Book 2B: Chapter 9 – Laws of Integral Indices
Revision
1.
Zero and negative integral indices
If a ≠ 0 , then
a0 = 1 ,
1
(ii) a − n = n , where n is a positive integer.
a
(i)
2.
Laws of integral indices
If m and n are integers and a, b ≠ 0 , then
(i)
a m × a n = a m+n
(ii) a m ÷ a n = a m − n
(iii) (a m )n = a mn = (a n )m
(iv) (ab) n = a nb n
n
an
a
(v)   = n
b
b
Example
1.
Assuming that b, p and q are non-zero numbers, simplify the following
expressions.
(a)
b 2 × b 7 ÷ b3
(b) (2c 2 d )(3c 3 d 2 )
(c)
8 p7q2
2 p5 q
Solution:
(a) b 2 × b 7 ÷ b3 = b 2 + 7 ÷ b3 = b9 ÷ b3 = b9−3 = b 6
(b) (2c 2 d )(3c3 d 2 ) = 2 × 3 × c 2 × c 3 × d × d 2 = 6c 2 +3 d 1+ 2 = 6c 5 d 3
(c)
2.
8 p7 q 2 8 p7 q 2
= × 5 × = 4 p 7 −5 q 2−1 = 4 p 2 q
5
2p q 2 p
q
Simplify the following expressions.
(a)
4
3 2
(y y )
 b15 
(b)  7 
b 
Solution:
(a) ( y 4 y 3 ) 2 = ( y 4+3 )2 = ( y 7 ) 2 = y 7×2 = y14
3
 b15 
(b)  7  = (b15−7 )3 = (b8 )3 = b8×3 = b 24
b 
1
3
Book 2B: Chapter 9 – Laws of Integral Indices
3.
4.
Simplify the following expressions.
(b) (−3c 2 ) 4
(a) (5a ) 2
(2ab) 4
(c)
 ab 2 


 4b 
(c)
 a 4   −2b5 
 3  × 2 
b   a 
Simplify the following expressions.
(a)
5.
(c)
a
 
2
4
 −a 2 
(b) 

 b 
3
2
Simplify the following expressions.
(a)
 3a 5 × a 


3
 c

2
(b)
(3a 4 ) 2 × (2a )3
12a10
2
3
5
Book 2B: Chapter 9 – Laws of Integral Indices
6.
Simplify the following expressions.
(a)
 4a 4 × a 2 


2
 b

3
(b)
(2a 3 ) × (6a 2 )3
48a 5
3
(c)
 4 a 4   − 8a 3 
 2  ÷ 4 
 b   b 
4 x +1 × 23 x −1
.
8x
7.
Given that x is a positive integer, simplify
8.
Find the values of the following expressions without using a calculator.
(Leave your answers in fractions if necessary.)
(a) 110 × 7 2
(c) (−3) −2 ÷ (−2) −3
(b) 23 × 40 ÷ 3−1
(d) 8−5 ×163
3
2
Book 2B: Chapter 9 – Laws of Integral Indices
9.
Simplify the following expressions and express your answers with positive
indices.
(a)
a 3 × a −5
a −2
 3a 2 
(b)  − −1 0 
 b a 
4
−3
(c)
( a 4 b 2 ) −3
(a −1b −3 )2
Book 2B: Chapter 9 – Laws of Integral Indices
Exercise
1.
Find the values of the following.
2
3
2
(a) (–3)
2.
(a) 3 × 3
3
(c) (–1) × (–2)
1
(d)   × (−5) 3 × 2 4
5
3
4
(m 4 ) 2
m3
3
(b) 2a × 4a (c)
(d) 21a3b2 ÷ 7a2b
Simplify the following expressions.
(a)
4.
4
Simplify the following expressions and write the answers in index notation.
6
3.
(b) 4 – 3
3
3x 3 × 4 x 4
6x5
2
3 3
(b) (–2m n )
(c)
 3a 2 
 3 
 b 
3
(d)
(x y )
2 2
3
y2 3
÷( )
x
Find the value of each of the following expressions without using a calculator.
Give the answers in integers or fractions.
(a) (4–1)0
5.
(b) 20 × 2–3
(c) 3–4 ÷ 3–1
(d) 55 × 5–3 + 50
Simplify the following expressions and express the answers with positive indices.
(All letters in the expressions represent non-zero numbers.)
(a) (a3)–2
6.
(b) (b0)–3
(c) (c–4)0
(d) (d–1)–2
Find the value of each of the following expressions without using a calculator.
Give the answers in integers or fractions.
–3
–5
0
(a) 5 ÷ 5 × 5
7.
–3
0 –2
(b) (2 ÷ 2 )
(c)
1
6 × 
6
−3
−4
( )
− 6 −8
0
Find the value of each of the following expressions without using a calculator.
Give the answers in integers or fractions.
−2
–3
–5
(a) 4 ÷ 2
(b)
(− 9)
−3
 1 
 1
5
×   (c) 25 − 2 × (− 5) ÷  − 
 27 
 5
5
3
Book 2B: Chapter 9 – Laws of Integral Indices
8.
Simplify the following expressions and express the answers with positive indices.
(All letters in the expressions represent non-zero numbers.)
–7
3
–5
(a) a × a ÷ a
9.
 b4
(b)  0
c



