Exponents


What is Exponents ? 
When a number is written in the form of an, the number is said to written in the exponential form.  Here a is called the base of the number and n is called the power or exponents.. For example 38 Here 3 is the base and 8 is the exponents.

Multiplicative Inverse
For non- zero integer a , a-m = 1/am, where m is a positive integer.
And  a-m is called the multiplicative inverse of  a.

Laws of Exponents 

For any non zero integer a and integer m and n, the laws of exponents are 

1. am x an = am+n 
2. am/an = am-n
3. (am)n = amn
4. am x bm = (ab)m
5. am/bm = (a/b)m
6  (a-m) = 1/am
7  a0 = 1


So according to laws 
if (2)-3 = (1/2)3 =1 / 8
if (1/3)-3 = (3/1)3 = 27
if (-3)4 x (-3)-12 = (-3) 4-12 = (-3)-8 = (1/-3)8

Some More examples
Proof 
1.  (a + b)-1 (a-1 + b-1) = (ab)-1
L.H.S = (a + b)-1 (a-1 + b-1)
         =  1 /(a +b) [1/a + 1/b]
         = 1/(a + b) [(b + a)/ab]
         = 1/(a + b) x (a + b)/ab 
         = 1/ab  = (ab)-1 = R.H.S   

Simplify
[ 2-1 + 30 + 51 + 72 + 93]  x (2/3)
= [1/2 + 1 + 5 + 49 + 729] x (2/3)
= [1/2 + 784] x (2/3)
= (1569/2) x (2/3)
= 1569/3 
= 523

Solution :-
1.  0.0000000003904 = 3.904 x 10-10 

 
time: 0.0140490532