Section2.8Simplifying Expressions
¶We know that if we have two apples and add three more, then our result is the same as if we'd had three apples and added two more. In this section, we'll formally define and extend these basic properties we know about numbers to variable expressions.
Subsection2.8.1Identities and Inverses
We will start with some definitions.
The number \(0\) is called the additive identity. If the sum of two numbers is the additive identity, \(0\text{,}\) these two numbers are called additive inverses. For example, \(2\) is the additive inverse of \(-2\text{,}\) and the additive inverse of \(-2\) is \(2\text{.}\)
Similarly, the number \(1\) is called the multiplicative identity. If the product of two numbers is the multiplicative identity, \(1\text{,}\) these two numbers are called multiplicative inverses. For example, \(2\) is the multiplicative inverse of \(\frac{1}{2}\text{,}\) and the multiplicative inverse of \(-\frac{2}{3}\) is \(-\frac{3}{2}\text{.}\) The multiplicative inverse is also called reciprocal.
Subsection2.8.2Introduction to Algebraic Properties
Commutative Property
When we compute the area of a rectangle, we generally multiply the length by the width. Does the result change if we multiply the width by the length?
We can see \(4\cdot2=2\cdot4\text{.}\) If we denote the length of a rectangle with \(L\) and the width with \(W\text{,}\) this implies \(LW=WL\text{.}\) This is referred to as the commutative property of multiplication.
The commutative property also applies to addition, as in \(1+2=2+1\text{,}\) which is called the commutative property of addition. However, there is no commutative property of subtraction or division, as \(2-1\ne1-2\text{,}\) and \(\frac{4}{2}\ne\frac{2}{4}\text{.}\)
Associative Property
Let's extend that example to a rectangular prism with length \(L=4\text{ cm}\text{,}\) width \(W=3\text{ cm}\text{,}\) and height \(H=2\text{ cm}\text{.}\) To compute the volume of this solid, we multiply the length, width and height, which we write as \(LWH\text{.}\)
In the following figure, on the left side, we multiply the length and width first, and then multiply the height; on the right side, we multiply the width and height first, and then multiply the length. Let's compare the products.
We can see \((LW)H=L(WH)\text{.}\) This is known as the associative property of multiplication.
The associative property also applies to addition, as in \((1+2)+3=1+(2+3)\text{,}\) which is called the associative property of addition. However, there is no associative property of subtraction, as \((3-2)-1\ne3-(2-1)\text{.}\)
Distributive Property
The final property we'll explore is called the distributive property, which involves both multiplication and addition. To conceptualize this property, let's consider what happens if we buy 3 sets that each contain one apple and one pear. This will have the same total cost as if we'd bought 3 apples and 3 pears. We write this algebraically as
\begin{equation*} 3(a+p)=3a+3p \end{equation*}Visually, we can see that it's just a means of re-grouping:
\begin{equation*} 3(\apple+ \pear) = 3(\apple)+ 3(\pear) \end{equation*}Subsection2.8.3Summary of Algebraic Properties
Let \(a\text{,}\) \(b\text{,}\) and \(c\) represent real numbers, variables, or algebraic expressions. Then the following properties hold:
- Commutative Property of Multiplication:
\(a\cdot b=b\cdot a\)
- Associative Property of Multiplication:
\(a\cdot(b\cdot c)=(a\cdot b)\cdot c\)
- Commutative Property of Addition:
\(a+b=b+a\)
- Associative Property of Addition:
\(a+(b+c)=(a+b)+c\)
- Distributive Property:
\(a(b+c)=ab+ac\)
Let's practice these properties in the following exercises.
