We know that,,8 = 2x2x2 625 = 5x5x5x5 -27 = (-3)x(-3)x(-3)
The same factor is repeated for several times So the aboveseveral times. So the above numbers can be denoted in a ‘ h t t ’‘shortcut way’
8 = 2x2x2 625 5 5 5 5625 = 5x5x5x5 -27 = (-3)x(-3)x(-3) • In 8, the factor 2 is repeated 3 times
Therefore, 8 = 23
• In 625, the factor 5 is repeated 4 timesTherefore, 625 = 54
I 27 the factor 3 is repeated 3 times• In -27, the factor -3 is repeated 3 timesTherefore, -27 = (-3)3
This way of writing a number is called y gThe Exponential form or The Base Index form.
In general, the product axaxa......(to n g p (factor) is denoted as an ,where a is called the base and n is the exponent por index of the number.
1)The exponential form provides a shortcut notation to represent theshortcut notation to represent the product of repeated factors
2)The product of the repeated factors is also known as the power of theis also known as the power of the factor
3)an is also called as the nth power of a or a is raised to the power na or a is raised to the power n
Exponents :Exponents : exponent
35power 5basebase
Example: 125 = 53 means that 53 is the exponential form of 125the exponential form of 125
There are 5 laws of exponents
First Law of Exponents:
Consider the following examples:1)33x32 = (3x3x3) x (3x3)1)33x32 = (3x3x3) x (3x3)
= 3x3x3x3x3= 35 = 33+2= 35 = 33 2
2)a3xa4 =(axaxa) x (axaxaxa)=axaxaxaxaxaxa=axaxaxaxaxaxa= a7 = a 3+4
From the above examples, we can li th l f llgeneralize the law as follows:
If i l b d ≠0If a is any real number and a≠0, m and n are +ve integers, then am x an = a m+n
Ex : 1) 314 x 312 = 314+12 = 326
2) 23 x 28 x 26 = 23+8+6 = 217
3) a10 x a12 = a 10+12 = a22
Second law of exponents:
Case 1:F ll l l f ( ≠0) dFor all real values of a (a≠0) and
m and n are +ve integers, then when m>n nm
n
m
aaa −=
Example : 1) a10 = a 10-3 = a 7a3
2) x24 = x 24-20 = x 4x20
Case-2:
am = 1 when n>mn n man a n-m
E ample 1) 45 = 1 = 1Example : 1) 45 = 1 = 1 48 48-5 43
2) x12 = 1 = 12) x12 = 1 = 1 x20 x20-12 x8
Definition of a 0 :
We know that when m and n are +ve integers. nmn
m
aaa −=
Now if m=n
a
Now if m=n, the above result becomes nn
n
n
aaa −=
ie 1= a 0 ' a 0 =1 where a≠0ie 1= a . . a =1, where a≠0
Definition of a -n :Definition of a :We know that when a≠0, m and n are +ve integers
N i th lt f 0
nmn
m
aaa −=
Now assuming the result for m=0 we get n
n aaa −= 0
0
ie 1 = a-n ( ' .' a0 =1)
a
( )an
' a-n = 1 when a≠0. . a-n = 1 when a≠0an
Third law of exponents:Consider the following example:Consider the following example:
Find 32x32x32Find 32x32x32
By first law of exponentsBy first law of exponents, 32x32x32 = 32+2+2 = 36
we can write this also aswe can write this also as(32)3 = 36 =32x3
'.'. (32)3= 32x3
From the above example, we can generalisethe relationship asthe relationship as
(am)n = amn
for all real values of a (a≠0 ), and for all +ve integers m and n.g
Example: 1) (54)3 = 54x3 = 512
2) (33)6 = 33x6 = 318
Fourth law of exponents:
Consider the following example(3x5)2 = (3x5) (3x5) = (3x3)(5x5) = 32x52( ) ( ) ( ) ( )( )
So from the above example, we can li th l ti higeneralise the relationship as
(ab)m = am bm
for all real values of a and b a≠0 b≠0 andfor all real values of a and b a≠0, b≠0, andfor all +ve integer m.
