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Chapter 10
Surds (Radicals)
【Example 1】Calculate:
(1) ( 4)2 ;
(2) ( 5) 2 ;
2
10.1 Surds
The square root of a, a ( a ≥ 0 ) where a appears under a
square root sign (or radical sign) is called a Surd (or a radical). For
3
, b 2 + 1 , (a − b) 2 are surds. In the doman of
example, 3 ,
5
real numbers, we cannot take square root of negative numbers.
Therefore, −5 and a ( a < 0 ) are undefined. In this chapter, all
alphabets are positive numbers unless otherwise specified.
As mentioned in the previous chapter, the positive square root of
a positive number
is also called the principal square root, labelled
as a . This indicates that a is a positive number and a > 0 .
Sqaure root of 0 is also called the principal square root of zero,
labelled as 0 and we have 0 = 0 . According to the above
analysis, it can be observed that a ≥ 0 ( a ≥ 0 ), that is a ( a ≥ 0 )
always a non-negative number (a non-negative number is either zero
or positive.)
According to the definition of square root, if a number when
squared equals 2, then the number is called the square root of 2.
Therefore, we have the relaltionship that the square of the square root
of 2 is 2:
 3
(4) (2 3) 2 .
(3) 
 ;
 5
Solution (1) ( 4) 2 = 22 = 4 or ( 4)2 = 4 ;
(2) ( 5) 2 = 5 ;
2
 3 3
(3) 
 = ;
 5 5
(4) (2 3) 2 = 22 × ( 3) 2 = 4 × 3 = 12 .
Reversing the above formula ( a ) 2 = a ( a ≥ 0 ) will result in:
a = ( a ) 2 ( a ≥ 0 ).
By making use of this formula, we can rewrite any non-negative
number in the form of a square of another number.
【Example 2】Rewrite the following non-negative number in the
form of a square:
(1) 2 ;
1
;
(4) ab.
7
(2) 0.5 = ( 0.5)2 ;
(3)
2
1  1
(3) = 
 ;
7  7
(4) ab = ( ab ) 2 .
Practice
1. Calculate:
2
(1) ( 0.5) ;
( a ) 2 = a (a ≥ 0 )
3
means
2
 2
(2) 
 ;
 7
2
- 37 -
(2) 0.5 ;
(1) 2 = ( 2)2 ;
Solution
( 2)2 = 2 .
In general, if a number when squared equals a, then the number
is called the square root of a. Therefore, we have the relationship that
the square of the square root of a is a:
3
.
In general,
2
(3) (5 7 ) ;
means
- 38 -
.

1
(4)  −3  .
3

Practice
a 2 and | a | , we have:
By comparing
2. Rewrite the following non-negative number in the form of a
square:
(1) 9;
(2) 6;
(3) 2.5;
(4) 0.25;
(5) b;
(6) 4a.
We now examine the principal square root of a 2 , i.e. a 2 ,
when a takes on different values, namely a > 0 , a = 0 and a < 0 .
(1)
22 = 2 ,
a2 = a .
(1)
(−1.5) 2 ;
(2)
(a − 3)2 ( a < 3 ).
02 = 0 ;
In other words, when a = 0 ,
(3)
【Example 3】Calculate:
32 = 3 ;
In general, when a > 0 ,
(2)
 a (a > 0)

a =| a | =  0 (a = 0)
−a (a < 0)

2
a2 = a .
(−2) 2 = 4 = 2 = −(−2) since 2 and −2 are opposite
(2)
numbers
(−3)2 = 9 = 3 = −(−3) since 3 and −3 are opposite
a 2 = −a .
Summarising the above, we have:
 a (a > 0)

2
a =  0 (a = 0)
− a (a < 0)

and it is known that
 a (a > 0)

| a | =  0 (a = 0)
− a (a < 0)

- 39 -
(a − 3) 2 =| a − 3 |
∵ a<3
∴ a −3 < 0
∴
numbers
In general, when a < 0 ,
(−1.5) 2 =| −1.5 |= 1.5 ;
Solution (1)
(a − 3)2 =| a − 3 |= −(a − 3) = 3 − a
Practice
1. (Mental) Is the following equation correct? Why?
(1) ( 7 )2 = 7 ;
(2) (− 7 )2 = −7 ;
(4) (−6) 2 = −6 .
(3) 62 = 6 ;
2. (Mental) Fnd the value of each of the following expression:
(1) ( 0.8)2 ;
(2) 0.82 ;
(3) (−0.8) 2 ;
3. Simplify:
(4) − (−0.8) 2 .
2
(1)
(5 − 9) ;
(2)
 1

