Explainer: Multiplying Monomials

In this explainer, we will learn how to multiply monomials involving single and multiple variables.

Definition: Monomial

A monomial is an algebraic expression consisting of one term. So, all the following expressions are monomials: π‘₯, βˆ’31𝑦, 2π‘’π‘£οŠ¨, βˆ’34π‘₯π‘¦οŠ©οŠ¨.

A monomial may contain one or more variables raised to different powers. Multiplying monomials therefore may involve multiplying powers of different variables.

First, we need to remember what a power means. We write a power in the form π‘οŠ, where 𝑏 is called the base and 𝑛 the exponent. And this means that 𝑏, which can be a number or a variable, is multiplied by itself (π‘›βˆ’1) times:

Now, when we multiply two powers of the same base, we get:

This is known as the Product Rule.

The Product Rule

The product of two powers that have the same base is a power of this same base with an exponent equal to the sum of the exponents:

It is important to understand here why the product rule is true only if the two powers have the same base. Let us look at a simple example: 2β‹…5. We can write 2β‹…5=2β‹…2β‹…2β‹…5β‹…5, and here we see that we are stuck; we cannot simplify this expression because the bases are different (2 and 5).

So far, we have assumed that the exponent is positive. To understand the meaning of a negative exponent, let us look at what happens when we divide the power of a number by this number; for instance, 6 divided by 6. Given that 6=6β‹…6β‹…6, we see that 6Γ·6=1β‹…6β‹…6=6.

If we perform the same division again on 6, we get 6Γ·6=1β‹…6=6.

The number 6 is of course simply 6, but it is useful to write the exponent here, as you will see in the following.

So, we start seeing a pattern here: every time we divide by 6, the exponent decreases by 1.

If we carry out the division by 6 once more, we get 6÷6=1=6.

And then, 6÷6=1÷6=6.

And once more, 6÷6=1÷6÷6=6.

Let us summarize these findings in a table.

61β‹…6β‹…6
61β‹…6
61
61÷6=16
61÷6÷6=16

We have just found out something very interesting. The exponent, when positive, tells us how many times the base is used in a multiplication with 1. When it is negative, it tells us how many times the based is used in a division of 1. From this, it follows that any power of a number is the reciprocal of the power with the same base but with an opposite exponent. This is given by the β€œnegative exponent rule.”

Negative Exponent Rule

Any power can be expressed as the reciprocal (or multiplicative inverse) of a power having the same base but an opposite exponent:

Example 1: Applying the Product Rule

Simplify π‘Γ—π‘οŠ«οŠ¨.

Answer

We want to simplify this product of two powers. We see that the base is the same (𝑏).

Therefore, we can apply the product rule, saying that the product of two powers that have the same base can be expressed as a power of this base with an exponent equal to the sum of the exponents.

Hence, 𝑏×𝑏=𝑏=𝑏.

Example 2: Multiplying Monomials with Coefficients of Different Signs

Simplify 7π‘₯Γ—ο€Ήβˆ’8π‘₯οŠͺ.

Answer

In this question, we want to simplify the product of two monomials. To make it clearer, we can rewrite this product adding the multiplication signs that are implicit within the monomials, that is, between the coefficient and the variables: 7β‹…π‘₯β‹…(βˆ’8)β‹…π‘₯.οŠͺ

Now, we can use the commutativity of multiplication to rearrange our right-hand side expression: 7β‹…(βˆ’8)β‹…π‘₯β‹…π‘₯,οŠͺ and multiply the coefficients together: 7β‹…(βˆ’8)=βˆ’56, and the powers of π‘₯ together. For this, we apply the product rule: π‘₯β‹…π‘₯=π‘₯=π‘₯.οŠͺοŠͺ

Putting these two calculations together, we find 7π‘₯Γ—(βˆ’8π‘₯)=βˆ’56π‘₯.οŠͺ

Example 3: Multiplying Monomials with Two Variables and Coefficients of Different Signs

Simplify 6π‘₯π‘¦Γ—ο€Ήβˆ’9π‘₯π‘¦ο…οŠ¨οŠ¨οŠ¨οŠ«.

Answer

Here, we can rewrite this product as 6β‹…π‘₯⋅𝑦⋅(βˆ’9)β‹…π‘₯⋅𝑦.

Using the commutativity of multiplication, we can pair like terms together in our expression: 6β‹…(βˆ’9)β‹…π‘₯β‹…π‘₯⋅𝑦⋅𝑦, and then first evaluate 6β‹…(βˆ’9)=βˆ’54.

Then, we evaluate the product of the powers of π‘₯ by applying the product rule: π‘₯β‹…π‘₯=π‘₯=π‘₯.οŠͺ We do the same for the product of the powers of 𝑦: 𝑦⋅𝑦=𝑦=𝑦. The final answer is then 6π‘₯𝑦⋅(βˆ’9)π‘₯𝑦=βˆ’54π‘₯𝑦.οŠͺ

Example 4: Multiplying Monomials with Exponents of Different Signs

Simplify π‘₯Γ—π‘₯Γ—π‘₯οŠͺ, where π‘₯β‰ 0.

Answer

We have here the product of three powers with the same base, π‘₯. We can therefore apply the product rule: π‘₯Γ—π‘₯Γ—π‘₯=π‘₯=π‘₯.οŠͺοŠͺ

We can also use the negative exponent rule to rewrite our expression and better understand what is happening. We know that π‘₯=1π‘₯. Hence, π‘₯Γ—π‘₯Γ—π‘₯=π‘₯Γ—π‘₯π‘₯=π‘₯π‘₯=π‘₯π‘₯=π‘₯=π‘₯.οŠͺοŠͺοŠͺ

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