INTERNATIONAL BACCALAUREATE

Mathematics: analysis and approaches

MAA

EXERCISES [MAA 2.8]

EXPONENTS
Compiled by Christos Nikolaidis

A. Practice questions

1. [Maximum mark: 15] [without GDC]

a
Find the following values in the form of an integer or a fraction of integers.
b

33 = 3 2 = 3 3 =

2 2 3
1 2 2
  =   =   =
 3 3 3
2 2 3
1 2 2
  =   =   =
 3 3 3

1 3 1

42 = 42 = 4 2 =

1
1 1
  25  2
25 2 = 25 2 =   =
 4 

2. [Maximum mark: 6] [without GDC]

a
Find the following values in the form of an integer or a fraction of integers
b

32  23 = 3 5  3 3 = 3 5  3 3 =

54  23 74  72
30  03 = = =
2 2  53 7  77

2 1
3 1  2 3 = 3  2 3 = =
3 1

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 [MAA 2.8] EXPONENTS

3. [Maximum mark: 9] [without GDC]

a
Find the following values in the form of an integer or a fraction of integers
b
1
30 2 15 2 12 2
= = =
15 2 30  2 3
1
2

1 3
7 0.3  7 0.7 = 72 72 = 7 2  7 =

2 3  33 4 3  33
7 2  7 1 = = =
63 63

4. [Maximum mark: 15] [without GDC]

Express the following in the form of a single power ( x y )

1
a6a 2 = a6a = (a 2 ) 6 =

a6
a  =
6 2
a  =
2 6
a2
=

a2a6 a 2b 6
a 2a3a = = =
a5 b 2 a 3 b 4
5 10
 3  3
a 5  =
 
a5  =
 
a  1  2
=
   

a n 5 a 8
a x 3 a 1 x = = =
a n3 a 10

5. [Maximum mark: 8] [without GDC]

Given that A  0 , B  0 , C  0 , simplify the expressions

A 6 B 3 C 10

C 5 A2 B

A 2 B  AB 3

AB

2 A  A( 4 B )  (2 A) 2

2A

A 4 B 3  A3 B 4

A B

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 [MAA 2.8] EXPONENTS

6. [Maximum mark: 5] [without GDC]

Consider the following powers of e :

1
x
A. e 2 B. e 2 C. e x D. e  x E. e 2 F. e 2 x G. e x  2

Find the equivalent expressions in the table below, by stating the corresponding letter.

1 e2 x 1 ex
e 
x 2
ex ex e2 e e 2 ex

7. [Maximum mark: 8] [without GDC]

(a) The graphs of 6 functions are shown below

Match the graphs (a) (b) (c) (d) (e) and (f) to the following functions

y  2x y  5x y  ex y  2x y  5x y  e x

(b) Complete the following table (it contains the common details for all six graphs)

y – intercept

Horizontal asymptote

Domain

Range

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 [MAA 2.8] EXPONENTS

8. [Maximum mark: 10] [without GDC]

Let f ( x)  4 x .
The points A(-1,a), B(1,b) and C(2,c) of the graph are shown in the diagram below.

(a) Write down the coordinates of the y-intercept of the graph. [1]
(b) Write down the value of a in decimal form. [2]
(c) Write down the values of b and of c . [2]
(d) Write down the equation of the horizontal asymptote. [1]
(e) On the diagram above, sketch the graph of f . [2]
(f) Write down the domain and the range of f . [2]

9. [Maximum mark: 6] [without GDC]

The graph of f ( x )  2  2 passes through the points A(1, a ), B(2, b ) and C(3, c ).
x

(a) Find the values of a , b and c . [2]

(b) Sketch the graph of the function. Indicate the y -intercept, the horizontal
asymptote and the points A, B, C of the graph. [4]

a  …………

b  …………

c  …………

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 [MAA 2.8] EXPONENTS

10. [Maximum mark: 6] [without GDC]

The graph of f ( x )  2 x  2 passes through the points A(1, a ), B(2, b ) and C(3, c ).
(a) Find the values of a , b and c . [2]
(b) Sketch the graph of the function. Indicate the y -intercept, the horizontal
asymptote and the points A, B, C of the graph. [4]

a  …………

b  …………

c  …………

11. [Maximum mark: 18] [without GDC]