3 − 2 ;
 2

(3)
(b − 4) 2 ( b > 4 );
(4)
(m − n) 2 ( m < n ).
2
- 40 -
In general, we have:
Practice
4. Two persons, A and B, calculated the value of a + 1 − 2a + a 2 .
They obtained different answers when a = 5 .
A calculated the result as:
a + 1 − 2a + a 2 = a + (1 − a ) 2 = a + 1 − a = 1
B calculated the result ast:
ab = a i b (a ≥ 0 , b ≥ 0 )
That is to say, the principal square root of a product is equal
to the product of the principal square roots of the constituent
factors.
【Example 1】Calculate:
a + 1 − 2a + a = a + (a − 1) = a + a − 1
2
2
= 2a − 1 = 2 × 5 − 1 = 9
Which answer is correct? For the answer that is incorrect, which
part of the calculation is incorrect? Why?
Solution (1)
(1)
16 × 81 ;
(3)
17 − 8 .
2
(2)
0.09 × 0.25 ;
2
16 × 81 = 16 × 81 = 4 × 9 = 36 ；
(2)
0.09 × 0.25 = 0.09 × 0.25 = 0.3 × 0.5 = 0.15 ；
10.2 Basic property of Surd
(3)
17 2 − 82 = (17 + 8)(17 − 8) = 25 × 9 = 5 × 3 = 15 .
As we understand, the surd a ( a ≥ 0 ) is the principal square
root of a. Therefore, when we study the property of Surd, it is suffice
to study the property of the principal square root.
Practice
1. Calculate:
(1)
1.
Product of principal square roots
49 × 121 ;
(2)
81×169 ;
(3)
9 × 25 × 225 ;
(4) 262 − 102 ;
(5) 0.652 − 0.162 ; (6) 25a 4b6 c 2 .
2. Is each of the following equation correct? Why?
Let us examine the following example:
( 4 × 9)2 = 4 × 9 = 36
(1)
32 + 42 = 3 + 4 = 7 ;
(2)
412 − 402 = 41 − 40 = 1 .
( 4 × 9)2 = ( 4) 2 × ( 9) 2 = 4 × 9 = 36
From the first equation, we know that
4 × 9 is a square root of
36. Since 4 × 9 is positive, it is the principal square root of 36.
From the second equation, we know that 4 × 9 is a square
root of 36. Since 4 × 9 is positive, it is the prinicipal square root
of 36.
As the principal square root of a number is unique, so 4 × 9
and 4 × 9 must be the same. That is
4×9 = 4 × 9 .
- 41 -
【Example 2】Simplify:
Solution (1)
(1)
102 × 2 ;
(3)
4a 2b3 ;
(2)
(4)
48 ;
x4 + x2 y 2 .
102 × 2 = 102 × 2 = 10 2 ;
(2)
48 = 42 × 3 = 42 × 3 = 4 3 ;
(3)
4a 2b3 = 2 2 i a 2 i b 2 i b = 2ab b ;
(4)
x4 + x2 y 2 = x2 ( x2 + y 2 ) = x x2 + y2 .
- 42 -
It can be observed from Example 2 that, for any squared factors
inside the radical sign, we can take the principal square root of them
and move them outside the radical sign. On the other hand, we can
also transform any non-negative factor(s) from the outside of the
radical sign to the insdie of the radical sign by squaring them.
【Example 3】 For the following expression, move the factors from
the outside of the radical sign to the inside of the
radical sign without changing the value of the
expression:
(1) 5 3 ;
(2) −3 a ;
(3) 4b bc .
Practice
2. For the following expression, move the factors from the outside
of the radical sign to the inside of the radical sign without
changing the value of the expression:
(2) −7 3 ;
(3) 6 5 ;
(1) 5 2 ;
c
b
;
(6) a
.
2
a
3. (Mental) Is the following equation correct? Why?
(5) −12
(4) 2 0.5 ;
(2) −3 2 = (−3)2 × 2 = 18 ;
(1) 2a b = 2a 2b ;
Solution (1) 5 3 = 52 × 3 = 75 ;
(3) 3
(2) −3 a = − 32 i a = − 9a ;
a
= a.
3
(3) 4b bc = (4b) 2 i bc = 16b3c .
Think for a while: Why is it not correct to write −3 a = (−3) 2 a
2.
= 9a ?
【Example 4】For the following expression, move the factors from
the outside of the radical sign to the inside of the
radical sign without changing the value of the
1
expression: (1) 10 0.1 ;
(2) 5
.
5
Solution (1) 10 0.1 = 102 × 0.1 = 10 ;
(2) 5
(7)
9x ;
1
9a 2bc3 ;
6
2
 2
2

