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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.

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

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

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