Negative Means Reciprocal

Now take the quotient rule one step past zero. In x^2 / x^5, two factors of x cancel from the top and the bottom, and three factors of x are left in the denominator: 1 / x^3. The quotient rule says to subtract the exponents, 2 - 5 = -3, so the same expression is x^-3. The two results must be equal, so x^-3 = 1/x^3.

A negative exponent means a reciprocal. For example, 2^-3 = 1/2^3 = 1/8. It does not mean that the value is negative.

You can also see it as a pattern. Each time the exponent of 2 goes down by 1, the value is divided by 2: 2^3 = 8, 2^2 = 4, 2^1 = 2. Continuing the pattern gives 2^0 = 1, then 2^-1 = 1/2, 2^-2 = 1/4 and 2^-3 = 1/8.