 =
5
 5
2
 2  ( 2) 2 2
=

 =
( 5)2 5
 5
(2) − 27 × 15 ;
(3)
212 − 42 ;
2
is a square root of
5
2
2
2
. Since
is positive, it is the principal square root of .
5
5
5
2
From the second equation, we know that
is a square root
5
(5)
5a 3 ;
(6)
8x2 y3 ;
of
(8)
16( x + 2)3 .
Practice
(4)
Let us look at the following example;
From the first equation, we know that
1
1
= 52 × = 5 .
5
5
1. Simplify:
(1) 18 ;
Quotient of principal square roots
- 43 -
2
. Since
5
2
2
is positive, it is the principal square root of .
5
5
- 44 -
As the principal square root of a number is unique, so
2
and
5
2
are the same. That is
5
The above example shows a very useful transformation. There
is a surd b in the denominator which we would like to eliminate.
We obserse that, by multiplying both the numerator and the
denominator by the same surd b , the denominator is converted to
2
2
=
.
5
5
In general, we have:
a
a
=
(a ≥ 0 ， b > 0 )
b
b
That is to say, the principal square root of a quotient is equal
to the quotient of the principal square root of the dividend
divided by the principal square root of the divisor.
【Example 5】Calculate:
Solution (1)
(1)
4
;
9
(2)
1
(3)
3
;
100
(4)
25 x 4
.
81 y 2
4
4 2
=
= ;
9
9 3
Let us look at another example:
a
aib
ab 1
=
=
=
ab .
b
bib
b2 b
15
;
49
b 2 = b . which does not fall under a radical sign and is no longer a
a
1
is thus simplified to
ab which does
b
b
not have a surd in its denominator.
surd. The expression
【Example 6】For the following expression, transform the surd to
eliminate the denominator under the radical:
2
1
(1)
;
(2)
1 ;
3
7
a −5
4x
(3)
;
(4)
( a > 5 ).
3y
a+5
Solution (1)
2
2×3 1
=
=
6;
3
3× 3 3
(2)
1
8
8× 7
22 × 2 × 7 2
1 =
=
=
=
14 ;
7
7
7×7
7×7
7
3
3
1
=
=
3;
100
100 10
(3)
4x
4x i 3y
2
=
=
3xy ;
3y
3y i 3y 3y
25 x 4
25 x 4 5 x 2
=
=
.
81y 2
9y
81y 2
(4)
a −5
(a − 5)(a + 5)
1
a 2 − 25 .
=
=
2
a+5
(a + 5)
a+5
(2)
1
(3)
(4)
15
64
64 8
1
=
=
= =1 ;
49
49
7
49 7
- 45 -
- 46 -
Practice
1. Calculate:
25
(1)
;
64
36 × 9
(4)
;
121
b
ab a 2
= a2 2 =
ab = a ab
a
a
a
b
Although the surds
a 3b and a 2
are different in
a
appearance, they can both be transformed into a comparatively
a2
(2)
(5)
4
;
225
0.04 × 144
;
0.49 × 169
(3)
(6)
0.01
;
0.16
1
4 ;
9
simplified form as a ab . When we compare a ab with
a 3b
6
49m 2 n
;
(8)
.
4a 2
9c 2
2. Transform the surd by eliminating the denominator under the
radical sign:
1
7
1
(1)
;
(2)
;
(3) 5 ;
2
12
3
5
27
n
(4) 6
;
(5)
;
(6)
;
6
2x
3m 2
a
a−b
(7)
;
(8)
( a > b ).
50
a+b
3. (Mental) Is following equation correct? Why?
3
3 1
(1)
(2)
3;
=2 3;
=
4
2 2
8
a
1
(3)
= 2;
(4)
=
a.
2
9b 3b
b
, we note that the surd in the form of a ab
a
satisfies the following 2 conditions:
(1) The power of every factor under the radical sign is less
than 2;
(2) There is no denominator under the radical sign.
A surd satisfying these 2 conditions is said to be expressed in its
y
simplest form. For example: 4 5a ,
, a 2 + b are surds in
2
c
simplest form, while 4a 3 ,
and 8 are not.
3
For a surd which is not presented in its simplest form, we can
transform the surd to its simplest form using the method described
above (i) to eliminate the denominator under the radical, and (i) to
move the squared factors to the outside of the radical.
After transforming the surd to its simplest form, the surd will
consist of two parts: (i) a part under the radical sign, and (ii) a part
outside the radical sign which is the coefficient of the surd.
10.3 Transforming Surd to Simplest Form and
Identifying Like Surds
【Example】Transform the following surd to its simplest form:
(7)
1.
and with a 2
(1)
Surd in its Simplest Form
Let us look at the following example:
a 3b = a 2 i ab = a 2 i ab = a ab
- 47 -
12 ;
(2)
1
(3) 4 1 ;
2
(4)
- 48 -
1
;
3
x2
y
.
x
Solution (1)
2.
12 = 22 × 3 = 2 3 ;
1
3
3
=
=
;
3
3× 3
3
1
3
6
(3) 4 1 = 4
=4
=2 6;
2
2
4
y
xy
= x 2 2 = x xy .
(4) x 2
x
x
Note: When transforming a surd to its simplest form, an essential
step is to factorise the values under the radical sign into prime
factors.
(2)
Practice
1. Distinguish which surd is in its simplest form. If it is not,
transform it to its simplest form.
ab
(3)
;
(1) 45 ;
(2) 25a3 ;
4
b
2
(5)
;
;
(6)
(4) 14 ;
a
2
4y
(8)
;
(7) 6a 2b3 ;
(9) a + b 2 .
5x
2. Transform the following surd into its simplest form:
(1) 3 216 ;
(4)
1
1 ;
3
(7) x 2
1
;
8x3
32 ;
(3)
(5)
20a 2b
;
a
(6) 2 a 3b3 ;
ab
;
( a + b) 2
- 49 -
By simplifying
1
a−b
(a > b)
(9) (a − b)
12 and
1
to the simplest form, we have
3
12 = 22 × 3 = 2 3
1
3
1
=
=
3
3
3× 3 3
1
3 are the
3
same number, namely 3. If there are a number of surds, which after
having been transformed to the simplest form, have the same number
1
inside the radical, we call them Like Surds. For example, 12 ,
3
1
and
3 are like surds. a ab and 3 ab are also like surds.
2
However, 2 and 3 are unlike surds. Similarly, a and 3a
are also unlike surds.
We have learnt that, when adding like terms under a polynomiial,
we can group and combine like terms together. In a similar manner,
when adding like surds under an expression, we can group and
combine like surds together.
The number under the radical sign for 2 3 and
【Example 1】
8
;
9
(2)
(8)
Identifying Like Surds
Solution ∵
Idnetify from the following surds, which of them
are like surds?
1
1
a
2
2 , 75 ,
,
, 3,
8ab3 , 6b
.
50
27
3
2b
75 = 52 × 3 = 5 3
1
2
1
=
=
2
50
50 × 2 10
1
3
1
3
=
=
27
27 × 3 9
- 50 -
2
2
4b
8ab3 = i 2b 2ab =
2ab
3
3
3
a
a i 2b
6b
= 6b
= 3 2ab
2b
2b i 2b
∴
1
are like surds.
50
1
75 ,
, 3 are like surds.
27
a
2
8ab3 , 6b
are like surds.
3
2b
2,
【Example 2】Group and combine like surds in the following
expression
1
1
(1) 2 2 −
3+
2− 2+ 3；
2
3
(2) 3 xy − a xy + b xy 。
1
1
Solution (1)
2 2−
3+
2− 2+ 3
2
3
4
1
1 

