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7.5: Simplify Products or Quotients of Single Variable Expressions

Difficulty Level: At Grade Created by: CK-12
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Laine is helping his mother decorate the patio by putting tiles on a rectangular wall that is 20 tiles wide and 32 tiles high. If each tile is a square with side s, then the total area is \begin{align*}20s\times 30s\end{align*}. How can Laine figure out the total area to be tiled in simplified form?

In this concept, you will learn to simplify products or quotients of single variable expressions.

Simplifying Products or Quotients of Single Variable Expressions

With expressions, terms are separated from each other by addition or subtraction, while factors are separated by multiplication or division. For example, the expression \begin{align*}3ab-7ba\end{align*} is composed of two terms, \begin{align*}3ab\end{align*} and \begin{align*}7ba\end{align*}, and each of those terms is composed of three factors.

When adding or subtracting terms in an expression, you can only combine like terms, which are composed of only the same variables. However, you can multiply or divide terms whether they are like terms or not.

For example, \begin{align*}3ab\end{align*} and \begin{align*}7ba\end{align*} are like terms - both terms include only the variables of a and b, regardless of the order in which they appear in each term. However, \begin{align*}3a\end{align*} and \begin{align*}7b\end{align*} are not like terms, since one contains the variable a, and the other contains the variable b. Since \begin{align*}3a\end{align*} and \begin{align*}7b\end{align*} are not like terms, they can't be added together: 

\begin{align*}3a+7b=3a+7b\end{align*}

There is nothing you can do to simplify the expression.

However, \begin{align*}3a\end{align*} and \begin{align*}7b\end{align*} can be multiplied by each other:

\begin{align*}3a\times 7b=21ab\end{align*}

Clearly, \begin{align*}21ab\end{align*} is simpler than \begin{align*}3a\times7b\end{align*}.

Two rules will help you multiply expressions that contain variables. The Commutative Property of Multiplication states that two terms can be multiplied in any order. The Associative Property of Multiplication states that the grouping of terms does not change your answer.

It is also helpful to remember that when multiplying like variables together, you add the exponents.

For example, remember that \begin{align*}x\end{align*} is the same as \begin{align*}{x}^{1}\end{align*}:

\begin{align*}\begin{array}{rcl} x(x) &=& x^2 \\ x(x)(x) &=& x^3\\ {x}^{2}(x) &=& {x}^{3} \end{array}\end{align*}

Look at the following example.

Simplify \begin{align*}6a(3a)\end{align*}.

First, multiply the number parts.

\begin{align*}6 \times 3 = 18\end{align*}

Next, multiply the variables.

\begin{align*}a \cdot a = a^2\end{align*}

The answer is \begin{align*}18a^2\end{align*}.

Here is another example.

Simplify \begin{align*}5 \times (8y)\end{align*}.

These are not like terms, since they contain different variables, but they can still be multiplied.

First, multiply the numbers.

\begin{align*}5\times8=40\end{align*}

Next, multiply the variables.

\begin{align*}x \cdot y=xy\end{align*}

The answer is \begin{align*}40xy\end{align*}.

Here is one more example.

Find the product \begin{align*}4z \times \frac{1}{2}\end{align*}.

\begin{align*}4z\end{align*} and \begin{align*}\frac{1}{2}\end{align*} are not like terms, however, you can multiply terms even if they are not like terms.

Use the commutative and associative properties to rearrange the factors to make it easier to see how they can be multiplied.

According to the commutative property, \begin{align*}z \left( \frac{1}{2} \right) = \frac{1}{2} (z)\end{align*}.

\begin{align*}4z \times \frac{1}{2} = 4 \times \frac{1}{2} (z)\end{align*}

According to the associative property, the grouping of the factors does not change the answer. Group the factors so that the numbers are multiplied first.

\begin{align*}4 \times \frac{1}{2} (z) = 4 \times \frac{1}{2} \times z = \left(4 \times \frac{1}{2}\right) \times z\end{align*}

Now, multiply.

\begin{align*}\left( 4 \times \frac{1}{2} \right) \times z = (2) \times z = 2z\end{align*}

The answer is \begin{align*}2z\end{align*}.

Here is an example using division.

Find the quotient \begin{align*}42c \div 7\end{align*}.

First, rewrite the problem like this \begin{align*}\frac{42c}{7}\end{align*}.

Then separate out the numbers and variables like this.

\begin{align*}\frac{42 c}{7} = \frac{42 \cdot c}{7} = \frac{42}{7} \cdot c\end{align*}

Now, divide 42 by 7 to find the quotient.

\begin{align*}\frac{42}{7} \cdot c = 6 \cdot c = 6c\end{align*}

The answer is \begin{align*}6c\end{align*}.

