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.
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.)
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].
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.
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
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 [latex]2[/latex] 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.
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 is multiplied by an expression inside parentheses, you can distribute the multiplier to each term of the expression individually.
To simplify [latex]3\left(3+y\right)-y+9[/latex], it may help to see the expression translated into words:
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 –
Note that the sign belongs to the term it precedes, which is why the [latex]2[/latex] is multiplied by a [latex]-b[/latex]. Remember that 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!
Absolute value expressions are another 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.
Take the absolute value of [latex]\left|4.5\right|[/latex].
[latex]\dfrac{7}{2(4.5)-\left(-3\right)}[/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].
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.
Licenses and Attributions
CC licensed content, Original
Revision and Adaptation. Provided by: Lumen Learning. License: CC BY: Attribution
Simplify an Expression in Fraction Form with Absolute Values. Authored by: James Sousa (Mathispower4u.com) for Lumen Learning. Located at: https://youtu.be/6wmCQprxlnU. License: CC BY: Attribution