 1 
=  2 + − 1 2 +  − + 1 3 =
2+
3
3
2
3 
 2 

(2) 3 xy − a xy + b xy = (3 − a + b) xy
1. Identify which of the following surds are unlike surds?
63 ,
(2)
28 ;
(3)
4x3 , 2 2x ;
(5)
2x ,
2a 2 x3 ,
3
−3 5 ;
3
(3) 6 3 + 0.12 + 48 ;
xy
5
(4)
xy − 2 xy −
.
2
2
(2)
(4)
50xy 2 .
12 ,
18 ,
1
;
3
2
50 , 2
;
9
27 , 4
1. For what value of real number a would the following expression
be defined in the domain of real numbers?
a−2 ,
2−a ,
a+2 ,
(a − 2)2
2. Rewrite the following expressions into difference of two squares,
and further factorise it.
(1) x 2 − 9 ; (2) a 2 − 3 ; (3) 4a 2 − 7 ; (4) 16b 2 − 11 .
3. Calculate:
(1) ( 11)2 ;
(−13) 2 ;
(2)
(3) − (5 × 6) 2 ;
2
6
a ;

2
(5)  −7
 ;
7

(7)
x 2 − 2 x + 1 ( x ≥ 1 );
(9)
x 2 − 4 x + 4 ( x < 2 );
(6)
4. Calculate:
9 × 25 ;
(1)
25 × 81× 289 ;
( x + 5) 2 ;
(8) ( x − y )2 ( x ≥ y );
(10) x + y + x 2 − 2 xy + y 2 ( x < y ).
(3)
- 51 -
5+ 3+2 5−
Exercise 3
(4)
Practice
(1)
Practice
2. Group and combine like surds in the following expression:
(1) 6 a + 2 b − 4 a + 3 b ;
(2)
36 × 256 ;
(4)
132 − 122 ;
- 52 -
(5)
652 − 162 ;
(6)
9a 2 ;
(7)
( x + y)2 c2 ;
(8)
(a + b)2 (a − b) 2 ( a < b ).
5. Simplify:
(1)
56 × 3 ;
(2)
242 × 49 ;
(3)
(−32)(−15) ;
(4)
4x3 ；;
(5)
7a 4 ;
(6)
5a ( x + a ) 3 ;
(7)
8(a + b) 4 (c − d ) 4 ;
a 2 n (n is an integer).
(8)
6. Transferm the expression by moving the positive factors outside
the radical sign to the inside of the radical sign without changing
the value of the expression
1
(3) 4
;
(2) −5 7 ;
(1) 2 6 ;
2
2
1 1
(5)
3;
(6) ab
+ .
(4) −2a b ;
3
a b
7. Calculate:
9
(1)
;
49
(2)
34
2 ;
81
(4)
0.01× 64
;
0.36 × 4
(5)
27
;
100
(7)
18a 2
;
4b 2
 1  2
(8) 1  −  
 25   5 
2
(3)
0.16
;
0.0225
(6)
25 y 4
;
121x 6
2
(9)
1302 − 662 .
8. Transform the surd by eliminating the denominator under the
rdical sign:
1
3
127
(1)
;
(2) 2 ;
(3)
;
6
11
32
(4) 8
3
;
128
(5)
n3
;
9m
- 53 -
(6) a
b
;
a5
(7)
7(a − b)
( a > b );
27(a + b)
(9)
a+b
( a < b ).
( a − b) 2
1 1
−
( a < b );
a 2 b2
(8) a
9. Transform the surd to the simplest form:
1
(2) 6
;
(1) 72 ;
8
4
(3) 10 1 ;
5
2
(−8) − 4 × (−4) (5)
2
(4)
 1  + 1
3   
 2 2
25m3 + 50m2 ;
(7)
(8)
2
(6)
2a 2
3b
x
;
y
b3 b 2
−
( b > 1 );
a4 a4
a
a 2b − 4ab 2 + 4b3
( a < 2b ).
a − 2b
a
(9)
10. When a = 1 , b = 10 , c = −15 , find the value of the expression
−b + b 2 − 4ac
(express the surd in the simplest form in the
2a
answer).
11. When a = 2 , b = −8 , c = 5 , find the value of the expression
−b − b 2 − 4ac
(express the surd in the simplest form in the
2a
answer).
12. In the following surds, which of them are like surds?
5
1
4
1
8 , 20 , −
,
, 3
− 121a 3 , a
, 2 a 3b 3 c ,
16
18
5
a
c
3 a bc , 4
,
ab
3
3
1
( m > p ),
mn − np
- 54 -
n3
( m > p ).
m− p
13. Group and combine the like surds in the following expression.
3
2
(1)
2 + 3 +3 2 −
−
;
3
2
2
1
(2)
125 + 3
− 4 216 − 3
;
27
5
(3) 5 xy − 7 x − 3 yx + 4 x ;
(4)


2a 3ab 2 −  b 27a 3 − 2ab 3a  .
4 
5
10.4 Addition and subtraction of Surds
Addition and subtraction of Surds is the grouping and combining
of like surds, similar to the addition and subtraction of like terms in
polynomials. Before we can combine like surds, we shall need to
transform all surds to the simplest form first. In other words, to
perform addition and subtraction of expressions involving surds, we
shall need to transform all surds into the simplest form, then we can
group and combine like surds together.
1
【Example 1】Calculate 2 12 − 4
+ 3 48 .
27
1
4
+ 3 48 = 4 3 −
3 + 12 3
Solution 2 12 − 4
27
9
4


=  4 − + 12  3
9


140
=
3
9
- 55 -
2
x
1
9x + 6
− 2x
.
3
4
x
2
x
1
Solution
9x + 6
− 2x
= 2 x +3 x −2 x = 3 x .
3
4
x
【Example 2】Calculate


