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Study Guides > Intermediate Algebra

Read: Use the Distributive Property to Simplify

Learning Objectives

  • Simplify expressions with fraction bars, brackets, and parentheses
  • Use the distributive property to simplify expressions with grouping symbols
  • Simplify expressions containing absolute values
In this section, we will use the skills from the last section to simplify mathematical expressions that contain many grouping symbols and many operations. We will also use the distributive property to break up multiplication into addition.  Additionally, you will see how to handle absolute value terms when you simplify expressions.

Example

Simplify 5[3+(2(6))]32+2\Large\frac{5-[3+(2\cdot (-6))]}{{{3}^{2}}+2}

Answer: This problem has brackets, parentheses, fractions, exponents, multiplication, subtraction, and addition in it. Grouping symbols are handled first. The parentheses around the 6-6 aren’t a grouping symbol; they are simply making it clear that the negative sign belongs to the 6. Start with the innermost set of parentheses that are a grouping symbol. In this example, the innermost set of parentheses would be in the numerator of the fraction, (2(6))(2\cdot(−6)). Begin working out from there. (The fraction line acts as a type of grouping symbol, too; you simplify the numerator and denominator independently, and then divide the numerator by the denominator at the end.)

5[3+(2(6))]32+25[3+(12)]32+2\Large\begin{array}{c}\frac{5-\left[3+\left(2\cdot\left(-6\right)\right)\right]}{3^{2}+2}\\\\\frac{5-\left[3+\left(-12\right)\right]}{3^{2}+2}\end{array}

Add 33 and 12-12, which are in brackets, to get 9-9.

5[3+(12)]32+25[9]32+2\Large\begin{array}{c}\frac{5-\left[3+\left(-12\right)\right]}{3^{2}+2}\\\\\frac{5-\left[-9\right]}{3^{2}+2}\end{array}

Subtract 5[9]=5+9=145–\left[−9\right]=5+9=14.

5[9]32+21432+2\Large\begin{array}{c}\frac{5-\left[-9\right]}{3^{2}+2}\\\\\frac{14}{3^{2}+2}\end{array}

The top of the fraction is all set, but the bottom (denominator) has remained untouched. Apply the order of operations to that as well. Begin by evaluating 32=93^{2}=9.

1432+2149+2\Large\begin{array}{c}\frac{14}{3^{2}+2}\\\\\frac{14}{9+2}\end{array}

Now add. 9+2=119+2=11.

149+21411\Large\begin{array}{c}\frac{14}{9+2}\\\\\frac{14}{11}\end{array}

Answer

5[3+(2(6))]32+2=1411\Large\frac{5-\left[3+\left(2\cdot\left(-6\right)\right)\right]}{3^{2}+2}=\frac{14}{11}

The video that follows contains an example similar to the written one above. Note how the numerator and denominator of the fraction are simplified separately. https://youtu.be/xIJLq54jM44

The Distributive Property

Parentheses are used to group, or combine expressions and terms in mathematics.  You may see them used when you are working with formulas, and when you are translating a real situation into a mathematical problem so you can find a quantitative solution.
Combo Meal Distributive Property Combo Meal Distributive Property
For example, you are on your way to hang out with your friends, and call them to ask if they want something from your favorite drive-through.  Three people want the same combo meal of 22 tacos and one drink.  You can use the distributive property to find out how many total tacos and how many total drinks you should take to them.

   3(2 tacos +1 drink)=32 tacos +3 drinks   =6 tacos +3 drinks \begin{array}{c}\,\,\,3\left(2\text{ tacos }+ 1 \text{ drink}\right)\\=3\cdot{2}\text{ tacos }+3\text{ drinks }\\\,\,=6\text{ tacos }+3\text{ drinks }\end{array}

The distributive property allows us to explicitly describe a total that is a result of a group of groups. In the case of the combo meals, we have three groups of ( two tacos plus one drink). The following definition describes how to use the distributive property in general terms.

The Distributive Property of Multiplication

For real numbers a,ba,b, and cc:

a(b+c)=ab+aca(b+c)=ab+ac.

What this means is that when a number multiplies an expression inside parentheses, you can distribute the multiplication to each term of the expression individually.

To simplify  3(3+y)y+93\left(3+y\right)-y+9, it may help to see the expression translated into words:

multiply three by (the sum of three and yy), then subtract yy, then add 99

To multiply three by the sum of three and yy, you use the distributive property -

         3(3+y)y+9         =33+3yy+9=9+3yy+9\begin{array}{c}\,\,\,\,\,\,\,\,\,3\left(3+y\right)-y+9\\\,\,\,\,\,\,\,\,\,=\underbrace{3\cdot{3}}+\underbrace{3\cdot{y}}-y+9\\=9+3y-y+9\end{array}

Now you can subtract yy from 3y3y and add 99 to 99.

