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
Example
Simplify [latex]\Large\frac{5-[3+(2\cdot (-6))]}{{{3}^{2}}+2}[/latex]Answer: This problem has brackets, parentheses, fractions, exponents, multiplication, subtraction, and addition in it. Grouping symbols are handled first. The parentheses around the [latex]-6[/latex] 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, [latex](2\cdot(−6))[/latex]. 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.)
[latex]\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}[/latex]
Add [latex]3[/latex] and [latex]-12[/latex], which are in brackets, to get [latex]-9[/latex].[latex]\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}[/latex]
Subtract [latex]5–\left[−9\right]=5+9=14[/latex].[latex]\Large\begin{array}{c}\frac{5-\left[-9\right]}{3^{2}+2}\\\\\frac{14}{3^{2}+2}\end{array}[/latex]
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 [latex]3^{2}=9[/latex].[latex]\Large\begin{array}{c}\frac{14}{3^{2}+2}\\\\\frac{14}{9+2}\end{array}[/latex]
Now add. [latex]9+2=11[/latex].[latex]\Large\begin{array}{c}\frac{14}{9+2}\\\\\frac{14}{11}\end{array}[/latex]
Answer
[latex-display]\Large\frac{5-\left[3+\left(2\cdot\left(-6\right)\right)\right]}{3^{2}+2}=\frac{14}{11}[/latex-display]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[latex]\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}[/latex]
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 [latex]a,b[/latex], and [latex]c[/latex]:
[latex]a(b+c)=ab+ac[/latex].
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.
multiply three by (the sum of three and [latex]y[/latex]), then subtract [latex]y[/latex], then add [latex]9[/latex]
To multiply three by the sum of three and [latex]y[/latex], you use the distributive property -
[latex]\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}[/latex]
Now you can subtract [latex]y[/latex] from [latex]3y[/latex] and add [latex]9[/latex] to [latex]9[/latex].
[latex]\begin{array}{c}9+3y-y+9\\=18+2y\end{array}[/latex]
The next example shows how to use the distributive property when one of the terms involved is negative.
Example
Simplify [latex]a+2\left(5-b\right)+3\left(a+4\right)[/latex]Answer: This expression has two sets of parentheses with variables locked up in them. We will use the distributive property to remove the parentheses.
[latex]\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}[/latex]
Note how we placed the negative sign that was on b in front of the [latex]2[/latex] when we applied the distributive property. When you multiply a negative by a positive the result is negative, so [latex]2\cdot{-b}=-2b[/latex]. It is important to be careful with negative signs when you are using the distributive property.
[latex]\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}[/latex]
We combined all the terms we could to get our final result.
Answer
[latex-display]a+2\left(5-b\right)+3\left(a+4\right)=4a+22-2b[/latex-display]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 [latex]0[/latex]. 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 [latex]\Large\frac{3+\left|2-6\right|}{2\left|3\cdot1.5\right|-\left(-3\right)}[/latex]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 [latex]\left|2–6\right|[/latex].
[latex]\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}[/latex]
Take the absolute value of [latex]\left|−4\right|[/latex].[latex]\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}[/latex]
Add the numbers in the numerator.[latex]\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}[/latex]
Now that the numerator is simplified, turn to the denominator. Evaluate the absolute value expression first. [latex]3 \cdot 1.5 = 4.5[/latex], giving[latex]\Large\begin{array}{c}\frac{7}{2\left|{3\cdot{1.5}}\right|-(-3)}\\\\\frac{7}{2\left|{ 4.5}\right|-(-3)}\end{array}[/latex]
The expression “[latex]2\left|4.5\right|[/latex]” reads “[latex]2[/latex] times the absolute value of [latex]4.5[/latex].” Multiply [latex]2[/latex] times [latex]4.5[/latex].[latex]\Large\begin{array}{c}\frac{7}{2\left|4.5\right|-\left(-3\right)}\\\\\frac{7}{9-\left(-3\right)}\end{array}[/latex]
Subtract.[latex]\Large\begin{array}{c}\frac{7}{9-\left(-3\right)}\\\\\frac{7}{12}\end{array}[/latex]
Answer
[latex-display]\Large\frac{3+\left|2-6\right|}{2\left|3\cdot1.5\right|-3\left(-3\right)}=\frac{7}{12}[/latex-display]Licenses & Attributions
CC licensed content, Original
- Revision and Adaptation. Provided by: Lumen Learning License: CC BY: Attribution.
- Combo Meal Image. Provided by: Lumen Learning License: CC BY: Attribution.
CC licensed content, Shared previously
- Simplify an Expression in Fraction Form. Authored by: James Sousa (Mathispower4u.com) for Lumen Learning. License: CC BY: Attribution.
- Simplify an Expression in Fraction Form with Absolute Values. Authored by: James Sousa (Mathispower4u.com) for Lumen Learning. License: CC BY: Attribution.
- Unit 9: Real Numbers, from Developmental Math: An Open Program. Provided by: Monterey Institute of Technology and Education Located at: https://www.nroc.org/. License: CC BY: Attribution.
- Ex 3: Combining Like Terms Requiring Distribution. Authored by: James Sousa (Mathispower4u.com). License: CC BY: Attribution.
