How To Multiply Monomials

In mathematics, a monomial is any single term with at least one variable in it: For example, 3x, a², 5x²y³ 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.

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: 3x × 2y².

1. Write Each Monomial out as Its Component Factors

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 3x × 2y², that works out to:

3 × x × 2 × y²

2. Group Coefficients and Alphabetize Variables

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.

3. Multiply Coefficients Together

Multiply the coefficients together. This gives you:

6 × x × y²

Which can be rewritten simply as:

6xy²

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 6b² × 5b⁷, you would calculate as follows:

1. Multiply the Coefficients

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

6 × 5 × b² × b⁷

Which can be simplified to:

30b²b⁷

2. Add the Exponents

Because all the exponents in your term have the same base, you can add the exponents together. In other words, b²b⁷ works out to b^{2 + 7} or b⁹. This gives you:

30b⁹