9+3yy+9=18+2y\begin{array}{c}9+3y-y+9\\=18+2y\end{array}

The next example shows how to use the distributive property when one of the terms involved is negative.

Example

Simplify a+2(5b)+3(a+4)a+2\left(5-b\right)+3\left(a+4\right)

Answer: This expression has two sets of parentheses with variables locked up in them.  We will use the distributive property to remove the parentheses.

a+2(5b)+3(a+4)=a+252b+3a+34\begin{array}{c}a+2\left(5-b\right)+3\left(a+4\right)\\=a+2\cdot{5}-2\cdot{b}+3\cdot{a}+3\cdot{4}\end{array}

Note how we placed the negative sign that was on b in front of the 22 when we applied the distributive property. When you multiply a negative by a positive the result is negative, so 2b=2b2\cdot{-b}=-2b.  It is important to be careful with negative signs when you are using the distributive property.

a+252b+3a+34=a+102b+3a+12=4a+222b\begin{array}{c}a+2\cdot{5}-2\cdot{b}+3\cdot{a}+3\cdot{4}\\=a+10-2b+3a+12\\=4a+22-2b\end{array}

We combined all the terms we could to get our final result.

Answer

a+2(5b)+3(a+4)=4a+222ba+2\left(5-b\right)+3\left(a+4\right)=4a+22-2b

https://youtu.be/STfLvYhDhwk

Absolute Value

Absolute value expressions are one final method of grouping that you may see. Recall that the absolute value of a quantity is always positive or 00. When you see an absolute value expression included within a larger expression, treat the absolute value like a grouping symbol and evaluate the expression within the absolute value sign first. Then take the absolute value of that expression. The example below shows how this is done.

Example

Simplify 3+26231.5(3)\Large\frac{3+\left|2-6\right|}{2\left|3\cdot1.5\right|-\left(-3\right)}

Answer: This problem has absolute values, decimals, multiplication, subtraction, and addition in it. Grouping symbols, including absolute value, are handled first. Simplify the numerator, then the denominator. Evaluate 26\left|2–6\right|.

3+26231.5(3)3+4231.5(3)\Large\begin{array}{c}\frac{3+\left|2-6\right|}{2\left|3\cdot1.5\right|-\left(-3\right)}\\\\\frac{3+\left|-4\right|}{2\left|3\cdot1.5\right|-\left(-3\right)}\end{array}

Take the absolute value of 4\left|−4\right|.

3+4231.5(3)3+4231.5(3)\Large\begin{array}{c}\frac{3+\left|-4\right|}{2\left|3\cdot1.5\right|-\left(-3\right)}\\\\\frac{3+4}{2\left|3\cdot1.5\right|-\left(-3\right)}\end{array}

Add the numbers in the numerator.

3+4231.5(3)7231.5(3)\Large\begin{array}{c}\frac{3+4}{2\left|3\cdot1.5\right|-\left(-3\right)}\\\\\frac{7}{2\left| 3\cdot 1.5 \right|-(-3)}\end{array}

Now that the numerator is simplified, turn to the denominator. Evaluate the absolute value expression first. 31.5=4.53 \cdot 1.5 = 4.5, giving

 7231.5(3)724.5(3)\Large\begin{array}{c}\frac{7}{2\left|{3\cdot{1.5}}\right|-(-3)}\\\\\frac{7}{2\left|{ 4.5}\right|-(-3)}\end{array}

The expression “24.52\left|4.5\right|” reads “22 times the absolute value of 4.54.5.” Multiply 22 times 4.54.5.

724.5(3)79(3)\Large\begin{array}{c}\frac{7}{2\left|4.5\right|-\left(-3\right)}\\\\\frac{7}{9-\left(-3\right)}\end{array}

Subtract.

79(3)712\Large\begin{array}{c}\frac{7}{9-\left(-3\right)}\\\\\frac{7}{12}\end{array}

Answer

3+26231.53(3)=712\Large\frac{3+\left|2-6\right|}{2\left|3\cdot1.5\right|-3\left(-3\right)}=\frac{7}{12}

The following video uses the order of operations to simplify an expression in fraction form that contains absolute value terms. Note how the absolute values are treated like parentheses and brackets when using the order of operations. https://youtu.be/6wmCQprxlnU

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