LAWS OFLAWS OF EXPONENTSEXPONENTS

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 3:

Fill up the blank using laws of exponents

Answer:

z15_____zz)3 52

715

=zxzx

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 !

‘CD’‘CD’

Concentration

and

Dedication

Take it as aTake it as a challenge!!!!

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