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# 12.7: Multiplying Monomials

Difficulty Level: At Grade Created by: CK-12
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Practice Exponent Properties with Variable Expressions
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Do you know how to multiply monomials? Take a look at this one.

\begin{align*}(6y^3)(8y^5)(-1xy)\end{align*}

This Concept will show you how to multiply monomial expressions.

### Guidance

You have already seen exponents, but let’s review the definitions of some words so that we can get started. First, let’s look at some of the parts of a term.

In the monomial above, the 7 is called the coefficient, the \begin{align*}x\end{align*} is the variable, and the 3 is the exponent.

We can say that the monomial \begin{align*}7x^3\end{align*} has a power of 3 or is to the \begin{align*}3^{rd}\end{align*} power.

Remember what we said about the coefficient of a variable like \begin{align*}x\end{align*}—if there is no visible coefficient, then the coefficient is an unwritten 1. You could write “\begin{align*}1x\end{align*}” but it is not necessary. Similarly, if there is no exponent on a coefficient or variable, then you can think of it as having an unwritten exponent of 1. So 7 could be written as \begin{align*}7^1\end{align*}. The constant 7 then, is to the \begin{align*}1^{st}\end{align*} power.

Also, the exponent is applied to the constant, variable, or quantity that is directly to its left. That value is called the base. In the monomial above, the base is \begin{align*}x\end{align*}. The exponent, in this case, is not applied to the 7 because it is not directly to the left of the exponent.

What is the exponent? It’s a shortcut. It’s a way of writing many multiplications in a simpler way. In the monomial above, \begin{align*}7x^3\end{align*}, the 3 indicates that the variable \begin{align*}x\end{align*} is multiplied by itself three times.

\begin{align*}7x^3=7 \cdot x \cdot x \cdot x\end{align*}

You can see the amount of space that is saved by using the exponent. When we write all of the multiplications instead of using the exponent, it is called the expanded form. You can see that it is, indeed, expanded—it takes much more space to write. Imagine if the exponent were greater, like \begin{align*}7x^{27}\end{align*}.

\begin{align*}7x^{27}=7 \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x \end{align*}

The exponent truly saves a lot of space!

Let’s look at how to write an expression out into expanded form.

\begin{align*}& (7x^3)(4x^5)\\ &=(7 \cdot x \cdot x \cdot x)(4 \cdot x \cdot x \cdot x \cdot x \cdot x)\end{align*}

Here we have written the expression out into expanded form.

Now, use the commutative property of multiplication to change the order of the factors so that similar factors are next to each other.

\begin{align*} &=7 \cdot 4 \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x && \text{Parentheses disappear.} \\ &=28x^8 && \text{Multiply coefficients 7 and 4.}\\ & && \text{Use exponent as a shortcut for the variables.}\end{align*}

This may seem like a cumbersome long way to work, but it is accurate!

We could do this one another way too.

When we multiply the expressions, we multiply the coefficients, but we add the exponents. Take a look.

\begin{align*}(7x^3)(4x^5)\end{align*}

\begin{align*}7 \times 4 = 28\end{align*}

\begin{align*}3 + 5 = 8\end{align*}

\begin{align*}28x^8\end{align*}

Notice that we got the same answer as doing it out the long way!

Multiply the following monomials.

#### Example A

\begin{align*}(6x^2)(4x^4)\end{align*}

Solution: \begin{align*}14x^6\end{align*}

#### Example B

\begin{align*}(2x^3)(4x^9)\end{align*}

Solution: \begin{align*}8x^{12}\end{align*}

#### Example C

\begin{align*}(6y^3)(8y^5)\end{align*}

Solution: \begin{align*}48y^8\end{align*}

Now let's go back to the dilemma from the beginning of the Concept.

\begin{align*}(6y^3)(8y^5)(-1xy)\end{align*}

First, let's multiply the coefficients.

\begin{align*}6 \times 8 \times -1 = -48\end{align*}

Now we can add the \begin{align*}x\end{align*} because there isn't another \begin{align*}x\end{align*} to multiply with this one.

\begin{align*}-48x\end{align*}

Next, we multiply the \begin{align*}y's\end{align*} and we do this by adding the exponents.

\begin{align*}-48xy^9\end{align*}

### Vocabulary

Monomial
a single term of variables, coefficients and powers.
Coefficient
the number part of a monomial or term.
Variable
the letter part of a term
Exponent
the little number, the power, that tells you how many times to multiply the base by itself.
Base
the number that is impacted by the exponent.
Expanded Form
write out all of the multiplication without an exponent.

### Guided Practice

Here is one for you to try on your own.

Multiply the following monomials.

\begin{align*}(-6x^3)(8y^5)\end{align*}

Solution

In this one, we can begin by multiplying the coefficients.

\begin{align*}-6 \times 8 = -48\end{align*}

Now we simply put the terms together. We can't add the exponents because the terms are not alike.

\begin{align*}-48x^3y^5\end{align*}

### Practice

Directions:Multiply the following monomials.

1. \begin{align*}(5x)(6xy)\end{align*}
2. \begin{align*}(5x^2)(-6xy)\end{align*}
3. \begin{align*}(-5x^2y)(2xy^2)\end{align*}
4. \begin{align*}(-5x)(-9yz)\end{align*}
5. \begin{align*}(18xy)(2xy^2z)\end{align*}
6. \begin{align*}(2y^4)(6y^5)\end{align*}
7. \begin{align*}(5x^3)(-5x^4y^3)\end{align*}
8. \begin{align*}(-2y^5)(6y^3)(2y^2)\end{align*}
9. \begin{align*}(5xy)(-2xy)(-x^2y^2)\end{align*}
10. \begin{align*}(2ab)(6ab)(-4ab)\end{align*}
11. \begin{align*}7x(6xy)\end{align*}
12. \begin{align*}(15x^2)(-10x^3)\end{align*}
13. \begin{align*}(5x)(6xy)(-9xy^5)\end{align*}
14. \begin{align*}(-2x^3)(-4xy)(-5x^2y^4)\end{align*}
15. \begin{align*}(-4abc)(-8a)(-4c)(d^2)\end{align*}

### Vocabulary Language: English

Base

Base

When a value is raised to a power, the value is referred to as the base, and the power is called the exponent. In the expression $32^4$, 32 is the base, and 4 is the exponent.
Coefficient

Coefficient

A coefficient is the number in front of a variable.
Expanded Form

Expanded Form

Expanded form refers to a base and an exponent written as repeated multiplication.
Exponent

Exponent

Exponents are used to describe the number of times that a term is multiplied by itself.
Power

Power

The "power" refers to the value of the exponent. For example, $3^4$ is "three to the fourth power".
power to a power

power to a power

Power to a power is a number raised to an exponent which in turn is raised to another exponent.
Variable

Variable

A variable is a symbol used to represent an unknown or changing quantity. The most common variables are a, b, x, y, m, and n.

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