 7.
√50 3.
√20 8.
√63 4.
√28 9.
√45 5.
√27
10.
√300

11. Write in surd form 1. √8
= 2√2 6.
√32
= 4√2 2.
√12
= 2√3 7.
√50
= 5√2 3.
√20
= 2√5 8.
√63
= 3√7 4.
√28
= 2√7 9.
√45
= 3√5 5.
√27
= 3√3
10.
√300
= 10√3