2. contents
Chapter 1 EXPONENTS AND POWERS
1.1 INTRODUCTION
1.2 EXPONENTS
1.3 LAWS OF EXPONENTS
1.3.1 Multiplying Powers with the same Base
1.3.2 Dividing powers with the same base
1.3.3 Taking power of a power
1.3.4 Multiplying powers with the same
exponents
1.3.5 Dividing powers with the same
exponents
1.4 MISCELLANEOUS EXAMPLES USING THE
LAWS OF EXPONENTS
1.5 VIDEO LINK
1.6 WHAT WE HAVE DISCUSSED?
1.7 SOME RELATED WEBSITES
3. 1.1 INTRODUCTION
Do you know what the means of earth is ? It is
5,970,000,000,000,000,000,000,000 kg! Can you read this
number?
Mass of Uranus is 86,800,000,000,000,000,000,000,000 kg.
Which has greater mass, Earth or Uranus?
Distance between sun and Saturn is 1,433,500,000,000 m
and distance between Saturn and Uranus is
1,439,000,000,000.can you read these numbers? Which distance
is less?
These very large numbers are difficult to read, understand
and compare. To make these numbers easy to read, understand
and compare, we use exponents. In this chapter , we shall learn
about exponents and also learn how to use them.
1.2 EXPONENTS
We can write large numbers in a shorter form using
exponents.
Chapter 1
Exponents and Powers
4. Observe 10,000=10×10×10×10=104
The short notation 104 stands for the product 10×10×10×10 .
here 10 is called the base and 4 is called the exponent.
The number 104 is read as the 10 raised to the power of 4 or
simply as fourth power of 10.104 is called the exponential for of
10,000.
We can similarly express 1,000 as power of 10. Since 1,000 is
10 multiplied by itself three times.
1000=10×10×10=103
Here again 103 is the exponential form of 1,000
Similarly 1, 00,000=10×10×10×10×10=105
105 is the exponential form of 1,00,000
In both these examples , the base is 10; in case of 103 , the
exponent is 3and in case of 105 the exponent is 5.
In all the above given examples, we have seen numbers whose
base is 10.However the base can be any other number also.For
example:
81=3×3×3×3 can be written as 81=34 , here 3 is the base and 4 is
the exponent.
25 =2×2×2×2×2=32, which is the fifth power of 2.
In 25 2 is the base and 5 is the exponent.
5. In the same way. 243=3×3×3×3×3=35
64= 2×2×2×2×2×2=26
625=5×5×5×5=54
TRY THESE
Find five more such examples, where a number is expressed in exponential form.Also identify the base and
the exponent in each case.
EXAMPLE 1 Express 256 as a power 2.
SOLUTION we have 256=2×2×2×2×2×2×2×2=28
So we can say that 256=28
EXAMPLE 2 which one is greater23 or 32 ?
SOLUTION we have ,23 =2×2×2=8,
And 32 =3×3=9
Since 9>8,so 32 is greater than 23
EXAMPLE 3 which one is greater 82 or28 ?
SOLUTION 82 =8×8=64
28 =2×2×2×2×2×2×2×2=256
Clearly, 28 >82
EXAMPLE 4 Expand a3 b2 ,a2 b3, b2 a3 ,b3 a2, are they all
same?
