How to Multiply Monomials

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In mathematics, a monomial is any single term with at least one variable in it: For example, 3_x_, a2, 5_x_2y3 and so on. When you're asked to multiply monomials together, you'll deal first with the coefficients (the non-variable numbers), and then with the variables themselves. You can use the same technique to multiply any quantity of monomials together, although it's easiest to practice with just two.

Multiplying Monomials

The following process works to multiply any monomials, whether they all have the same variable or different variables. For example, imagine that you're asked to calculate the product of two monomials: 3_x_ × 2_y_2.

  1. Write Each Monomial out as Its Component Factors

  2. With a little practice, you'll be able to skip this step. But when you first start multiplying monomials together, it can help to write each monomial out as its component factors. If you're calculating 3_x_ × 2_y_2, that works out to:

    3 × x × 2 × y2

  3. Group Coefficients and Alphabetize Variables

  4. Group the coefficients, or the numbers that aren't variables, together at the front of your expression, and then write the variables after them in alphabetical order. (This is possible because the commutative property states that changing the order in which you multiply numbers won't affect the result.) This gives you:

    3 × 2 × x × y2

    With a little practice you'll be able to skip this step, too, but when you're first learning, it's good to break things down into the simplest steps possible.

  5. Multiply Coefficients Together

  6. Multiply the coefficients together. This gives you:

    6 × x × y2

    Which can be rewritten simply as:


A Shortcut for the Same Variable

If the monomials you're asked to multiply all have the same variable in them – for example, b – you can take a shortcut. For example, if you've been asked to multiply 6_b_2 × 5_b_7, you would calculate as follows:

  1. Multiply the Coefficients

  2. Group the coefficients of the two terms together, followed by the variables. This gives you:

    6 × 5 × b2 × b7

    Which can be simplified to:


  3. Add the Exponents

  4. Because all the exponents in your term have the same base, you can add the exponents together. In other words, b2b7 works out to b2 + 7 or b9. This gives you:



About the Author

Lisa studied mathematics at the University of Alaska, Anchorage, and spent several years tutoring high school and university students through scary -- but fun! -- math subjects like algebra and calculus.