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What is relation between x^-n and 1/x^n...?
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Explanation

You should must be careful about order of operations when using negative numbers or negative exponents.

      -n    n     -n
If   x  , -x  , -x    then 

-n x means x raise to power -n n -x means -x raise to power n -n -x means -x raise to power -n

Any number x raise to a negative power -n is same as 1 over the number raised to positive power n, i.e.

     -n      1 
    x   = ------
             n  
            x

Similarly any number x raise to a power n is same as 1 over the number raised to negative power n, i.e.

     n      1
    x  = ------
            -n
           x 

When apply the addition rule of exponents will still work with negative exponents.

     n    -n    n+(-n)    n-n    0
    x  × x   = x       = x    = x  = 1

     -n    n    -n+n      0
    x  × x   = x       = x  = 1

Therefore x^n and x^-n is are reciprocal of each other and multiplcative inverse 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. x^n and x^-n are reciprocal of each other.
2. x^n and x^-n are multiplcative inverse of each other.

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               /     /        ____
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             /  4  /  4  .  / 4  
            /    \/       \/
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          /            3
        \/            2    
    
        
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