How to Divide a Monomial With Negative Exponents

By Amy Harris
Use a pencil and paper to decrease your chance of making errors when dividing monomials.

Dividing monomials with negative exponents requires a degree of algebraic background knowledge. For starters, you must know the definition of a base. A base is the number or variable attached to the left side of the exponent – in 3^5, the base is 3, and in x^6, the base is x. You must also know the rule for dividing monomials with positive exponents: When dividing powers with the same base, subtract the bottom exponent from the top exponent. For instance, y^7 / y^2 produces a result of y^5.



Numerator’s Exponent is Positive and Denominator’s Exponent is Negative

Identify the base. Consider the problem 8t^5 / 4t^-3. The base here is t.

Divide the coefficients, if any exist. In 8t^5 / 4t^-3, the coefficients are 8 and 4, so divide 8 by 4 to get 2. Hence, the problem becomes 2t^5 / t^-3.

Perform subtraction on the exponents, which in this case becomes addition. Take the numerator’s exponent minus the denominator’s exponent. In the previous example, perform 5 – -3. When a minus sign appears directly in front of a negative sign, change both the negative and minus signs into a single plus sign. In this example, the problem becomes 5 + 3, yielding 8. Recall that this number is the exponent. In problems where the original numerator’s exponent is positive and the denominator’s exponent is negative, this exponent will always be positive. Integrate this result back into the problem, producing a final answer of 2t^8.

Numerator’s Exponent is Negative and Denominator’s Exponent is Positive

Identify the base. Take the expression 12b^-8 / -3b^2. The base here is b.

Divide any coefficients. In the case of 12b^-8 / -3b^2, divide 12 by -3 to obtain -4, yielding -4b^-8 / b^2.

Subtract the exponent in the denominator from the exponent in the numerator. This will always produce a negative exponent in problems of this type. In the example, perform -8 – 2, resulting in -10. Hence, in total thus far, you have -4b^-10.

Rewrite the solution with only positive exponents. This means that if there is a negative exponent in the numerator, move that exponent and its base to the denominator; if there is a negative exponent in the denominator, move that exponent and its base to the numerator. In the previous example, -4b^-10 becomes -4 / b^10.

Both Exponents Are Negative

Recognize the base. In the problem 22x^-7 / 2x^-4, the base is x.

Divide coefficients, if they are present. Here, divide 22 by 2 to get 11, simplifying the problem to 11x^-7 / x^-4.

Subtract the exponents, following the rule for subtracting negatives outlined in the first section. In -7 – -4, change the minus and negative signs into a plus sign, yielding -7 + 4, which results in -3. Thus, the previous example becomes 11x^-3. In this case, the exponent is negative, necessitating an additional step, but if the exponent is positive, you can stop here.

Rewrite the answer without negative exponents, as described in the second section. In 11x^-3, move the exponent and its base to the denominator, producing a final answer of 11 / x^3.

Tip

A coefficient is a constant multiplied by a variable and written to the left of the variable. For example, in 5x^-3, the number 5 is a coefficient.

If the bases in the numerator differ, you cannot perform division, but you can rewrite the expression without negative exponents. For instance, c^4 / d^-6 can be written as (c^4)(d^6) but cannot be simplified any further.

About the Author

Based in western New York, Amy Harris began writing for Demand Media and Great Lakes Brewing News in 2010. Harris holds a Bachelor of Science in Mathematics from Penn State University; she taught high school math for several years and has also worked in the field of instructional design.