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SURDS
1. Definition and Manipulation
A surd is an expression involving a square root, cube root etc., whose value is irrational.
Examples are 2 , 3 10 etc. Note that 9 and 20.25 are not surds, because they have rational
values, namely 3 and 4.5 respectively.
We use surds rather than decimals because the surd is exact, whereas the decimal is infinite, and
rounded to a certain number of decimal places can only be an approximation. For example, the
length of x in the triangle below is exactly 21 , and this can be used in further calculations, for
example to find y.
5
y
x
2
6
If we used 4.58 for x, our value for y would be inaccurate. Also, anyone who was told that x was
4.58 would not be able to say for sure it was 21 , but anyone who was told it was 21 could
work out it was 4.58 to two decimal places
Surds can also be manipulated, as follows.
a)
ab  a b
Proof : square both sides...
(LHS)2 = ab × ab = ab
(RHS)2 = a b  a b  a a b b  ab
Therefore (LHS)2 = (RHS)2  LHS = RHS
This property helps us to simplify surds (the smaller the number being rooted, the simpler a
surd is considered to be). We do this by taking out the largest square number possible.
Example 1 :
45  9  5
 9 5
3 5
Example 2 :
50  25  2
 25  2
5 2
Example 3 :
80  2 500  16 5  2 100 5
 4 5  20 5
 24 5
Example 4 :
5 27  2 75  5 9 3  2 25 3
 15 3  10 3
5 3
Example 5 :
63
9 7

3
3
 7
We can also reverse this process.
Example 6 : Write 7 3 as the square root of a single number.
7 3  49 3
 49  3
 147
b)
a
a

b
b
The proof is identical to the proof in a). Feel free to have a go!
This property can help us find square roots of rational numbers...
2 14 
9
4

9
4
3
2
 1 12

...and simplify surd expressions...
3

4
3
4

3
2
Note that a  b  a  b , just as (a  b)2  a 2  b2 . You can test this by substituting numbers
for a and b, or by squaring both sides and finding that they give different results.
C1 p10 Ex 1G
2. Multiplication
Treat these just like expanding brackets containing algebra.
Example 1 :
1  2 3  2 2   3  2
2 3 2 2 2 2
 3 2 2 3 2  4
 1  2
It is a good idea to draw a grid to help you multiply the brackets together.
3
2 2
1
3
2
3 2
2 2
–4
Example 2 :

 
2
3 1 

3 1

3 1
3
1
 3 3  3  3 1
3
3
3
 3  3  3 1
1
3
1
 42 3
3. Division
15
7
4
,
and
so that the surds appear only in
3
2
5 1
the numerator, and not the denominator. This process is known as rationalising the denominator.
We do this by multiplying top an bottom by the same thing.
It is better to simplify expressions such as
• If the denominator is of the form
a , multiply top and bottom by
a.
b . (In other
• If the denominator is of the form a  b , multiply top and bottom by a
words, change the sign within the denominator). This is because the denominator
becomes a  b a  b  a 2  b , which is rational.



• If the denominator is of the form
denominator becomes

a b

a  b , multiply top and bottom by

a  b  a  b , which is rational.
Example 1 :
15 15 3

3
3 3

15 3
3
5 3
a
b . The
Example 2 :
7
3 2

7 2
3 2 2

7 2
6
Example 3 :
4
4
5 1


5 1
5 1
5 1


4
4



5 1
5 1

5 1
4
 5 1
At this point it would be a good idea to check the question and answer have the same decimal
representation by entering them into a calculator.
Example 4 :
7 3
7 3 52 3


52 3 52 3 5 2 3

35  14 3  5 3  6
25  12

41  19 3
13
Example 5 :
5 3
5 3 2 5 3


2 5 3 2 5 3 2 5 3
C1 p11 Ex 1H

10  15  2 15  3
20  3

13  3 15
17
Topic Review : Indices and Surds