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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.

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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1/3 -2/3 2n - 2n -------------- = ? -2/3 2n

_____________________ / _____________ / / ____ / / / 2 / 4 / 4 . / 4 / \/ \/ / -------------------- / 3 \/ 2

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