−2
(c) (m–3 n2)–1
Simplify the following expressions and express the answers with positive indices.
(All letters in the expressions represent non-zero numbers.)
–2
–3
–2 4 –3 –1
(a) (–3a b)
(b) (2 x y )
(c)
 − 7 m −3

2
 n



−2
10. Find the value of each of the following expressions without using a calculator.
Give the answers in integers or fractions.
(a) 16–1 × (4–2)–1 ÷ (2–5)0
(b) 3–4 × 123 ÷ (–36)–2
(c) 53 – (–5)–2 × (20–1)–2
11. Simplify the following expressions and express the answers with positive indices.
(All letters in the expressions represent non-zero numbers.)
(a)
 a 4 b −2
 −3 0
a b



−1
(b)
(4m n )
(2n m )
2
− 3 −2
−1
− 2 −1
12. Simplify ( x 3 y 4 ) 2 • ( xy ) 5 • ( xy 4 ) 3 .
2
 3p7
13. Simplify  2
 q
6
  p
 ÷   .
 q
14. Simplify ( xy 2 ) • (−3 x 4 ) 2 .
 x2 
15. Simplify  
 y
 r 
16. Simplify  2 
s 
3
•
 4 y4 

 .
 x 
•
s7
.
r2
3
2
 c4   c2 
17. Simplify  2  ÷  4  .
 4d   d 
6
(c) (x5y–2)–3(x–2y)–2
Book 2B: Chapter 9 – Laws of Integral Indices
18. Simplify (a 4b 2 ) 3 × b 3 ÷ (−a 2b) 2 .
19. Simplify (4a 5b 3 ) 2 ÷ (−3a 2b) 3 × 3a 4 .
20. Simplify
(m 2 n) 3 (−8mn 2 ) 2
.
32m 5 n 7
21. Simplify
(− x 2 y )(3xz ) 2 ( yz ) 3
.
(6 xyz ) 2
22. Simplify 9 2 m + 3 ÷ (33 ) m+1 .
(4 3 x )(16 x +1 )
23. Simplify
.
2 5 x −2
3n − 4 × 91− n
24. Simplify
.
27 n + 2
4n + 22n
25. Simplify
.
8n
26. Find the value of (1253 ) −2 ÷ (−25−2 ) 5 without using a calculator.
 3− 3 

27. Find the value of (24) × 
 −8
−5
−2
without using a calculator.
28. Simplify (−5mn 2 ) −3 • (−5m 0 n) −2 and express your answer with positive indices.
29. Simplify (r −4 s 5 ) −3 • (rs 2 ) 0 and express your answer with positive indices.
 x 4−3 y
30. Simplify  3− y
 x
−3

 , where y is an integer.

31. Simplify
( m 3 ) 3 n −2
and express your answer with positive indices.
(mn) 2
32. Simplify
x6 y 0
and express your answer with positive indices.
( xy 2 )2 • x 4
33. Simplify
( p 2 q 5 ) −2
and express your answer with positive indices.
− p −4 q −1
34. Simplify (−3a 5 b −2 ) 2 ÷ (a 3 b) 3 and express your answer with positive indices.
 x 
35. Simplify  3 
y 
−1
 5a −1 
36. Simplify  0 
 8b 
37. Simplify
•
−2
 0  and express your answer with positive indices.
 3x 
−2
and express your answer with positive indices.
(a 2 ) 5 b 6
and express your answer with positive indices.
a11 (b −3 ) 2
7
Book 2B: Chapter 9 – Laws of Integral Indices
38. Simplify
(4d 2 ) 0 (c −3 d ) −2
and express your answer with positive indices.
(c −4 d 2 ) 3
 10u 2 
39. Simplify  −3 
 v 
40. Simplify
−2
 u4 
 −5 
 8v 
−1
(2 g 0 h 4 ) −1
and express your answer with positive indices.
(5 g − 6 h5 ) − 2
−3
 r −4 s   − 9 
  4 −5 
41. Simplify 
 4  r s 
42. Let M =
and express your answer with positive indices.
−2
and express your answer with positive indices.
a −5 b 3
a3
and
N
=
.
b −7 a − 2
(b −5 ) 2
(a) Simplify M and N. Express your answers with positive indices.
(b) Hence, find M 2, N 2 and
N
.
M
8