Exercise2.8.6
Exercise2.8.7
Exercise2.8.8
Exercise2.8.9
Exercise2.8.10
Subsection2.8.4Applying the Commutative, Associative, and Distributive Properties
Like Terms
One of the main ways that we will use the commutative, associative, and distributive properties is to simplify expressions. In order to do this, we need to recognize like terms. Two terms are considered like terms if they contain the same variable raised to the same exponent. Here's a table comparing like terms and non-like terms:
| Like Terms | Non-Like Terms |
| \(7x,2x\) | \(7x,7y\) |
| \(-5xy,\frac{1}{3}xy\) | \(-5x^2,-4x\) |
| \(4y^3,y^3\) | \(3x,4\) |
| \(3,-10\) | \(2xy,5y\) |
Combining Like Terms
We combine like terms when we take \(2a+3a\) and write the result as \(5a\text{.}\) The formal process actually involves invoking the distributive property, as we obtain:
\begin{align*} 2a+3a \amp=(2+3)a\\ \amp=5a \end{align*}In practice, it's helpful to think of this as having \(2\) of an object and then an additional \(3\) of that same object. In total, we then have \(5\) of that object.
Example2.8.11
Where possible, simplify the following expressions by combining like terms.
\(6c+12c-5c\)
\(-5q-3q\)
\(-3x-5y+4x\)
\(2x-3y+4z\)
Solution
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We will combine all three like terms: \(6c\text{,}\) \(12c\) and \(-5c\text{:}\)
\begin{align*} 6c+12c-5c \amp=18c-5c\\ \amp=13c \end{align*} -
We will combine like terms \(-5q\) and \(-3q\text{:}\)
\begin{align*} -5q-3q \amp=-8q \end{align*} -
We will combine like terms \(-5x\) and \(4x\text{:}\)
\begin{align*} -3x-5y+4x \amp=-3x+4x+(-5y)\\ \amp=x-5y \end{align*} The expression \(2x-3y+4z\) cannot be simplified as there are no like terms.
Remark2.8.12
The expression \(x\) represents \(1x\) and the expression \(-x\) represents \(-1x\text{,}\) but we don't write either the “\(1\)” or the “\(-1\)” as each is implied. However, it's helpful when combining like terms to remember that \(x=1x\) and \(-x=-1x\text{.}\)
Adding Expressions
When we add an expression like \(4x-5\) to an expression like \(3x-7\text{,}\) we write them as follows:
\begin{equation*} (4x-5)+(3x-7) \end{equation*}In order to remove the given sets of parentheses and apply the commutative property of addition, we will rewrite the subtraction operation as “adding the opposite”:
\begin{equation*} 4x+(-5)+3x+(-7) \end{equation*}At this point we can apply the commutative property of addition and then combine like terms. Here's how the entire problem will look:
\begin{align*} (4x-5)+(3x-7)\amp=4x+(-5)+3x+(-7)\\ \amp=4x+3x+(-5)+(-7)\\ \amp=7x+(-12)\\ \amp=7x-12 \end{align*}Remark2.8.13
Once we get more comfortable simplifying such expressions, we will simply write the following:
\begin{equation*} (4x-5)+(3x-7)=7x-12 \end{equation*}Example2.8.14
Use the associative, commutative, and distributive properties to simplify the following expressions as much as possible.
