Created by T.

Madas

SURDS
PRACTICE

Created by T. Madas
 Created by T. Madas

Question 1
Write each of the following surds in its simplest form.

a) 45

b) 48

c) 80

d) 50 + 3 8

e) 5 12 − 2 75

Question 2
Write each of the following surds in its simplest form.

a) 150 + 54

b) 250 − 40

c) 450 + 2 50

d) 243 + 27

e) 343 − 28

Created by T. Madas
 Created by T. Madas

Question 3
Write each of the following surds in its simplest form.

a) 512 + 18

b) 245 − 45

c) 216 − 24

d) 392 + 8

e) 432 − 48 − 12

Question 4
m
Express each of the following surds in the form k , where k , m and n are
n
positive integers.

a) 4.5

b) 12.5

c) 22.5

d) 24.5

e) 62.5

Created by T. Madas
 Created by T. Madas

Question 5
m
Express each of the following surds in the form k , where k , m and n are
n
positive integers.

8
a)
3

5
b)
6

5
c)
8

3
d)
10

8
e)
27

Question 6
Find the value of each of the following surd expressions, writing the final answer in
its simplest form.

a) 2 2 × 3

b) 3 2 × 2 3

c) 5 3 × 3

2
d) (2 2 )
2
e) (4 3)

Created by T. Madas
 Created by T. Madas

Question 7
Find the value of each of the following surd expressions, writing the final answer in
its simplest form.

a) 2 6 × 24

b) 2 2 × 18

c) 3 2 × 6

d) 2 5 × 5 10

e) 2 6 × 4 3

Question 8
Find the value of each of the following surd expressions, writing the final answer in
its simplest form.

a) 6 × 27

b) 10 × 125

c) 3 8 × 10

d) 14 × 42

e) 2 6 × 3 10 × 30

Created by T. Madas
 Created by T. Madas

Question 9
Find the value of each of the following surd expressions, writing the final answer in
its simplest form.

a) (
3 3+ 2 3 )
b) 2 2 ( 2 −3 )
c) (2 2 −3 5 × 5 )
d) (2 6 −3 × 3)
e) 2 5 ( 10 − 3 5 )

Question 10
Find the value of each of the following surd expressions, writing the final answer in
its simplest form.

a) ( 2 + 2 )(1 + 2 )
b) ( 2 + 3 )(3 + 3 )
c) ( )(
7 + 2 1+ 7 )
d) ( )(
5 + 2 3− 5 )
e) ( )(
11 + 2 5 − 11 )

Created by T. Madas
 Created by T. Madas

Question 11
Find the value of each of the following surd expressions, writing the final answer in
its simplest form.

a) ( )(
7 +3 2 7 −3 )
b) (2 )(
3 −1 3 − 3 3 )
c) (5 − 2 5 )( 2 + 3 5 )
d) (2 )(
2 +3 3 3 2 −2 3 )
e) (3 )(
3−2 2 2 2 +3 3 )

Question 12
Find the value of each of the following surd expressions, writing the final answer in
its simplest form.

a) (3 + 3 )( 2 − 3 )
b) (1 − 5 )(3 + 5 )
c) (1 + 2 2 )(1 + 2 )
d) (1 − 2 2 )( 2 − 3 2 )
e) (3 + 2 2 )(1 − 3 2 )

Created by T. Madas
 Created by T. Madas

Question 13
Find the value of each of the following surd expressions, writing the final answer in
its simplest form.

a) ( 2 − 3 )(1 − 3 )
b) ( 4 − 5 )(3 + 2 5 )
c) ( 4 − 7 )(5 + 2 7 )
d) (3 2 +1)( 2 −1 )
e) ( 2 + 3 )( 4 − 12 )

Question 14
Find the value of each of the following surd expressions, writing the final answer in
its simplest form.

a) ( 2 − 3 )(1 − 3 )
b) ( 4 − 5 )(3 + 2 5 )
c) ( 4 − 7 )(5 + 2 7 )
d) (3 2 +1)( 2 −1 )
e) ( 2 + 5 )(5 − 20 )

Created by T. Madas
 Created by T. Madas

Question 15
Find the value of each of the following surd expressions, writing the final answer in
its simplest form.

a) 24 + 6

b) 98 − 50

c) 63 + 2 28

d) 5 2 × 4 3 − 6 24

18
e) − 2 27
3

Question 16
Find the value of each of the following surd expressions, writing the final answer in
its simplest form.

a) 2 32 + 18 − 3 8

10
b) 3 20 +
5

6
c) 5 8 +
2

6
d) 48 + 27 −
3

Created by T. Madas
 Created by T. Madas

Question 17
Find the value of each of the following surd expressions, writing the final answer in
its simplest form.