1  1
【Example 3】Calculate  32 + 0.5 − 2  − 
− 75  .
3  8




1  1
Solution  32 + 0.5 − 2  − 
− 75 
3  8


1
1
1
= 32 +
−2
−
− 75
2
3
8
1
2
1
=4 2+
2−
3−
2 +5 3
2
3
4
2
1 1
=  4 + −  2 +  5 −  3
3

2 4

17
13
=
2+
3
4
3
Practice
1. Calculate:
(1) 5 2 + 8 − 7 18 ;
(2)
2
1
−2
;
(4)
5
10
1
8 1
(5)
32 +
−
50 ;
(6)
3
2 5
1
x
1
(7) x
+ 4y −
+y
;
x
y
2
(3) 3 40 −
28 + 9 112 ;
12 +
1
1
−
;
27
3
2 x − 8 x 3 + 2 2 xy 2 ;
−b + b 2 − 4ac −b − b 2 − 4ac
(8)
+
( b 2 > 4ac ).
2a
2a
- 56 -
10.5 Multiplication of Surds
Practice
2. Calculate:
(1) 18 − ( 98 − 2 75 + 27) ;
(2)
(3)
(4)
(5)
(6)
 1

( 45 + 108) +  1 − 125  ;
 3



2  1
− 6;
 24 − 0.5 − 2
−
3  8


1
3
( 2 + 3) − ( 2 − 27 ) ;
2
4
a
a 2 8a + 3a 50a 3 −
18a 3 ;
2


a 2 3  
b
+
a b  −  3a
+ 9ab  .
 4b
b a
a

 

3. Find the value of the following expression (correct to the nearest
0.01):
2
2 1
−
(1) 2 +
54 ;
3
3 5
 1 1
 5 4

(2)  5
+
20  − 
− 45  .
 5 2
 4 5

By reversing the equation
a i b = ab .
We can apply this formula to calculate the Multiplication of
Surds. From the formula, we know that the product of two surds is a
surd with the value under the radical sign being the product of the
values under the radical sign of the original surds.
【Example 1】Calculate:
Solution (1)
8 + 18
= 4 + 9 = 2+3=5.
2
(1)
14 i 7 ; (2) 3 5a i 2 10b .
14 i 7 = 14 × 7 = 7 2 × 2 = 7 2 ;
(2) 3 5a i 2 10b = 3 × 2 5a i 10b = 30 2ab .
Note: If the result of an operation on surds gives rise to some surds,
in general, we shall need to transform the surds to the simplest form
to see if they can be further simplified.
【Example 2】Calculate:
Solution (1)
4. Is the following equation correct? Why?
(1) 2 + 3 = 5 ;
(2) 2 + 2 = 2 2 ;
(3) a x − b x = (a − b) x ;
(4)
ab = a i b , we get
(2)
 8
(1) 
−5
 27
(2) (5 + 6)(5

3 i 6 ;

2 − 2 3) .
 8

8
i 6 −5 3 i 6
−5 3 i 6 =

27
 27

8
=
× 6 − 5 3× 6
27
4
= − 15 2
3
(5 + 6)(5 2 − 2 3) = 25 2 − 10 3 + 5 12 − 2 18
= 25 2 − 10 3 + 10 3 − 6 2
= 19 2
Multiplication of a sum of surds by another sum of surds is
similar to the multiplication of polynomials. Whatever multiplicative
- 57 -
- 58 -
rules that can be applied to the multiplication of polynomials can also
be applied to the multiplication of the sum of surds.
【Example 3】Calculate:
(1) (2 3 + 3 2)(2 3 − 3 2) ;
(2 3 + 3 2)(2 3 − 3 2) = (2 3) 2 − (3 2)2
= 12 − 18
= −6
2
2
(4 + 3 5) = 4 + 2 i 4 i 3 5 + (3 5)2


1

(2)  5 − 2 3   3 5 −
3;
2 
 3

(3) ( a + b )( a − c ) ;
(4) (2 x + y )( x − y ) .
4. Calculate:
(1) (4 − 3 5)(4 + 3 5) ;
(2) (7 2 + 2 6)(2 6 − 7 2) ;
= 16 + 24 5 + 45
(3)
= 61 + 24 5
( 6 − 3 3) 2 = ( 6)2 − 2 i 6 i 3 3 + (3 3) 2
(3) ( 4 x + 3 − 2 x )( 4 x + 3 + 2 x ) ;
= 6 − 18 2 + 27
2
= 33 − 18 2
【Example 4】Calculate:
(2)
(4) ( 3 + 2 2) ;