Solve the equations
(a) 2 2 x  21 x [3]
(b) 2 2 x  81 x [3]
(c) 4 2 x  81 x [3]
(d) 8 x  3  16 x [3]
1
(e) x
 4 x 3 [3]
2

(f) 2 x  41 x [3]

12. [Maximum mark: 15] [without GDC]

Solve the equations
(a) 25 x1  53 [3]
1
(b) 25 x 1  [3]
5x

(c) 25 x 1  5 x [3]
2
(d) 25 x  125 x [3]
2
5 x
(e) 7x 1 [3]

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 [MAA 2.8] EXPONENTS

B. Exam style questions (SHORT)

13. [Maximum mark: 9] [with GDC]

Let f ( x )  e x  2 . The points A(1, e +2) and B(2,b) of the graph are shown in the
diagram below.

(a) Write down value of b

(i) in exact form (ii) correct to 3s.f. [2]
(b) Write down, correct to 3.s.f.,
(i) the value of f (1) (ii) the value of f 1 (6) [2]
(c) On the diagram above, sketch the graph of f . Indicate the y-intercept and the
horizontal asymptote of the graph. [3]
(d) Write down the domain and the range of f . [2]

14. [Maximum mark: 5] [without GDC]

Find the exact solution of the equation 9

2x
 27 (1 x ) .

15. [Maximum mark: 5] [without GDC]

2x
1
Solve the equation 9 x 1   
3
16. [Maximum mark: 5] [without GDC]
2
Solve the equation 25 x  5

17. [Maximum mark: 5] [without GDC]

2
Solve the equation 4 x  8 x .

18*. [Maximum mark: 5] [without GDC]

2 x 1
x2 3  1 
Solve the equation 5   .
 25 

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 [MAA 2.8] EXPONENTS

C. Exam style questions (LONG)

19. [Maximum mark: 12] [without GDC]

For each of the following functions sketch the graph and complete the table
(a) f ( x )  2e x  3 (the point A lies on the curve).

y - intercept:

horizontal asymptote: Exact value of b :

Domain: Range:
[6]
(b) f ( x )  2e x  3 (the point A lies on the curve).

y - intercept:

horizontal asymptote: Exact value of b :

Domain: Range:
[6]
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 [MAA 2.8] EXPONENTS

20*. [Maximum mark: 12] [with GDC]

Consider the function y  f ( x) with f ( x)  10e0.3 x  5
(a) Write down the domain of f . [1]
(b) Find the y-intercept of the graph. [2]
1
(c) Find (i) f (5) correct to 3sf. (ii) f (100) correct to 3sf. [3]
(d) Find the first integer value of x for which the value of y will exceed 120. [2]
(e) Find the value of f (20) and deduce the equation of the horizontal asymptote of
the graph. [2]
(f) Write down the range of f . [2]

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 [MAA 2.8] EXPONENTS
SOLUTIONS
Compiled by: Christos Nikolaidis

A. Practice questions
1.
1 1
3 3 = 27 3 2 = 3 3 =
9 27
2 2 3
1 1 2 4 2 8
  =   =   =
 3 9 3 9 3 27
2 2 3
1 2 9 2 27
  =9   =   =
 3 3
  4 3
  8
1
1 3  1
42 =2 42 =8 4 2 =
2
1 1
1  1  25  2 5
25 2 =5 25 2 =   =
5  4 2

2.
1
3 2  2 3 = 72 3 5  3 3 = 9 3 5  3 3 =
9
54  23 74  72
30  0 3 = 1 = 10 =1
2 2  53 7  77
8 3 2 1 3
3 1  2 3 = 3  2 3 = 1
=
3 8 3 2

3.
1
30 2 15 2 12 2
=4 =4 =2
15 2 30  2 3
1
2

1 3 1
7 0.3  7 0.7 = 7 7 2  7 =
7 2  7 2 = 49 7
2 3  33 4 3  33
7 2  7 1 = 7 =1 =8
63 63

4.
1
a6a2 = a8 a6a = a7 (a 2 ) 6 = a 3
a6
a 
6 2
= a 12 a  = a
2 6 12
a 2
= a4

a2a6 a 2b 6
a 2 a3a = a 6 = a3 = a5
a5 b 2 a 3 b 4
5 10
 3  3
a 5  = a3
 
a 5  = a6
 
a 1  2
= a2
   
a n5 a 8
a x  3 a 1 x = a 4 n 3
= a2 10
= a2
a a

1
5.

