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
3 5 , 4 3 , 4 5 , 11 2 , 0
Created by T. Madas
Created by T. Madas
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
8 6 , 3 10 , 25 2 , 12 3 , 5 7
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
19 2 , 4 5 , 4 6 , 16 2 , 6 3
Created by T. Madas
Created by T. Madas
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
3 2 , 5 2 , 3 10 , 7 2 , 5 10
2 2 2 2 2
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
2 6 , 1 30 , 1 10 , 1 30 , 2 6
3 6 4 10 9
Created by T. Madas
Created by T. Madas
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)
2 6 , 6 6 , 15 , 8 , 48
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
24 , 12 , 6 3 , 50 2 , 24 2
Created by T. Madas
Created by T. Madas
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
9 2 , 25 2 , 12 5 , 14 3 , 180 2
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 )
6 + 3 3 , 4 − 6 2 , −15 + 2 10 , 6 2 − 3 3 , −30 + 10 2
Created by T. Madas
Created by T. Madas
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 )
4 + 3 2 , 9 + 5 3 , 9 + 3 7 , 1 + 5 , −1 + 3 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 )
5 + 3 7 , −21 + 9 3 , −20 + 11 5 , −6 + 5 6 , 19
Created by T. Madas
Created by T. Madas
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 )
3 − 3 , −2 − 2 5 , 5 + 3 2 , 14 − 7 2 , −9 − 7 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 )
5−3 3 , 2+5 5 , 6+3 7 , 5−2 2 , 2
Created by T. Madas
Created by T. Madas
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 )
5−3 3 , 2+5 5 , 6+3 7 , 5−2 2 , 5
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
3 6 , 2 2 , 7 7 , 8 6 , 0
Created by T. Madas
Created by T. Madas
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
5 2 , 8 5 , 13 2 , 5 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
4 2 , 4 3, 2 2 , 3 3
Created by T. Madas
Created by T. Madas
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
3 7 , 20 3 , 3 7 , 7 5 , 11 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
4, 2
Created by T. Madas
Created by T. Madas
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
2, 3+ 2 , 5+2 2 , 1
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
3 , 18 , 12 , 48 , 9 2
Created by T. Madas
Created by T. Madas
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 )
5 + 3 3 , 22 − 12 2 , 6 + 3 , 10
23
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 )
17 − 12 2 , 22 − 12 2 , 26 − 15 3 , 3 − 5 7
Created by T. Madas
Created by T. Madas
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
8+ 2 5 , 8− 2 3 , 2+ 3 , 4+8 3
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
6 + 2 7 , 2 + 3 , 11 + 4 7 , 3 + 7
Created by T. Madas
Created by T. Madas
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
2 − 2 , −4 + 3 3 , 11 − 4 7 , −1 + 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
10 + 2 7 , 5 − 2 5 , 1 − 2 2 , 1 + 2 3
7 7 11 11
Created by T. Madas
Created by T. Madas
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
3 + 5 , 8 + 21 , 7 + 3 5 , 7 + 5 2
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
11 + 2 30 , 14 + 6 5 , 7 − 3 5 , 2 + 3 3
2 2
Created by T. Madas
Created by T. Madas
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
3 + 21
3+ 2 7 , 5− 2 2 , 5+ 2 7 ,
4
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
4 + 15 , 4 − 11 , −3 + 2 5 , −11 + 5 5
Created by T. Madas
Created by T. Madas
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
5 + 3 3 , 22 − 12 2 , 6 + 3 , 3 − 5 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
10 , 3 + 3 3 , 13 , 2
23 2
Created by T. Madas
Created by T. Madas
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
x = 3 + 2 3 , y = 2 + 2 2 , z = 3 2 , w = 1+ 2
Created by T. Madas
Created by T. Madas
Created by T. Madas
