mathlinE lesson
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Example 4 Express as entire surd. (a) 7 5
(b)  5 8 (c) 9 2 x .
Surds (Suitable for year 8 to 10)
Irrational numbers involving the radical (root) signs,
, 3 , 4 etc, are called surds.
12 , 3 4 are surds but
Example 1
because they are not irrational.
9  3 and
3
9 , 3 8 are not
8  2.
(a) 7 5  49 5  49  5  245 .

The multiplication or division of surds can sometimes
result in a rational number, e.g.
2 18
Simplify (a)
(b) 2 5  3 10 (c)
3 8
3 15
15 3


23 2
3 2 2
1
3 7
6  2 (d)
3 15
15 3
Addition and subtraction of like surds
.
Like surds have the same radical factor when they are
expressed in standard form, e.g. 7 , 5 7 and  2 7
are like surds because they all have the same radical
factor 7 .
(a) 3  7  3  7  21 .
(b) 2 5  3 10  2  3  5 10  6 50 .
(c) 6  2  6  2  3 .
(d)

3 5  2 125  6 625  150
Multiplication and division of surds
Example 2

(b)  5 8  25 8  25  8  200 .
(c) 9 2 x  81 2 x  81  2 x  162 x .
5
3 15 1
.

5 or
5
15 3 5
Change surds to standard form in order to determine
whether they are like or unlike surds.
Only like surds can be added or subtracted.
Entire surds and standard form
243 and 9 3 are equal surds. The former is called
an entire surd and the latter is written in standard
form.
3 27 is also equal to 243 and 9 3 , but it is
neither an entire surd nor a surd in standard form.
A surd in standard form has the number inside the
square (or cube etc) root sign not divisible by a
perfect square (or cube etc) number greater than 1.
Example 5
Express in standard form, 12  4 3 and
2 75  10 3 ,  they are like surds.
Example 6 Simplify (a) 5 7  7 7  3 7
(b) 18  2 2  98 .
(a) 5 7  7 7  3 7  5  7  3 7  7 .
(b)
Example 3 Express in standard form. (a)
(b) 5 32 .
(a) 125  25  5  25 5  5 5 .
(b) 5 32  5 16  2  5 16 2  20 2 .
Are 12 and 2 75 like surds?
18  2 2  98  3 2  2 2  7 2  8 2 .
125
The addition or subtraction of unlike surds cannot be
simplified and it is left in exact form unless you are
instructed to change it to a terminating decimal as an
approximation.
1

Example 11 Rationalise the denominators of the
11
5
following expressions. (a)
(b)
6 2
3 1
3
(c)
.
3 5 2 7
Mixed surd operations

2 .
Example 8 Expand and simplify (a) 5 2 3  2

6 5 32


(c)  8 6  2

6 5 3  2  5


(a) 5 2 3  2  5  2 3  5 2  10 3  5 2 .
(b)
Multiply each denominator by its conjugate surd to
rationalise it.
5 3 1
5 3 1
5
(a)
.


2
3 1
3 1 3  1
3  6  2 6  5 18  2 6
 5 9  2  2 6  15 2  2 6 .
(c)  8 6  2 2   48  2 16   16  3  8

11
(b)
Rationalising the denominator of an expression
6 2
Example 9 Rationalise the denominator of each of
the following expressions.
3
2
2
(a)
(b)
(c)
.
5
7
2 3
7
3
(b)
2 7

7 7
3 5

5
2
(c)

2 7
.
7

15
.
5
5 5
2 3

2 3

2 3 3
3 5 2 7
6
6
.

23 6






(c) 3 5  2 7 3 5  2 7 .
(a)
(b)


3 1

11 6  2

33
6 2



6 2

52 7




11 6  2
.
4
3 5  2 7 3 5  2 7 
33 5  2 7 

.
17
Example 12 Calculate 8721 3  10681 2 accurate
to the 14th decimal place. (Source: one of the first
year uni. textbooks)
8721 3  10681 2
1
8721 3  10681 2 8721 3  10681 2
8721 3  10681 2 =
Example 10 Expand the following pairs of conjugate
3  1 3  1 (b)
6 2 6 2
surds. (a)



Using an indirect approach as shown below, higher
accuracy can be obtained with the calculator.
3 5  2 7 and 3 5  2 7 form a pair of conjugate
surds. The product of a pair of conjugate surds gives a
rational number.




8721 3  10681 2  3.3101  10 5  0.000033101 .
When the difference is calculated directly by means
of a graphics calculator, a value accurate to the 9th
decimal place is obtained.
Conjugate surds


3
(c)
Sometimes it is necessary to change the denominator
of an expression to a rational number.
2



 4 3  8.
(a)

9 5 5 6 5 7 6 7 5 4 7 7
 9  5  4  7  17 .
3 2  27  8  3 2  3 3  2 2  5 2  3 3 .
(b)

(c) 3 5  2 7 3 5  2 7
Simplify 3 2  27  8 .
Example 7

3 1  3 3  3  3 1 3 1 2.
 6  2  6  2  
6 6 6 2 2 6 2 2






1 8721 3  10681 2
228167523  228167522

 3.310115066  10 5
8721 3  10681 2
 0.00003310115066 .
More mixed operations
Example 13 Simplify (a)
1
2

2
(b)
3
3
5

2
.
5 3
= 6 – 2 = 4.
2
Rationalise the denominator(s) before adding or
subtracting.
(a)
1 2
1 2
2
2 2 3 2 4 3 2 4
 
 
 
 
2 3
6
6
6
2 3
2 2 3
3
(b)
2


3 5
2 3


5 5 3
5 5 5 3 3
9 5 2 3 9 52 3



.
15
15
15
Example 14 Simplify
3
Exercise
(1) Express in standard form. (a)
(2) Express as entire surd. (a) 5 7 (b)
3 5 2 3

5
15




 




3  3 4  6 2  7 3  21 8  12 2



2
7
14
14
 7 3  12 2  13

.
14

Locating the exact positions of
number line
2 , 3 , 5 etc on a

(6) Rationalise the denominators of the following
expressions.
2
3
6 3
(a)
(b)
(c)
.
10
2
15
(7) Rationalise the denominators of the following
27
1
expressions. (a)
(b)
3 2
2 1
3
3 7 2 5
3
1
1
2
5
1
2
Start from the origin on a number line with unit
scales. Draw a vertical line segment of unit length at 1
on the number line. Use the length of the line segment
from the origin (as the centre) to the top end of the
vertical line as the radius draw an arc to cut across the
number line. The intercept is 2 .
Draw another vertical line segment of unit length at
2 on the number line. Use the length of the line
segment from the origin (as the centre) to the top end
of the vertical line as the radius draw an arc to cut
across the number line. The intercept is 3 .
Locate 5 , 6 and 7 on the number line below.
0
1 2
(9) Simplify
(10) Simplify
.
1
2
5
2
(b)
.


2
3
2 5 5 2
(8) Simplify (a)
1

(5) Expand and simplify 3 5 2 3  20 .
(c)
Use Pythagoras’ theorem to determine the length of
2 , 3 , 5 etc.
2
8 5.
(4) Simplify 3 24  54  6 8 .
.
2 3
2 2 2 3
3
2 2



1 3
2  3 1 3 1 3
2 3 2 3
1 3
3 1 3

(3) Simplify (a) 15 2  8 2  2
(b) 12  2 3  75 .
2 2

500 (b) 5 27 .
3
33
3 2
3 1

2 3

2 3
3 3
2 1
.
.
(11) Locate 10 , 11 and 17 on the same number
line.
3
3