C1
1
Worksheet D
INDICES AND SURDS
a Express ( 23 )−2 as an exact fraction in its simplest form.
(2)
b Solve the equation
3
x 2 − 27 = 0.
2
(2)
a Find the value of x such that 2x − 1 = 16.
(3)
y
b Find the value of y such that 2(3 − 10) = 34.
3
Simplify
12 −
a
b
4
(2)
5
3
(3)
(4 x )3
16 x
(2)
7
3
a Express 2 2 − 2 2 in the form k 2 .
(2)
b Show that
( x + 6)2 + ( 2 x − 3)2
can be written in the form ax + b where a and b are integers to be found.
5
(3)
Solve the equation
x 12 + 9 = x 3 ,
giving your answer in the form k 3 , where k is an integer.
6
(3)
B
A
C
M
The diagram shows triangle ABC in which AB = BC = 4 +
3 and AC = 4 + 4 3 .
Given that M is the mid-point of AC,
7
8
a find the exact length BM,
(4)
b show that the area of triangle ABC is 6 + 2 3 .
(2)
a Find the value of x such that 82x − 1 = 32.
(3)
b Find the value of y such that ( 13 )y − 2 = 81.
(3)
Express each of the following in the form a + b 2 , where a and b are integers.
a
b
9
48 − 600
12
(3)
2
4+3 2
(3)
Find the non-zero value of x for which
(2 x )3 = 4x.
(4)
 Solomon Press
C1
INDICES AND SURDS
10
a Express 5 3 in the form
Worksheet D continued
k.
(2)
b Hence find the integer n such that
n < 5 3 < n + 1.
11
a Express (12 14 )
− 12
(2)
as an exact fraction in its simplest form.
(2)
b Solve the equation
3x−3 = 7 19 .
12
Simplify
a
192 − 2 12 +
b (2 +
13
(3)
75
(3)
3 )(5 − 2 3 )
(2)
a Write down the value of x such that 2x = 32.
(1)
b Solve the equation
32y + 1 = 4y.
14
(3)
a Find the value of x such that
3
x 2 = 64.
b Find the values of the rational constants a and b such that
3 +1
2 3 −3
15
16
17
= a + b 3.
Solve the equation
42y + 7 = 8y + 3.
Giving your answer in the form k 2 , solve the equation
2(x −
32 ) =
2
3 2 −4
−
98 − x.
(4)
Express
3− 2
2 +1
(6)
Given that 5x + 1 = 25y − 3,
a find an expression for y in terms of x.
Given also that 16
x−1
a Express (1 −
(3)
z
=4,
b find an expression for z in terms of y.
19
(4)
(4)
in the form a + b 2 , where a and b are integers.
18
(2)
5 )2 in the form a + b 5 .
(3)
(2)
b Hence, or otherwise, solve the equation
y2 = 3 −
5,
giving your answers in the form c 2 + d 10 , where c and d are exact fractions.
 Solomon Press
(4)