SOLUTION a3 b2 = a3 x b2
TRY THESE
Express:
1. 729 as a power of 3
2. 128 as a power of 2
6. = (a x a x a) x (b x b)
= a x a x a x b x b
a2 b3 =a2 x b3
=a x a x b x b x b
b2 a3 =b2 x a3
=b x b x a x a x a
b3 a2 =b3 x a2
=b x b x b x a x a
EXAMPLE 5 Express the following numbers as a product of
powers of prime factors:
(1) 72 (2) 432 (3) 1000 (4) 16000
SOLUTION
(1) 72 = 2x36 = 2x2x18 2 72
= 2x2x2x9 2 36
= 2x2x2x3x3 2 18
= 23x32 3 9
3
Thus , 72 =23x32 (required prime factor product form)
(2) 432 = 2x216=2x2x108=2x2x2x54
7. = 2x2x2x2x27=2x2x2x2x3x9=2x2x2x2x3x3x3
Or 432 =24 x 33 (required form)
(3) 1000 = 2 x 500 = 2 x 2 x 250 = 2 x 2 x 2 x 125
= 2 x 2 x 2 x 5 x 25 = 2 x 2 x 2 x 5 x 5 x 5
Or 1000 = 23 x 53
(4) 16,000 = 16x 1000 =(2x2x2x2)x1000= 24 x103 (as 16
=2x2x2x2)
=(2x2x2x2)x(2x2x2x5x5x5)=24 x 23 x 53 (since
1000=2x2x2x5x5x5)
=(2x2x2x2x2x2x2)x(5x5x5)
Or 16000=27x53
EXAMPLE 6 Work out (1)5,(-1)3,(-1)4, (-10)3,(-5)4
SOLUTION
(1) We have (1)5=1x1x1x1x1=1
In fact, you will realize that 1 raised to any powers is 1.
(2) (-1)3 =(-1)x(-1)x(-1)=1x(-1)=-1
(3) (-1)4=(-1)x(-1)x(-1)x(-1)=1x1=1
You may check that (-1) raised to any odd power is (-1),
And (-1)raised to any even power is (+1)
(4) (-10)3=(-10)x(-10) x(-10)=100x(-10)= -1000
(5) (-5)4=(-5)x(-5)x(-5)x(-5)=25x25=625
(-1)odd number = -1
(-1)even number = +1
8. Exercise 1.1
1. Find the value of
(i) 26 (ii) 93 (iii) 112 (iv) 54
2. Express the following in exponential form:
(i) 6 x 6 x 6 x 6 (ii) t x t (iii) b x b x b x b
(iv) 5 x 5 x 7 x 7 x 7 (v) 2 x 2 x a x a (vi) a x a x a
x c x c x c x c x d
3. Express each of the following numbers using exponential
notation:
(i) 512 (ii) 343 (iii) 729 (iv) 3125
4. Identify the greater number, wherever possible, in each of
the following?
(i) 43 or 34 (ii) 53 or 35 (iii) 28 or 82
(iv) 1002 or 2100 (v) 210 or 102
5. Express each of the following as product of powers of their
prime factors:
(i) 648 (ii) 405 (iii) 540 (iv) 3600
6. Simplify :
(i) 2 x 10 (ii) 72 x 22 (iii) 23 x 5 (iv)3 x 44
(v)0 x 102 (vi) 52 x 33 (vii) 24 x 32 (viii) 32 x 104
7. Simplify :
(i) (-4)3 (ii) (-3)x(-2)3 (iii) (-3)2 x (-5)2
9. 1.3 LAWS OF EXPONENTS
1.3.1 Multiplying Powers with the Same Base
(i) Let us calculate 22 x 23
22 x 23 = (2x2) x (2x2x2)
= 2x2x2x2x2 =25=22+3
Note that the base in 22 and 23 is same and the sum of the
exponents, i.e., 2 and 3 is 5
(ii) (-3)4 x (-3)3= [(-3) x (-3) x (-3) x (-3)] x [(-3) x (-3) x (-3)]
= (-3) x (-3) x (-3) x (-3) x (-3) x (-3) x (-3)
= (-3)7
= (-3)4+3
Again, note that the base is same and the sum of exponents,
4 and 3, is 7
(iii) a2 x a4= (a x a ) x (a x a x ax a)
= a x a x a x a x a x a = a6
(Note: the base is the same and the sum of exponents is 2+4
= 6)
From this we can generalize that for any non-zero integer a,
where m and n are whole numbers,
TRY THESE
Simplify and write in exponential form
i) 25x23 ii) p3x p2 iii)43x 42 iv) a3xa2xa7
am x an = am+n
10. 1.3.2 Dividing Powers with the Same Base
Let us simplify 37 ÷ 34 ?
