Simplify Variable Expressions Involving Integer Multiplication

Multiply integers with and without variables to create a new term.

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Simplify Variable Expressions Involving Integer Multiplication
License: CC BY-NC 3.0

Mr. Myers teaches geometry at Rosewood High School. The school year is about to start and he's thinking about how many tests he will have to grade over the course of the year. He isn't sure exactly how many tests he will be giving, but each of his classes has 25 students and he has 5 geometry classes. If \begin{align*}t\end{align*} represents the number of tests he gives each student that year, how can Mr. Myers write and simplify an expression to represent the total number of tests he will have to grade over the course of the year?

In this concept, you will learn how to simplify terms within variable expressions using integer multiplication.

Simplifying Expressions

A variable expression is a math phrase that has numbers, variables, and operations in it. Variable expressions are made up of terms that are separated by addition or subtraction.

Here is an example:

\begin{align*}2x \cdot (-4)+3x-1\end{align*}

In this variable expression there are 3 terms. The first term is \begin{align*}2x \cdot (-4)\end{align*}, the second term is \begin{align*}3x\end{align*}, and the third term is -1.

Sometimes individual terms can be simplified if they contain more than one number.

For example, in the variable expression above, the term \begin{align*}2x \cdot (-4)\end{align*} can be simplified. Let's look at how you would simplify that term.

The first step is to rewrite \begin{align*}2x \cdot (-4)\end{align*} as \begin{align*}(-4) \cdot 2x\end{align*}.

You can flip the order of the multiplication due to the commutative property of multiplication which states that the order in which factors are multiplied does not matter.

The next step is to group the integers together.

Rewrite \begin{align*}(-4) \cdot 2x\end{align*} as \begin{align*}(-4 \cdot 2) \cdot x\end{align*}.

You can change the way the parts of the term are grouped due to the associative property of multiplication which states that you can group the factors being multiplied in any order.

Then, you can multiply the integers. Remember that a negative times a positive equals a negative so

\begin{align*}-4 \cdot 2=-8\end{align*}

\begin{align*}(-4 \cdot 2) \cdot x=-8x\end{align*}

The answer is \begin{align*}2x⋅(-4)=-8x\end{align*}.

Keep in mind that whenever you are simplifying a term, you are combining all the integers in the term that are being multiplied by performing the multiplication. When you write your answer, the variables that were in the term at the beginning will appear at the end of the term.

Let's look at another example.

Simplify \begin{align*}(-5)(-2m)(n)\end{align*}.

This variable expression has only one term. It can be simplified by combining the integers within the term together through multiplication.

First, rewrite this expression by grouping all of the integers together and leaving the variables at the end.

\begin{align*}(-5)(-2m)(n)=(-5 \cdot -2)mn\end{align*}

Next, multiply the integers so that you have one integer in your expression instead of two. Remember that a negative times a negative equals a positive.

\begin{align*}-5 \cdot -2=10\end{align*}

\begin{align*}(-5 \cdot -2)mn=10mn\end{align*}

The answer is \begin{align*}(-5)(-2m)(n)=10mn\end{align*}.

Examples

Example 1

Earlier, you were given a problem about Mr. Myers and his geometry classes. He teaches 5 geometry classes and each class has 25 students. If he gives each student \begin{align*}t\end{align*} tests during the school year, how many tests will he have to grade over the course of the year?

If \begin{align*}t\end{align*} represents the number of tests that he gives each student, then the number of tests he will have to grade from one geometry class is \begin{align*}25t\end{align*}.

He teaches 5 geometry classes, so the total number of tests he will have to grade is:

\begin{align*}(25t) \cdot 5\end{align*}

You can simplify this expression by combining the integers together through multiplication.

First, rewrite the expression by grouping all of the integers together and leaving the variable at the end.

\begin{align*}(25t)\cdot 5=(25 \cdot 5)t\end{align*}

Next, multiply the integers so that you have one integer in your expression instead of two.

\begin{align*}25 \cdot 5 =125\end{align*}

\begin{align*}(25 \cdot 5)t=125t\end{align*}

The answer is \begin{align*}(25t) \cdot 5=125t\end{align*}.

