Suppose you wish to multiply a number, variable or expression a repeated number of times. Exponents allow you to write these products in a simpler form. For example, 5 ∙ 5 ∙ 5 ∙ 5 = 54, y ∙ y ∙ y ∙ y = y⁴ , (x - 3) ∙ (x - 3) ∙ (x - 3) ∙ (x - 3) ∙ (x - 3) ∙ (x - 3) = (x - 3)⁶ . Each of the simplified expressions are called exponential expressions. The number being raised to the exponent is known as the base and the exponent is known as the power. But how do we deal with the concept of negative exponents?

The general rules for positive exponents hold true for negative exponents. If x is an integer and n ≠ 0, then x^-n = 1/xⁿ. To see why this holds true, multiply the equation by xⁿ to get x^-n ∙ xⁿ = 1. Remember when multiplying exponential expressions, you add the exponents, so -n + n = 0 and x⁰ = 1.

You can think of this as the reciprocal of x^-n and changing the exponent from negative to positive. Generally speaking, you want to make sure you have all positive exponents in the expression in simplest form.

Examples: Write each expression with only positive exponents.

3^-4 , (2a)^-2 , 5x^-3

3^-4 = 1/(3⁴) = 1/81
(2a)^-2 = 1/(2a)² = 1/4a²
5x^-3 = 5/(x³)

Note when raising a fraction to a negative exponent, take the reciprocal of the fraction and change the exponents to positive.

Example: Simplify the expression using only positive exponents.

(x/y)^-3

Take the reciprocal first then change the negative -3 to 3.
(y/x)
(y/x)³

Then apply the rule for raising an exponent to an exponent (remember y is y¹ and x is x¹). Therefore, (x/y)^-3 = y³/x³.

Example: Simplify the expression using only positive exponents.

4x^-3/(x¹⁰)

There are two ways to solve this. You can apply the rule for dividing exponential expressions and subtract the exponents.

4x^-3/(x¹⁰) = 4x^-(3-10) = 4x^-13

Then change the negative exponent to positive by changing x^-13 to 1/x¹³. The simplified form is 4/x¹³. You can change the negative exponent to positive first to get 4/(x³x¹⁰), then apply the rule for multiplying exponential expressions to get 4/x¹³.

Example: Simplify the expression using only positive exponents.

(x⁵x⁶)/(x^-4)

Keep the common base x in the numerator and add the exponents to get x¹¹/x^-4
Subtract the exponents to get x^{(11 + 4)} = x¹⁵. You could also use the rules for negative exponents to get x⁵x⁶x⁴ = x¹⁵ .

The examples above should help students who are having trouble understanding the concept of negative exponents.