6
a) 2 8 + 18 −
2

6
b) 48 − + 6× 2
3

14 3
c)
2
− 18 − ( 2)

60
d) 2 5 × 15 − 75 −
5

Question 18
Find the value of each of the following surd expressions, writing the final answer in
its simplest form.

21
a)
7

3 12
b) (2 3) −
3

63 14
c) +
3 7

20
d) 45 +
5

3+ 3
e) 2 75 + − 2× 2
3− 3

Created by T. Madas
 Created by T. Madas

Question 19
Find the value of each of the following surd expressions, writing the final answer in
its simplest form.

50 + 18
a)
8

14
b) 50 + 3 × 6 −
2

Question 20
Find the value of each of the following surd expressions, writing the final answer in
its simplest form.

7 5 − 45
a)
20

3+ 6
b)
3

5 + 2 10
c)
5

75 + 48
d)
3 27

Created by T. Madas
 Created by T. Madas

Question 21
Find the value of each of the following surd expressions, writing the final answer in
its simplest form.

2
a) ( 75 − 48 )
2
b) ( 8+ 2 )
2
c) ( 27 − 3 )
2
d) ( 108 − 12 )
2

e)
7 ( 50 − 8 )
18 + 32

Question 22
Find the value of each of the following surd expressions, writing the final answer in
its simplest form.

2
a) (4 − 5 )
2
b) ( 3− 2 )
2
c) (2 + 3)
2
d) (2 − 3 5 )

Created by T. Madas
 Created by T. Madas

Question 23
Find the value of each of the following surd expressions, writing the final answer in
its simplest form.

2
a) (3 − 8 )
2
b) (2 − 3 2 )
2

c)
(2 − 3)
2+ 3

3
d) (1 + 2 )

Question 24
Simplify each of the following surd expressions, giving the final answer in the form
p + q r , where p , q and r are integers

22
a)
4− 5

26
b)
4+ 3

12 + 2
c)
12 − 2

44
d)
2 3 −1

Created by T. Madas
 Created by T. Madas

Question 25
Simplify each of the following surd expressions, giving the final answer in the form
p + q r , where p , q and r are integers

4
a)
3− 7

3 +1
b)
3 −1

5+ 7
c)
3− 7

7 +1
d)
7 −2

Question 26
Simplify each of the following surd expressions, giving the final answer in the form
p + q r , where p , q and r are integers

2
a)
1+ 2

5− 3
b)
3 +1

2 7 −1
c)
2 7 +5

4 + 28
d)
3+ 7

Created by T. Madas
 Created by T. Madas

Question 27
Simplify each of the following surd expressions, giving the final answer in the form
p + q r , where p , q and r are integers

36
a)
5− 7

5
b)
2+ 5

1− 2
c)
3− 2

3
d)
6− 3

Question 28
Simplify each of the following surd expressions, giving the final answer in the form
p + q r , where p , q and r are integers

4
a)
3− 5

5 7− 3
b)
7− 3

6+2 5
c)
3− 5

2 +2
d)
3 2 −4

Created by T. Madas
 Created by T. Madas

Question 29
Simplify each of the following surd expressions, giving the final answer in the form
p + q r , where p , q and r are integers

6 + 30
a)
6 − 30

2 5+2
b)
5−2

2
c)
3 5+7

10 3 − 1
d)
4− 3

Question 30
Simplify each of the following surd expressions, giving the final answer in the form
p + q r , where p , q and r are integers

8− 7
a)
7 −2

6+ 2
b)
2+ 2

1+ 7
c)
3− 7

4 3+3 7
d)
3 3+ 7

Created by T. Madas
 Created by T. Madas

Question 31
Simplify each of the following surd expressions, giving the final answer in the form
p + q r , where p , q and r are integers

5 + 15
a)
5 − 15

2 11 − 3
b)
2 + 11

5 5−2
c)
4+ 5

d)
(3 − 5 )
1+ 5

Question 32
Simplify each of the following surd expressions, giving the final answer in the form
p + q r , where p , q and r are integers

a)
2 ( 3+2 )
3 +1

98
b) 2
(3 + 2 )
2 2

c)
( 2 + 3 ) − (1 − 3 )
3

6 4
d) −
3+ 7 3− 7

Created by T. Madas
 Created by T. Madas

Question 33
Simplify each of the following surd expressions, giving the final answer in the form
p + q r , where p , q and r are integers

1 1
a) +
5+ 2 5− 2

5 1
b) −
3 −1 1 − 3

2+ 5 5
c) +
3− 5 3+ 5

1+ 7 8 − 7
d) −
3 − 7 −2 + 7

Question 34
Solve each of the following equations, giving the answer as an exact simplified surd.

a) ( )
3 x− 3 = x+ 3

2+ y
b) = 2
y

2z
c) z 8 − 6 =
2

1+ w
d) = 2
w

Created by T. Madas