(5)  −1 − 3  ;
 2 
(6) (4 7 − 7 3) ;
 a
b
(7) 
+
 ;
a
 b
2
(1) ( 3 + 6)( 3 − 6) ;
2
2
(2) (2 ax − 5 by )(2 ax + 5 by ) .
Solution (1)
(2) 2 5( 10 + 4 12) ;
(3) ( 2 + 2 12 − 6) i 2 3 ; (4) 3 6(3 2 − 15) .
3. Calculate:
(1) (2 3 − 2)(3 2 − 3) ;
(3) ( 6 − 3 3) 2 .
(2)
2. Calculate:
(1) ( 12 − 3 75) i 3 ;
(2) (4 + 3 5) 2 ;
Solution (1)
Practice
( 3 + 6)( 3 − 6) = ( 3)2 − ( 6) 2 = 3 − 6 = −3 ；
(2 ax − 5 by )(2 ax + 5 by ) = (2 ax ) − (5 by )
= 4ax − 25by
2
2
(8) ( x + y ) 2 + ( x − y ) 2 ;
(9) ( 2 + 3 − 6) 2 − ( 2 − 3 + 6) 2 ;
(10) (1 + 2 − 3)(1 − 2 + 3) .
Practice
1. Calculate:
(1)
5 i 3;
3
2
2 ;
2
3
a b b a
(5)
i
;
b a a b
(3) 9 45 ×
(2) 6 27 i (−2 3) ;
(4)
6x i 2x ;
(6) 10x y i
- 59 -
1
.
x
10.6 Division of Surds
By reversing the equation
a
a
=
, we get
b
b
a
a
=
.
b
b
- 60 -
We can apply this formula to perform the operation of division
of surds. For example, if we need to find the quotient of a ÷ b ,
we can apply the formula to find the square root of the quotient
a
a ÷ b , which is
.
b
【Example 1】Calculate:
(1)
72 ÷ 6 ；
(2)
a
, we can apply fraction
b
reduction process to reduce the fraction by cancelling common
factors between a and b. If the denominator cannot be totally
removed after the fraction reduction process, we shall change the
a
surd back to the form
and apply another process to eliminate
b
the surd in the denominator, .which is to multiply both the numerator
and denominator by b .
When the surd is in the form of
For example, to remove the surd in the denominator of
multiply both numerator and denominator by 2 as follows:
3
3× 2
6
1
=
=
=
6.
2
2
2
2 × 2 ( 2)
- 61 -
3+ 2
3−2
= 3+ 2
The process of eliminating the surd in the denominator is called
Rationalising the Denominator (or Denominator Rationisation).
If two expressions with surds are multiplied together, the
resulting expression does not contain any surd, we say that the two
expressions are rationalising factors to each other.
In the above example, 2 and 2 , 3 + 2 and 3 − 2
are rationalisation factors to each other or mutually rationalisation
factors.
=
1
1
1 ÷
.
2
6
72
72
72 ÷ 6 =
=
= 12 = 2 3 ;
6
6
3
1
1
3
1 ÷
= 2 =
×6 = 9 = 3.
2
6
2
1
6
(2)
Solution (1)
1
，we multiply both the
3− 2
numerator and denominator by the same factor ( 3 + 2) , as
follows:
1
3+ 2
=
3 − 2 ( 3 − 2)( 3 + 2)
In a similar manner, to simplify
3
, we
2
【Example 2】Rationalize the denominator of the following
expression:
1
4
;
(2)
;
(1)
5
3 7
a
;
a+b
(3)
Solution (1)
(2)
(3)
(4)
5a
.
20a
1
5
5
=
=
;
5
5
5i 5
4
3 7
=
4i 7
4
=
7;
3 7 i 7 21
a
a i a+b
a
=
=
a+b ;
a+b
a+b i a+b a+b
- 62 -
(4)
5a
5( a ) 2 1
=
=
a.
20a 2 5 i a 2
It can be observed from Questions (1) to (3) of Example 2 that
the rationalising factor of a is a . Sometimes, in the fraction
reduction process, the denominator is totally removed, and there is
no need to perform the rationalisation process. This can be seen in
Question (4) of Example 2.
reduction process first to reduce the fraction by cancelling common
factors. If the denominator is not totally removed, we shall then
apply the denominator rationalisation process to eliminate the
denominator.
【Example 4】Calculate:
(2) ( 12 − 5 8) ÷ ( 6 + 2) .
Solution (1)
【Example 3】Rationalise the denominator of the following
expression:
1
;
2 +1
(1)
x− y
(3)
Solution (1)
(2)
x+ y
(2)
( x ≠ y );
(4)
2
3− 3
;
(1) (6 7 − 4 2) ÷ 3 ;
(2)
x− y
.
x+ y
(6 7 − 4 2) i 3
3i 3
4
= 2 21 −
6
3
12 − 5 8
( 12 − 5 8) ÷ ( 6 + 2) =
6+ 2
(2 3 − 10 2)( 6 − 2)
=
( 6 + 2)( 6 − 2)
(6 7 − 4 2) ÷ 3 =
6 2 − 2 6 − 20 3 + 20
4
3
1
=
2−
6 −5 3 +5
2
2
1
2 −1
2 −1
=
=
= 2 −1 ;
2 −1
2 + 1 ( 2 + 1)( 2 − 1)
2
3− 3
=
x− y
2(3 + 3)
(3 − 3)(3 + 3)
=
3 2+ 6 3 2+ 6
=
;
9−3
6
( x − y )2
(3)
x + y − 2 xy
=
=
;
x− y
x + y ( x + y )( x − y )
(4)
( x ) 2 − ( y ) 2 ( x + y )( x − y )
x− y
=
=
x+ y
x+ y
x+ y
= x− y
It can be observed from Questions (1) to (3) in the above
example that a x + b y and a x − b y are rationalising factors
of each other. Further, it can be observed from Question (4) that,
when a fraction is involved, it is usually simpler to apply fraction
- 63 -
=
In general, regarding the division of Surds, we can first write it
in fractional form, followed by denominator rationalisation.
Practice
1. Calculate:
3
1
1 ÷ 3 ;
5
5
1
1 2 1
(4)
6i4
÷ 1 ;
2
12 3 2
(1) − 54 ÷ 3 ;
(2)
(3) 6 3 ÷ 3 6 ;
(5) 4 6a 3 ÷ 2
a
;
3
(6) a 2 x ÷ ax3 .
- 64 -
2.
Practice
2. Rationalise the denominator of the following expression:
1
x2
3
(1)
;
(3)
;
(2)
;
3
4 xy
40
2n
a2 − b2
(4)
;
(5)
.
3 n
a+b
Calculate:
(1) ( 18 − 98) + (2 75 − 27 ) ;
(2)
(3)
(4)
3. Calculate:
(1)
3 ÷ (5 − 7) − (2 3 − 5 7 )2 ;
(5)
(2) (2 3 − 2)(3 6 + 2) + (3 3 + 5 2) ÷ (3 3 − 5 2) .
(6)
Exercise 4
1.
Calculate:
(1) 3 8 + 2 32 − 50 ;
(3)
6−
3.
(2) 9 3 − 7 12 + 5 48 ;
3
2
−
;
2
3
(4) 2
1
1 1
+ 2 20 − 4
−
5;
5
5 5
1
1 2
12 + 3 1 − 5 −
48 ;
(6)
3
3 3
3a
(7) 2a 27a + 6a
;
4
1 3
1
(8)
x −
;
a
x
b
3
27a 3 + 2ab a ;
(9) 2a 3ab 2 −
6
4
y
x
(10) 5 x 3 y − 2 y xy − 6 xy +
+
.
x
y
(5)
- 65 -
2
1 1
+
−
54 ;
3
6 5
4.
( 45 + 18) − ( 8 − 125) ;