\((2x+3)+(4x+5)\)
\((-5x+3)+(4x-7)\)
\((-2x-1)+(-4x-9)\)
Solution
-
We will remove parentheses, and then combine like terms:
\begin{align*} (2x+3)+(4x+5) \amp=2x+3+4x+5\\ \amp=2x+4x+3+5\\ \amp=6x+8 \end{align*} -
We will remove parentheses, and then combine like terms::
\begin{align*} (-5x+3)+(4x-7)\amp=-5x+3+4x+(-7)\\ \amp=7x+(-4)\\ \amp=7x-4 \end{align*} -
We will remove parentheses, and then combine like terms:
\begin{align*} (-2x-1)+(-4x-9) \amp=-2x+(-1)+(-4x)+(-9)\\ \amp=-2x+(-4x)+(-1)+(-9)\\ \amp=-6x+(-10)\\ \amp=-6x-10 \end{align*}
Applying the Distributive Property with Negative Coefficients
Applying the distributive property in an expression such as \(2(3x+4)\) is fairly straightforward, in that this becomes \(2(3x)+2(4)\) which we then simplify to \(6x+8\text{.}\) Applying the distributive property is a little trickier when subtraction or a negative constant is involved, for example, with the expression \(2(3x-4)\text{.}\) Recalling that subtraction is defined as “adding the opposite,” we can change the subtraction of positive \(4\) to the addition of negative \(4\text{:}\)
\begin{equation*} 2\big(3x+(-4)\big) \end{equation*}Now when we distribute, we obtain:
\begin{equation*} 2(3x)+2(-4) \end{equation*}As a final step, we see that this simplifies to:
\begin{equation*} 6x-8 \end{equation*}Remark2.8.15
We can also extend the distributive property to one involving subtraction, which states that \(a(b-c)=ab-ac\text{.}\) With this property, we would simplify \(2(3x-4)\) more efficiently:
\begin{align*} 2(3x-4) \amp=2(3x)-2(4)\\ \amp=6x-8 \end{align*}In general, we will use this approach.
Example2.8.16
Apply the distributive property to each expression and simplify it as much as possible.
\(-3(5x+7)\)
\(2(-4x-1)\)
\(-4(2x-3)\)
\(-7(-x-9)\)
Solution
-
We will distribute \(-3\) into the parentheses:
\begin{align*} -3(5x+7)\amp=-3(5x)+(-3)(7)\\ \amp=-15x-21 \end{align*} -
We will distribute \(2\) into the parentheses:
\begin{align*} 2(-4x-1)\amp=2(-4x)-2(1)\\ \amp=-8x-2 \end{align*} -
We will distribute \(-4\) into the parentheses:
\begin{align*} -4(2x-3)\amp=-4(2x)-(-4)(3)\\ \amp=-8x+12 \end{align*} -
We will distribute \(-7\) into the parentheses:
\begin{align*} -7(-x-9)\amp=-7(-x)-(-7)(9)\\ \amp=7x+63 \end{align*}
Exercise2.8.17
Exercise2.8.18
Subtracting Expressions
To subtract one expression from another expression, such as \((5x+9)-(3x+2)\text{,}\) we will again rely on the fact that subtraction is defined as “adding the opposite.” To add the opposite of an expression, we will technically distribute a constant factor of \(-1\) and simplify from there:
\begin{align*} (5x+9)-(3x+2)\amp=(5x+9)+(-1)(3x+2)\\ \amp=5x+9+(-1)(3x)+(-1)(2)\\ \amp=5x+9+(-3x)+(-2)\\ \amp=2x+7 \end{align*}Remark2.8.19
The above example demonstrates how we apply the distributive property in order to subtract two expressions. But in practice, it can be pretty cumbersome. A shorter (and often clearer) approach is to instead subtract every term in the expression we are subtracting, which is shown like this:
\begin{align*} (5x+9)-(3x+2)\amp=5x+9-3x-2\\ \amp=2x+7 \end{align*}In general, we'll use this approach.
Example2.8.20
Use the associative, commutative, and distributive properties to simplify the following expressions as much as possible.
\((-6x+4)-(3x-7)\)
\((-2x-5)-(-4x-6)\)
Solution
-
We will remove parentheses by the distributive property, and then combine like terms:
\begin{align*} (-6x+4)-(3x-7) \amp=-6x+4-3x-(-7)\\ \amp=-6x+4-3x+7\\ \amp=-9x+11 \end{align*} -
We will remove parentheses by the distributive property, and then combine like terms:
\begin{align*} (-2x-5)-(-4x-6)\amp=-2x-5-(-4x)-(-6)\\ \amp=-2x-5+4x+6\\ \amp=2x+1 \end{align*}
Subsection2.8.5The Role of the Order of Operations in Applying the Commutative, Associative, and Distributive Properties
When simplifying an expression such as \(3+4(5x+7)\text{,}\) we need to apply the order of operations and do multiplication \(4(5x+7)\) before doing addition. We cannot do \(3+4(5x+7)=7(5x+7)\text{,}\) which is not following the order of operations.