Examples

Example 1

Earlier, you were given a problem about Laine, who is decorating the patio by gluing tiles on a rectangular wall 20 tiles tall (long) and 30 tiles wide.

Each tile is a square with side \begin{align*}s\end{align*}. Since the area of a rectangle is length times width, the total area is \begin{align*}20s\times30s\end{align*}.

Laine needs to know what the total area is to be tiled in simplified form. So he will need to multiply 20s by 30s.

First, multiply the numbers.

\begin{align*}20\times30=600\end{align*}

Next, multiply the variables.

\begin{align*}s\times s={s}^{2}\end{align*}

The answer is \begin{align*}600{s}^{2}\end{align*}.

Example 2

Find the quotient of \begin{align*}50g \div 10g\end{align*}.

First, rewrite the problem.

\begin{align*}\frac{50g}{10g}\end{align*}

Then, separate the factors.

\begin{align*}\frac{50 \cdot g}{10 \cdot g}\end{align*} 

Next, reduce and cancel.

 \begin{align*}\frac{\bcancel {50} \ 5 \cdot \bcancel {g}}{\bcancel {10} \ 1 \cdot \bcancel {g}}=\frac{5}{1}=5\end{align*}

The answer is 5.

Example 3

Use the commutative and associative properties of multiplication to simplify \begin{align*}6a(9a)\end{align*}.

First, apply the associative property to separate the a and the 9.

\begin{align*}6a(9a)=6(a9)a \end{align*}

Next, apply the commutative property to put the numbers and variables next to each other.

\begin{align*}6(a9)a=6(9a)a\end{align*}

Then, apply the associative property again to group the similar factors.

\begin{align*}6(9a)a=(6 \cdot 9)(a \cdot a)\end{align*}

Finally, multiply the similar factors.

 \begin{align*}(6 \cdot 9)(a \cdot a)=54{a}^{2}\end{align*}

The answer is \begin{align*}54{a}^{2}\end{align*}.

Example 4

Use the commutative and associative properties of multiplication to simplify \begin{align*}15b \div 5b\end{align*}.

First, rewrite the problem in vertical format.

\begin{align*}\frac{15 b}{5b}\end{align*}

Next, separate the factors. 

\begin{align*}\frac{15 \cdot b}{5 \cdot b}\end{align*}

Then, identify and cancel similar factors.

\begin{align*}\frac{\bcancel {5} \cdot 3 \cdot \bcancel {b}}{\bcancel {5} 1 \cdot \bcancel {b}}\end{align*}

Finally, simplify to get the answer.

\begin{align*}\frac{3}{1}=3\end{align*}

The answer is 3.

Example 5

Simplify \begin{align*}\frac{20 c}{4}\end{align*}.

First, separate the factors.

\begin{align*}\frac{20 \cdot c}{4}\end{align*}

Next, identify and cancel similar factors.

\begin{align*}\frac{5 \cdot 4 \cdot c}{4}\end{align*}

Finally, simplify to get the answer.

\begin{align*}\frac{5c}{1}=5c\end{align*}

The answer is \begin{align*}5c\end{align*}.

Review

Simplify each product or quotient.

  1. \begin{align*}6a(4a)\end{align*} 
  2. \begin{align*}9x(2)\end{align*}
  3. \begin{align*}14y(2y)\end{align*}
  4. \begin{align*}16a(a)\end{align*}
  5. \begin{align*}22x(2x)\end{align*}
  6. \begin{align*}18b(2)\end{align*}
  7.  \begin{align*}21a\div7\end{align*}
  8.  \begin{align*}22b\div2b\end{align*}
  9.  \begin{align*}25x\div x\end{align*}
  10.  \begin{align*}45a \div 5a\end{align*}
  11.  \begin{align*}15x \div 3x\end{align*}
  12.  \begin{align*}18y\div9\end{align*}
  13.  \begin{align*}22y \div 11y\end{align*}
  14.  \begin{align*}\frac{15x}{3y}\end{align*}
  15.  \begin{align*}\frac{82x}{2x}\end{align*}

Review (Answers)

To see the Review answers, open this PDF file and look for section 7.5.

Resources

 

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Vocabulary

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).

Expression

An expression is a mathematical phrase containing variables, operations and/or numbers. Expressions do not include comparative operators such as equal signs or inequality symbols.

Product

The product is the result after two amounts have been multiplied.

Simplify

To simplify means to rewrite an expression to make it as "simple" as possible. You can simplify by removing parentheses, combining like terms, or reducing fractions.

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Dec 02, 2015
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