# 12.7: Multiplying Monomials

**At Grade**Created by: CK-12

**Practice**Exponent Properties with Variable Expressions

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*}

**This is our answer.**

### 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*}

**This is our answer.**

### Video Review

### Practice

Directions:Multiply the following monomials.

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

### Notes/Highlights Having trouble? Report an issue.

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Term | Definition |
---|---|

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 is the base, and 4 is the exponent. |

Coefficient |
A coefficient is the number in front of a variable. |

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

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

Power |
The "power" refers to the value of the exponent. For example, is "three to the fourth 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 |
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. |

### Image Attributions

Here you'll multiply monomials by expanding the expression, regrouping factors and multiplying the coefficients.