1
1  1
− 2  − 2
− 18  ;
 12 −
2
3  8


 1 1
 5 4

−
20  + 
− 45  ;
5
 5 2
 4 5

 1

7 a −a
− 4 ab 2  ;
 a

2
1
y  x
− x2
.
 x 9x + 6x
+
x
x  y
3
(1) When x = 7 , calculate the value of the following expression:
x + 5 + x − 4 − 4x −1 .
(2) When x = 4 and y = 16 , calculate the value of the
following expression:
1
1 2
x3 + x 2 y + xy 2 +
x y + xy 2 + y 3
4
4
Calculate:
2
3 
2
15 i 1 ;
(2) 6 1 i  −5 2  ;
(1)
5 
3
5
1
(3)
12 x i 2 3 x ;
2
1
(4) 10a 2 ab i 5
;
a
1
1 3
2
30 i 40
i
2 ;
(5)
3
2 2
3
a 
x b
(6) 3
i  −2
.
i
x 
a a
- 66 -
5.
Calculate:
(1) ( 12 + 5 8) i 3 ;
(2)
(3)
(4)

3 2 i  2 12 − 4


y
+
 xy − 2
x

2
(8)

1
+ 3 48  ;
8

x
 i xy ;
y
(9)
(10) ( x + y + x − y ) 2 + ( x + y − x − y ) 2 ( x > y ).
8.
( a 3b + ab3 − ab) i ab ;
3 5
3

(1 − 4 3 + 3 5) .
−2 3−

4 3
 2
Calculate:
(1) (2 3 − 2)(3 6 + 2) ;
(5)
6.
7.
(2)
( 27 + 28)( 12 − 63) ;
(3)
(2 3 − 3 2 + 6)( 6 − 5 3) ;
(4)
( 5 + 3 + 2)( 5 − 2 3 + 2) ;
(5)
( x + 3)(2 x + 3 2) ;
(6)
( x + y + 2 xy )( x − y ) .
Calculate:
(1) (5 3 + 4 2)(5 3 − 4 2) ;
(2)
(7 5 + 6 7 )(6 7 − 7 5) ;
(3)
(3 2 + 48)( 18 − 4 3) ;
(5)
 −b + b 2 − 4ac  −b − b 2 − 4ac 
2


 ( b − 4ac > 0 );
2a
2a



2
(7 3 + 2 7) ;
(6)
(4 − 5 3) ;
(7)

x
3 a + 2
 ;
a

(4)
9.
Rationalise the denominator of the following expression:
3
7n
2
(1)
;
(3)
;
(2)
;
5
3 n
3 40
(4)
x −1
( x > 1 );
x +1
(5)
1
;
5−2
(6)
5
;
3+ 2
(7)
3 + 15
;
3 − 15
(8)
3 5−2 3
;
3 5+2 3
(9)
Calculate:
1 3
2
(1)
÷
2 ;
(2)
45 2
3
2
(3)
15 i 1 ÷ 24 ;
3
a  b
1
(4)
i
÷
 ( x > 1 );
b  a
b
(5)
(6)
2
(7)
2
(8)
- 67 -

2
4
3 1 − 1  ;
3
5

( 2 + 3 − 6)( 2 − 3 − 6) ;

1 1
1  ÷ 27 ;
 48 +
2 2

3

7 − 6  ÷ ( 5 − 3) ;

4

(7 2 + 2 6) ÷ (2 6 − 7 2) ;
1 
 1
+
15 ÷ 
;
2
 3
- 68 -
2 x+2 +3 x−2
( x > 2 ).
x+2 + x−2
20a ÷
2
b;
3
10. Calculate the value of the following expression (correct to the
nearest 0.01):
5 −1
2
(1)
;
;
(2)
2
3− 3
(3)
3
2 2
2
;
5 +1
−
(4)
5− 3
;
5+ 3
2
(5)
(6)
 2  , where x = 3 ;


 x +1
a − 4b
, where a = 6 , b = 5 .
a −2 b
2
, find the value of x 2 − x + 1 ;
3 −1
3x 2 − 2 x + 5
.
(2) Given x = 2 + 3 , find the value of
2x − 7
11. (1)
Given x =
12. Simplify:
(1)
(2)
(3)
(4)
1
1
2
+
−
;
3+ 2
2 +1
3 +1
1
6
7 −5
−
−
;
2
4 − 11 3 + 7
7 −2
1
x
1
−
x− y +
( x − y )3 ( x > y );
y
x− y x− y
5
+
x +1 − x
x +1 + x
+
.
x +1 + x
x +1 − x
- 69 -
Chapter Summary
I. This chapter mainly teaches the basic property and operation
of Surds (also known as Radicals).
II. Basing on the property of principal square root being unique,
we have derived the following basic properties for Surds:
( a )2 = a
(a ≥ 0)
 a (a > 0)

a 2 =| a |=  0 (a = 0)
− a (a < 0)

