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Clearly, if we divide a quantity by another quantity having different base but the same exponent,
then the quotient is obtained by dividing the bases keeping the exponent same.

a m  a m
m
m
In general, for any two non-zero integers a and b: a ÷ b = m =   , where m is any whole
number. b  b
It is called V law of exponents.
Example 14: Simplify the following in exponential form:

3
3
6
(a) 5 ÷ 7 3 (b) (–5) ÷ (–3) 6 (c) (–a) ÷ b 3
5 3  5 3  m m a m  a m 
3
3
Solution: (a) 5 ÷ 7 = =    Qa ÷ b = m =   
7 3  7   b  b  

− ( ) 5 6  5 6   6  a m  a m 
5
− 
6
6
m
m
5
3
(b) − ( ) ÷− ( ) = =   =    Qa ÷ b = m =   
3
− 
− ( ) 3 6  3    b  b  

− ( ) a 3  −  a 3  a m  a m 
m
m
3
(c) (–a) ÷ b = =    Qa ÷ b = =   
3
b 3  b   b m  b  

Quick Check
Expression Expression written using On Multiplying Quotient powers
repeated multiplication Fractions
 3 4 3 3 3 3 3333 3 4
⋅⋅⋅
⋅⋅⋅
  2 2 2 2 2222 2 4
2

⋅⋅⋅
 −  x 3 − ( )⋅− ( )⋅− ( ) x
x
x
   ................... 333 ...................
⋅⋅
 3
 −  x 5
   ................... ................... ...................
 y
Numbers with Exponent Zero
××
2 3 222
0
3
3
Let us simplify, 2 3 – 3 = 2 ÷ 2 = = = 1 . Thus, 2 = 1
2 3 222
××
3
3
− ( ) 3 4 − ( ) ×− ( ) ×− ( ) ×− ( ) 3
3
4
4
Similarly, (–3) 4 – 4 = (–3) ÷ (–3) = = = 1.
3
3
3
− ( ) 3 4 − ( ) ×− ( ) ×− ( ) ×− ( ) 3
0
Thus, (–3) = 1
If we put m = n in II law of exponents, we get, Remember
0
a m 0 is an undefined term
m
m
a ÷ a = = a m – m = a 0 and 0 ≠ 1.
0
a m
a m a m aaaa××××... times
m
Or = = = 1 (Q m = n)
m
a n a m aaaa××××... times
Clearly, a = 1
0
Thus, we can say the value of any number (except 0) with exponent 0 is always 1.
87 Exponents and Powers