37 ÷ 34 = 37/34=3 x 3 x 3 x 3 x 3 x 3 x 3 / 3 x 3 x 3 x 3
= 3 x 3 x 3 = 33 = 3 7-4
Thus 37 ÷ 34 = 3 7-4
Similarly,
56 ÷ 52 = 56 / 52 =5 x 5 x 5 x 5 x 5 x 5 / 5 x 5
= 5 x 5 x 5 x 5 = 54 = 5 6-2
Or 56 ÷ 52 = 5 6-2
Let a be a non-zero integer, then,
a4 ÷a2=a4/a2 =a x a x a x a / a x a = a x a = a2 = a4-2
Or a4 ÷a2 = a4-2
In general, for any non-zero integer a,
am ÷ an = am-n
Where m and n are whole numbers and m>n.
1.3.3 Taking Power of a Power
Consider the following
T RY THESE
Simplify and write in exponential form:
1. 29 ÷23
2. 108÷104
3. 911÷97
4. 4
11. Simplify (23)2 ; (32)4
Now,(23)2 means 23 is multiplied two times with itself.
(23) = 23 x 23
= 23+3(Since am x an = am+n)
= 26 = 23x2
Thus (23)2=23x2
Similarly (32)4 = 32x 32x 32x 32
= 32+2+2+2
= 38 (Observe 8 is the product of 2 and 4)
= 32x4
Can you tell what would (72)10 would be equal to ?
So (23)2 = 23 x 2 = 26
(32)4 = 32x4 = 38
(72)10 = 72 x10 =720
(a2)3 = a2 x 3 = a6
(am)3 = amx3 = a3m
From this we can generalize for any non-zero integer ‘a’, where
‘m’ and ‘n’ are whole numbers,
(am)n = amn
TRY THESE
Simplify and write the answer in
exponential form:
i.(62)4 ii.(22)100 iii. (750)2 iv. (53)7
12. 1.3.4 Multiplying Powers With The Same Exponents
Can you simplify 23x33? Notice that here the two terms 23 and
33 have different bases, but the same exponents.
Now, 23 x 33 = (2x2x2)x(3x3x3)
= (2x3)x(2x3)x(2x3)
= 6x6x6
= 63 (Observe 6 is the product of bases 2 and 3)
Consider 44 x 34 = (4 x 4 x 4 x 4) x (3 x 3 x 3 x 3)
= (4 x 3) x (4 x 3) x (4 x 3) x (4 x 3)
= 12 x 12 x 12 x 12
= 124
Similarly, a4 x b4 = (a x a x a x a) x (b x b x b x b)
= (a x b) x (a x b) x (a x b) x (a x b)
= (a x b)4 (Note a x b =ab)
In general, for any non-zero integer a
(Where m is any whole number)
am x bm= (ab)m
TRY THESE
Put into another
form using
am x bn= (ab)m.
i.43 x 23
ii.25 x b5
iii.a2 x t2
iv.56x(-2)6
v.(-2)4x(-3)4
13. EXAMPLE 8. Express the following terms in the
exponential form:
(i). (2 x 3)5 (ii). (2a)4 (iii). (-4m)3
SOLUTION
(i) (2 x 3)5 = (2 x 3) x (2 x 3) x(2 x 3) x(2 x 3) x(2 x 3)
= (2 x 2 x 2 x 2 x 2 x 2) x (3 x 3 x 3 x 3 x 3)
= 25 x 35
(ii) (2a)4 = 2a x 2a x 2a x 2a
= (2 x 2 x 2 x 2) x (a x a x a x a)
= 24 x a4
(iii) (-4m)2 = (- 4 x m)3
= (-4 x m) x (- 4 x m) x (- 4 x m)
= (- 4) x (- 4) x (- 4) x (m x m x m) = (-4)3 x
(m)3
1. 3.5 Dividing Powers with the same Exponents
Observe the following simplifications:
(i) 24/34 = (2x2x2x2) ÷ (3x3x3x3)
= (2/3) x (2/3) x (2/3) x(2/3) = (2/3)4
(ii) a3/b3 = a x a x a / b x b x b
= a/b x a/b x a/b = (a/b)3
From these examples we may generalise
am ÷ bm = am/bm =(a/b)m where a and b are any
non integers and m is a whole number.
