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What is relation between positive and negative exponent...?

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Explanation

Each number has its reciprocal. For example reciprocal of 5 is 1/5 and reciprocal of 7 is 1/7. Also notice that multiplication of 5 and 1/5 is 1.

         1
    5 . --- = 1
         5
         1
    7 . --- = 1
         7


Now let we have 
     n
    x

Its reciprocal is 
      1
    ---- 
       n
     x

        -n
    = x

Which is a negative exponent.

Now multiplication of x^n and x^-n is also 1
     n   -n   n-n    0
    x . x  = x    = x = 1

    n                        -n
So x   is the reciprocal of x 


Now multiplication of x to power n and its reciprocal x to power -n is same as dividing x to power n by x to power n, i.e.

                   n
     n     -n     x
    x  ×  x   = ------ ........(i)
                   n
                  x 

Similarly multiplication of x to power -n and its reciprocal x to power n is same as division of x to power n by x to power n, i.e.

                   n
     -n     n     x
i.e x  ×  x   = ------........(ii) 
                   n
                  x 

From (i) & (ii) n -n n n -n x x × x = x × x = ------ = 1 n x Therefore x^n and x^-n is are reciprocal of each other.

In other words multiplication of x to power n and its reciprocal x to power -n or multiplication of x to power -n and its reciprocal x to power n is same as dividing x to power n by x to power n.

Conclusion

1. Positive exponent and negative exponet are reciprocal of each other.
2. Positive exponent and Negative exponent are multiplcative inverse of each other.

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  • Solve
             1/3      -2/3      
           2n   -   2n  
          -------------- = ?
                 -2/3 
               2n  
            
  • Solve
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               /     /        ____
              /     /        /  2  
             /  4  /  4  .  / 4  
            /    \/       \/
           /  --------------------
          /            3
        \/            2    
    
        
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