Explanation of the Origin of Negative Exponents

Mathematical Opposites

• Each algebraic operation has an opposite. Addition is the opposite of subtraction, and multiplication is the opposite of division. On a number line, the positive integers to the right of 0 have their negative opposites on the left of the 0. Because a positive exponent denotes multiplication, a negative exponent (which is the opposite of positive) denotes division.

Division and Exponents

• The standard exponential expression of 5^3 could also be written 5 * 5 * 5 = 125. The negative exponential expression of 5^-3 could also be written as 1 / 5 / 5 / 5. Note that the 5 itself was positive so there are no negative numbers involved in the division. The leading 1 comes into play because simply dividing the base of a negative exponent by itself would always produce a result of 1 or -1. The leading 1 changes it into a true opposite, or inverse, of the positive exponent.

Simplifying the Division

• While 1 / 5 / 5 / 5 and 5^-3 both produce the answer of 0.008, there is a neater way to write the division problem. Place the 1 in the numerator of a fraction and 5 * 5 * 5 in the denominator. Simplify even further by changing the 5 back into exponential form, though it would now be positive since there's multiplication. The division would thus become 1 / 5^3.

  This works because 1 / 5 / 5 / 5 is equivalent to 1 * (1/5) * (1/5) * (1/5). The simplification multiplies the numerators and denominators of the fraction for the final answer.

Working in Reverse

• Previous steps proved that 5^-3 is equivalent to 1 / 5^3. But in the case of a negative exponent that begins in the denominator of a fraction, such as in 1 / 3^-2, it would equal 3^2. This is because the negative exponent is already placed in an inverse, and reversing this requires using the opposite sign (multiplication) than is typically used for negative exponents (division). Thus 1 / 3^-2 becomes 1 * 3 * 3 = 9.