TRY THESE
Put into another
form using
am ÷ bm=(a/b)m
1. 45÷35
2. 25÷b5
3. (-2)3÷b3
4. p4÷q4
5. 56÷(-2)6
14. EXAMPLE 9 Expand: (i) (3/4)4 (ii) (4/7)5
SOLUTION
(i) (3/5)4 = 34/54 = (3 x 3 x 3 x 3) / (5 x 5 x 5 x 5)
(ii) (4/7)5 = (-4)5/75 =((-4) x (-4) x (-4) x (-4)) / (7 x
7 x 7 x 7 x 7 )
Number with exponent zero
Can you tell what 35/35 equals to ?
35/35 = 3 x 3 x 3 x 3 x 3 /3 x 3 x 3 x 3 x 3
= 1
By using laws of exponents
35 ÷ 35 = 35-5=30
So 30 = 1
Can you tell what 70 is equal to ?
73 ÷ 73 =73-3 =70
And 73 ÷ 73= (7 x 7 x7)/(7x7x7)
= 1
Therefore 70 = 1
Similarly a3 ÷ a3 = a3-3 = a0
And a3 ÷ a3 = a 3/ a3 = a x a x a / a x a x a
= 1
Thus a0 =1(for any non zero integer a)
So we can say that any number(except 0) raised to the power(or
exponent) 0 is 1.
15. 1. 4 MISCELLANEOUS EXAMPLES USING THE
LAWS OF EXPONENTS
Let us solve some examples using rules of exponents
developed.
EXAMPLE 10 Write exponential form for 8 x 8 x 8 x 8
taking base as 2.
SOLUTION:
We have 8x8x8x8 =84
But we know that 8= 2 x 2 x 2 =23
Therefore 84=(23)4 =23 x 23 x 23 x 23
= 23x4 (You may also use (am)n= amn )
= 212
EXAMPLE 11 Simplify and write the answer in the
exponential form.
(i) 23 x 22 x 55 (ii) (62x64) ÷ 63
(iii) [(22)3 x 36]x 56 (iv) 82 ÷ 23
SOLUTION
(i) 23 x 22 x 55=23+2 x 55
= 25 x 55=(2 x 5)5=105
(ii) (62x64) ÷ 63 = 62+4 ÷ 63
= 66/63= 66-3= 63
(iii) [(22)3 x 36]x 56 =[26 x 36] x 56
= (2x3)6 x 56
= (2x3x5)6 = 306
16. (iv) 8 = 2 x 2 x 2=23
Therefore 82 ÷ 23= (23)2 ÷ 23
= 26 ÷23=26-3=23
Check this Video
17. WHAT HAVE WE DISCUSSED ?
1. Very large numbers are difficult to read,
understand , compare and operate upon. To make
all these easier, we use exponents, converting
many of the large numbers in a shorter form.
2. The following are exponential forms of some
numbers?
10,000 =104(read as 10 raised to 4)
243 =35 , 128 =27.
Here, 10 ,3 and 2 are the bases, whereas 4,5 and 7
are their respective exponents . we also say, 10,000 is
the 4th power of 10, 243 is the 5th power of 3, etc…
3. Numbers in exponential form obey cetain laws,
which are:
For any non zero integers a and b and whole
numbers m and n ,
(a) am x an = am+n
(b) am ÷ an =am-n, m>n
(c) (am)n=amn
(d) amxbm=(ab)m
(e) am ÷ bm=(a/b)m
(f) a0=1
(g) (-1)even number =1
(h) (-1)odd number= -1
18. Some Related Websites
http://www.purplemath.com/modules/exponent.htm
http://www.mathsisfun.com/exponent.html
http://mentorminutes.com/
http://www.mathgoodies.com/lessons/vol3/exponents.html
http://www.platinumgmat.com/