Fifth law of exponents:Consider the below example:
5
55
34
3333344444
34
34
34
34
34
34
==⎠⎞
⎜⎝⎛⎠⎞
⎜⎝⎛⎠⎞
⎜⎝⎛⎠⎞
⎜⎝⎛⎠⎞
⎜⎝⎛=
⎠⎞
⎜⎝⎛ xxxx
5333333333333 ⎠⎝⎠⎝⎠⎝⎠⎝⎠⎝⎠⎝ xxxx
So from the above example, we can li th l ti higeneralise the relationship as
mm a=a ⎞⎜⎛
for all real values of a & b, a≠0, b≠0, for all
mb=
b ⎠⎜⎝
, , ,+ve integer m
Laws of Indices for -ve integral indices:With definition of n 1−With definition of
we can extend the five laws of exponents for any integral indices
nn
aa =
any integral indices.
Definition of rational index: a be any +vereal number, n be any +ve integer and m is any integer , then we define
( )mnn mnm
aa ==aif m=1, then nn aa =
1
Laws of exponents for rational indices:n m
m
By using the definition ofall the five laws can be extended to rational i di f ll
n mn aa =
indices as followsLet a>0 be a real number and p and q be rational numbers then we haverational numbers, then we have
.2.1 aaaaxa qpq
pqpqp == −+
( ) ( ) )0(.4.3 realisbbaabaaa
ppppqqp
q
>==
)0(.5 realisbba
ba
p
pp
>=⎟⎠⎞
⎜⎝⎛
Exercise 1:
Express 81 in exponential or
Answer:
exponential or base index form. 81 = (3x3x3x3) = 34
Express 216 in exponential or base
Answer:216exponential or base
index form216 = =(2x2x2)(3x3x3)23 X 33= 23 X 33
Exercise 2:
Fill up the blank using first law of
Answer:
indices 415 x 411 =
4264 x 4 = _____
Answer:
(2/3)5x(2/3)3 = ____(2/3)8
Exercise 3:
Fill up the blank using laws of exponents
Answer:
105‐2 = 103 _____1010)1 2
5
=
Answer:
10
2) 76 x 74 ÷ 73 =___ 77
Exercise 4:
Simplify by using the law of exponents
Answer:
k40( ) ____85 =k
Which is greater Answer:
( ) 2323 22 or 232
Exercise 5:
Fill up the blank using laws of
Answer:
indices1) (3x7)15 =
315 X 715) ( ) ____
Answer:
____)28
2 =⎟⎠⎞
⎜⎝⎛
cba
816
8
cba
Exercise 6:
Express the following as a rational number.
Answer:1) 64/49
1) (7/8)-2
) /2) 4/15
1
453)2
−
⎟⎠⎞
⎜⎝⎛ x
Express the following i t f
Answer:in root form:
1)23/2 2 8)
2) 53/44 125
Exercise 7:
Find
1) 161/4
Answer:1) 21) 161/4
2) (-125)1/3
)
2) ‐5
Express the following i b i d f
Answer:in base index form:
3 17)1 −
4 3/1)2 ⎟⎠⎞
⎜⎝⎛−
41
3)2 qp
⎟⎠⎞
⎜⎝⎛−
31
7)1)
a qp yx)33)2
aq
ap
yx)3
Exercise 8:
Simplify using laws of exponents
Answer:
45 x‐10y‐7572
23
35)1 −−
−
yxyx
Answer:52
2/8143
52
3.33.3.2)2 −
−
Indices leading to logarithm:g g
The laws of indices made an important b k th h i th d l t fbreak through in the development of logarithms.
In the first law of exponents, the product of the numbers is replaced by the sum of p ythe exponents and in the second law, the quotient of two numbers is replaced by the q p ydifference of Indices.
Indices leading to logarithm:g g
So this concept of product replacing to d ti t l i t diffsum and quotient replacing to difference
has led to the development of logarithms hich in t rn ill help in doing thewhich in turn will help in doing the
calculations in Physics & Engineering.
But after the invention of calculators theBut after the invention of calculators, the importance of logarithm tables has gone down !