A 6 B 3 C 10 4 2 5
 A B C
C 5 A2 B

A 2 B  AB 3 2
 A B
AB

2 A  A(4 B)  ( 2 A) 2
 1  2B  2 A
2A

A 4 B 3  A3 B 4 3 3
 A B
A B

6.
1 e2 x 1 ex
e 
x 2
ex ex e2 e e 2 ex
1
x
F. e 2 x D. e  x C. e x A. e 2 B. e 2 G. e x 2 E. e 2

7. (a)

y  2x y  5x y  ex y  2x y  5x y  ex

(c) (a) (b) (d) (f) (e)

(b)

y – intercept y 1
Horizontal asymptote. y0
Domain x R
Range y0

8. (a) y  1 (b) 0.25 (c) b  4 c  16 (d) y  0 (f) x  R y  0

9. (a) a  4, b  6, c  10
(b) y-intercept: (0,3), horizonta asymptote: y  2

2
10. (a) a  0, b  2, c  6
(b) y-intercept: (0, -1), horizonta asymptote: y  2

11. (a) 2 2 x  21 x  2 x  1  x  3 x  1  x  1 / 3

(b) 2 2 x  81 x  2 2 x  2 33 x  2 x  3  3 x  5 x  3  x  3 / 5

(c) 4 2 x  81 x  2 4 x  2 33 x  4 x  3  3x  7 x  3  x  3 / 7

(d) 8 x  3  16 x  2 3 x  9  2 4 x  3x  9  4 x  x  9

1
(e) x
 4 x 3  2  x  2 2 x  6   x  2 x  6  6  3 x  x  2
2

x
x
(f) 2 x  41 x  2 2  2 2 2 x   2  2 x  x  4  4 x  5x  4  x  4 / 5
2

12. (a) 25 x 1  5 3  5 2 x  2  5 3  2 x  2  3  2 x  1  x  1 / 2
1
(b) 25 x 1   5 2 x  2  5  x  2 x  2   x  3 x  2  x  2 / 3
5x
x
x
(c) 25 x 1  5 x  5 2 x  2  5 2  2 x  2   4 x  4  x  3x  4  x  4 / 3
2
2 2
(d) 25 x  125 x  52 x  53 x  2 x 2  3 x  x(2 x  3)  0

 x  3 / 2 or x  0
2
5 x 2
(e) 7 x 1  7x 5 x
 7 0  x 2  5 x  0  x ( x  5)  0

 x  0 or x  5

3
B. Exam style questions (SHORT)

13. (a) (i) b  e 2  2 (ii) b  9.39

(b) f (1)  e  2  4.72 , f 1 (6)  1.386294...  1.39
(c) y-intercept: (0,3), horizontal asymptote: y  2

(d) Domain: x  R , Range: y  2

3
14. 9 2 x  27 (1 x )  34 x  333 x  4 x  3  3 x  7 x  3 
x
7
2x
1
15. 9x–1 =    32x–2 = 3–2x
3
1
2x – 2 = –2x  x =
2
1
2 2 1 1
16. 25 x  5  52 x  5 2  2 x 2   x2 
2 4

1
x
2
2 2
17. 4 x  8 x  2 2 x  23 x  2 x 2  3 x  x(2 x  3)  0

x  3 / 2 or x  0
2 x 1
2
3  1  2
18. 5x    5x 3
 5 4 x  2  x 2  3   4 x  2  x 2  4 x  5  0
 25 

x  5 or x  1

4
C. Exam style questions (LONG)

19.
(a) y- intercept: (0,5) (b) y- intercept: (0,-1)
h. asymptote: y=3 value of b: 2e+3 h. asymptote: y= -3 value of b: 2e-3
Domain: xR Range: y>3 Domain: xR Range: y> -3

20. (a) xR

(b) y  15

(c) (i) f (5)  49.8 (ii) x  7.50

(d) x  8.14 so x  9
(e) y  5.02 , the horizontal line is y  5

(f) y 5