Explanatory slides on the topic SURDS

1. SURDS
for WASSCE and UTME

2. Introduction
A number whose square root gives a whole number is called
a perfect square.
For example, 4, 9, 16, 25, 36 are perfect squares
because their square roots equal 2, 3, 4, 5, 6 respectively.
When the square root is not a whole number, we refer to it
as a surd
For example, 2, 3, 5, 6 are called surds

3. Laws of Surds
1. 𝑚𝑛 = 𝑚 × 𝑛
2.
𝑚
𝑛
=
𝑚
𝑛
Recall from indices that
𝑎𝑏 2
= 𝑎2
𝑏2
𝑎 + 𝑏 2
≠ 𝑎2
+ 𝑏2
Verify this yourself (put 𝑎 = 2, 𝑏 = 3)

4. Simplification of Surds
Surds can be simplified by
breaking the number into a product of two factors where one of
them is perfect square and then applying the laws of surds
Example: Simplify the following surds
1. 18
2. 48

5. Simplification of Surds
18 = 9 × 2
= 9 × 2
= 3 2
48 = 16 × 3
= 4 3
( 6 × 3 won’t help us)

6. Addition and Subtraction of Surds
Surds can be added or subtracted together if and only if they have
the same basic form.
For example, 3 2 + 5 2 = 8 2
but 2 3 + 3 2 = 2 3 + 3 2
Example: Simplify 18 + 50 − 72
9 × 2 + 25 × 2 − 36 × 2
3 2 + 5 2 − 6 2
2 2
In other words, they can’t be
added

7. Addition and Subtraction of Surds
Example: Simplify 75 − 48 + 27
5 3 − 4 3 + 3 3
4 3

8. Multiplication of Surds
Surds can be multiplied together by multiplying number for
number and surd for surd
Example: Simplify 2 3 × 4 5
(2 × 4) 3 × 5
8 15
Example: Expand 2 + 4 3 5 − 2
10 − 2 2 + 20 3 − 4 6
Recall (𝑎 + 𝑏)(𝑥 + 𝑦)

9. Division of Surds
Example: Simplify
4
2
It is not normal practice to have a surd as the denominator
of a fraction. To remove the surd, multiply by another surd
that would make the denominator a whole number
4
2
×
2
2
4 × 2
2 × 2
=
4 2
2
= 2 2
Mathematically, we have only just
multiplied by 1 which is nothing new

10. Division of Surds
Example: Simplify
1
3− 2
Multiply by the conjugate of the denominator.
The conjugate of 3 − 2 is 3 + 2
1
3 − 2
×
3 + 2
3 + 2
3 + 2
3 + 6 − 6 − 2
3 + √2
3 − 2
3 + 2
Change the sign in the middle
to get the conjugate

11. Worked Examples
If 𝑎 = 5 2 and 2𝑎 = 2𝑥, what is the value of 𝑥
2 × 5 2 = 2𝑥
10 2 = 2𝑥
200 = 2𝑥
𝑥 = 100
take squares both sides

12. Worked Examples
Given that sin 60 =
3
2
, sin 30 =
1
2
, find the value of sin 60 +
sin 30
sin 60 + sin 30 =
3
2
+
1
2
=
3 + 1
2

13. Worked Examples
Rationalize the denominator
2 2 − 3
3 − 2
2 2 − 3
3 − 2
×
3 + 2
3 + 2
2 6 + 4 − 3 3 − 3 2
3 − 2
2 6 + 4 − 3 3 − 3 2