Mr. Myers will have to grade \begin{align*}125t\end{align*} tests over the course of the year.

Example 2

Simplify \begin{align*}-2x(4y)(6)\end{align*}.

First, rewrite this expression by grouping all of the integers together and leaving the variables at the end.

\begin{align*}-2x(4y)(6)=(-2 \cdot 4 \cdot 6)xy\end{align*}

Next, multiply the integers so that you have one integer in your expression instead of three. You can do this in two steps. First multiply \begin{align*}-2 \cdot 4\end{align*} and then multiply the result by 6.

\begin{align*}-2 \cdot 4=-8\end{align*}

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

\begin{align*}(-2 \cdot 4 \cdot 6)xy=-48xy\end{align*}

The answer is \begin{align*}-2x(4y)(6)=-48xy\end{align*}.

Example 3

Simplify \begin{align*}3x(4y)\end{align*}.

First, rewrite this expression by grouping all of the integers together and leaving the variables at the end.

\begin{align*}3x(4y)=(3 \cdot 4)xy\end{align*}

Next, multiply the integers so that you have one integer in your expression instead of two.

\begin{align*}3 \cdot 4=12\end{align*}

\begin{align*}(3 \cdot 4)xy=12xy\end{align*}

The answer is \begin{align*} 3x(4y)=12xy\end{align*}.

Example 4

Simplify \begin{align*}-6a(-4b)\end{align*}.

First, rewrite this expression by grouping all of the integers together and leaving the variables at the end.

\begin{align*}-6a(-4b)=(-6 \cdot -4)ab\end{align*}

Next, multiply the integers so that you have one integer in your expression instead of two.

\begin{align*}-6 \cdot -4=24\end{align*}

\begin{align*}(-6 \cdot -4)ab=24ab\end{align*}

The answer is \begin{align*}-6a(-4b)=24ab\end{align*}.

Example 5

Simplify \begin{align*}-4z(10)\end{align*}.

First, rewrite this expression by grouping all of the integers together and leaving the variable at the end.

\begin{align*}-4z(10)=(-4 \cdot 10)z\end{align*}

Next, multiply the integers so that you have one integer in your expression instead of two.

\begin{align*}-4 \cdot 10=-40\end{align*}

\begin{align*}(-4 \cdot 10)z=-40z\end{align*}

The answer is \begin{align*}-4z(10)=-40z\end{align*}.

Review

Simplify each variable expression.

1. \begin{align*}(-7k)(-6)\end{align*}
2. \begin{align*}(-8)(3a)(b)\end{align*}
3. \begin{align*}-6a(b)(c)\end{align*}
4. \begin{align*}-8a(6b)\end{align*}
5. \begin{align*}(12y)(-3x)(-1)\end{align*}
6. \begin{align*}-8x(4)\end{align*}
7. \begin{align*}-a(5)(-4b)\end{align*}
8. \begin{align*}-2ab(12c)\end{align*}
9. \begin{align*}-12ab(12c)\end{align*}
10. \begin{align*}8x(12z)\end{align*}
11. \begin{align*}-2a(-14c)\end{align*}
12. \begin{align*}-12ab(11c)\end{align*}
13. \begin{align*}-22ab(-2c)\end{align*}
14. \begin{align*}18ab(12)\end{align*}
15. \begin{align*}-21a(-3b)\end{align*}

Answers for Review Problems

To see the Review answers, open this PDF file and look for section 4.11.

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Vocabulary Language: English

TermDefinition
Associative property The associative property states that the order in which three or more values are grouped for multiplication or addition will not affect the product or sum. For example: $(a+b) + c = a + (b+c) \text{ and\,} (ab)c = a(bc)$.
Commutative Property The commutative property states that the order in which two numbers are added or multiplied does not affect the sum or product. For example $a+b=b+a \text{ and\,} (a)(b)=(b)(a)$.
Factors Factors are numbers or values multiplied to equal a product.
Integer The integers consist of all natural numbers, their opposites, and zero. Integers are numbers in the list ..., -3, -2, -1, 0, 1, 2, 3...
Product The product is the result after two amounts have been multiplied.
Variable Expression A variable expression is a mathematical phrase that contains at least one variable or unknown quantity.

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