ab = a i b
(a ≥ 0、b ≥ 0)
a
a
=
b
b
(a ≥ 0、b > 0)
III. From the basic property of Surds, we have derived the
operation rules and simplification process for Surds.
A Surd in its simplest form satisfies the following 2 conditions:
(1) The power of every factor unde the radical sign is less than
2;
(2) There is no denominator under the radical sign.
When Surds are transformed into the simplest form, if two surds
have the same values under the radical sign, we call them Like Surds,
and can combine them together into one surd.
For additions and subtractions, it is essential that we transform
all surds to the simplest form first. Then if there are Like Surds, we
can group them and combine them in the same way that we groupd
and combine like terms in a polynomial.
For multiplication of Surds, we operate using the formula
a i b = ab ( a ≥ 0 , b ≥ 0 ) and the multiplication rules for
polynomials.
- 70 -
For division of Surds, we operate using the formula
a
a
=
b
b
a
( a ≥ 0 , b > 0 ). For the fraction
under the radical sign, we first
b
apply fraction reduction process to reduce the fraction to its simplest
form to see if the denominator can be totally removed. If the
denominator cannot be totally removed, we shall transform the
a
radical back to the form
and simplify the surd by applying the
b
denominator rationalisation process to eliminate the denominator.
It is customary practice that, at the end of the operation, all surds
that result from the operation are transformed to the simplest form.
Revision Exercise 10
(2)
∵
−2 3 = (−2) 2 × 3 = 12 and
∴
∴
−2 3 = 2 3
−2 = 2
12 = 2 3
4. What is meant by a surd in the simplest form? Please transform
the following surd to the simplest form:
2
(2) 4 ;
(3) 12x ;
(1) 500 ;
3
2
y
;
(6) x 2
;
(4) 3a 2b 2 ( b < 0 ); (5)
2
3ab
8x
(7)
x2 − y2
( x > y );
a
(9)
(a 2 − b 2 )(a 4 − b 4 ) ( a > b ); (10)
(8) ( x − y )
b2
(x> y)
x2 − y2
a 2 n +1b3 .
1. For what value of x would the following expression be defined in
the domain of real numbers?
(1) x − 3 ;
(2) 3 − x ;
(3) 1 + x 2 ;
1
1
(6)
.
(4)
;
(5) x + − x ;
2
1− x
x
5. What is meant by like surds? Which of the following expressions
are like surds?
1
5
1
44 ,
, − 1 , x3 y 2 , 175 , 2 a 2 x ,
63 ,
x
11
2
4
m
− 99 , 5 3 ,
( x > 1 ), 225m3 .
7
1 − 2 x + x2
2. Factorise the following expression in the domain of real numbers.
(1) x 2 − 7 ;
(2) 4a 4 − 1 ;
(3) a 4 − 6a 2 + 9 ;
(4) m 4 − 10m 2 n 2 + 25n 4 .
6. Calculate:


1
2  1
(1)  24 −
+2
+ 6;
−
2
3  8


3. Explain how the incorrect result is derived wrongly from the
steps::
(1) ∵ (−3)2 = 32
∴
∵
∴
(−3)2 = 32
(−3)2 = −3 and
−3 = 3
32 = 3
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b2
b2 x
−6
;
(2) 7 a + 5 a x − 4
a
9
1
(3) 2 12 i
3 ÷5 2 ;
4
1 3
2
(4) 9 45 ÷ 3 ×
2 ;
5 2
3
2
- 72 -

 1
2
(5)  6 3 − 5 1  
8−
;
3
2  4
 2
x
(6) ( 2 x − 3 8 x3 ) ÷ 8
;
4
(7) (10 48 − 6 27 + 4 12) ÷ 6 ;
−b + b 2 − 4ac
−b −
9. Given x1 =
, x2 =
2a
are real numbers and b 2 − 4ac ≥ 0 ,
following expression:
(1) x1 + x2 ;
(2)
(3) ax12 + bx1 + c ;
(8) (2 3 + 3 6)(2 3 − 3 6) ;
(9) ( x + x − 1)( x − x − 1) ( x > 1 );
(10) (8 5 + 6 3)2 ;
(4) ax22 + bx2 + c .
6( x + 1) = 7( x − 1) ;
(2)
2
(13)
5
3
2+ 3
+
−
;
3 +1
5 − 3 2− 3
4 5 +3 6
5
+
;
3 5 −2 6
5+ 2
(15) (5 3 + 2 5) ÷ (2 3 − 5) ;
x1 i x2 ;
10. Solve the following equation:
(1)
3 2
1
(11)  1 − 1  ;
4
2 3
(12) ( 2 + 2 3 − 3 6)( 2 − 2 3 + 3 6) ;
b 2 − 4ac
, where a, b, c
2a
find the value of the
3x
2 2x
+1 =
.
2
3
11. Solve the following simultaneous linear equations:
 3x − 2 y = 1
 2 x + 3 y = 7
(1) 
(2) 
 6 x − 7 y = 5
 2 x − 3 y = 0
(14)
( This chapter is translated to English by courtesy of Mr. NG Luk Pan
and reviewed by courtesy of Mr. SIN Wing Sang, Edward.)
(16) ( 2 + 3 + 5)(3 2 + 2 3 − 30) ;
(17)
n + 2 + n2 − 4
n + 2 − n2 − 4
+
n + 2 − n2 − 4
n + 2 + n2 − 4
( n > 2 ).
7. When x = 2 − 3 , find the value of the expression
(7 + 4 3) x 2 + (2 + 3) x + 3 .
1
1
8. Given x = ( 7 + 5) , y = ( 7 − 5) , find the value of the
2
2
following expression:
x y
(2)
+ .
(1) x 2 − xy + y 2 ;
y x
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- 74 -