To simplify \(3+4(5x+7)\text{,}\) we will first apply the distributive property. After that, we will use the commutative and associative properties:
\begin{align*} 3+4(5x+7)\amp=3+4(5x)+4(7)\\ \amp=3+20x+28\\ \amp=20x+3+28\\ \amp=20x+31 \end{align*}Exercise2.8.21
Simplify the following expressions using the commutative, associative, and distributive properties.
\(4-(3x-9)\)
\(5x+9(-2x+3)\)
\(5(x-9)+4(x+4)\)
Solution
-
We will remove parentheses by the distributive property, and then combine like terms:
\begin{align*} 4-(3x-9) \amp=4-3x-(-9)\\ \amp=4-3x+9\\ \amp=-3x+13 \end{align*} -
We will remove parentheses by the distributive property, and then combine like terms:
\begin{align*} 5x+9(-2x+3)\amp=5x+9(-2x)+9(3)\\ \amp=5x-18x+27\\ \amp=-13x+27 \end{align*} -
We will remove parentheses by the distributive property, and then combine like terms:
\begin{align*} 5(x-9)+4(x+4)\amp=5x-45+4x+16\\ \amp=9x-29 \end{align*}
Exercise2.8.22
Subsection2.8.6Rules of Exponents and Simplifying
In the section Introduction to Exponent Rules, we introduced the three exponent rules for multiplication. It is important to understand how these three rules fit within the larger picture of simplifying expressions.
Example2.8.23
Simplify the following expressions using the rules of exponents.
\(-2t^3\cdot 4t^5\)
\(5\left(v^4\right)^2\)
\(-(3u)^2\)
\((-3u)^2\)
Solution
-
To simplify \(-2t^3\cdot 4t^5\text{,}\) we want to multiply and simplify the constant factors and then apply the Product Rule:
\begin{align*} -2t^3\cdot 4t^5 \amp=-2\cdot 4 \cdot t^3\cdot t^5\\ \amp=-8t^{3+5}\\ \amp=-8t^8 \end{align*} -
To start simplifying \(5\left(v^4\right)^2\text{,}\) we want to first recognize that the number \(5\) is outside the parentheses to which the exponent of \(2\) applies. According to the order of operations, we will first apply the exponent in \(\left(v^4\right)^2\text{,}\) and then multiply \(5\text{.}\) In applying this exponent, we'll use Power to a Power Rule:
\begin{align*} 5\left(v^4\right)^2 \amp=5\left(v^{4\cdot 2}\right)\\ \amp=5v^8 \end{align*} -
As we simplify \(-(3u)^2\text{,}\) we will again note that the exponent of \(2\) only applies to the part inside the parentheses. So we will first simplify \((3u)^2\) according to the Product to a Power Rule:
\begin{align*} -(3u)^2\amp=-\left(3^2\cdot u^2\right)\\ \amp=-\left(9u^2\right)\\ \amp=-9u^2 \end{align*} -
In this problem, the exponent of \(2\) applies to \((-3u)\text{.}\) We have:
\begin{align*} (-3u)^2\amp=(-3)^2\cdot u^2\\ \amp=9u^2 \end{align*}
When we add/subtract two expressions, we can only combine like terms. For example:
\(3x-x=2x\)
\(x^2+x^2=2x^2\)
\(x^2+x\) cannot be combined.
However, we can multiply two expressions regardless of whether they are like terms or not. For example:
\(x\cdot x=x^2\)
\(x^2\cdot x^2=x^4\)
\((x^2)(x)=x^3\)
Compare the following problems:
When we combine like terms, the variable's exponent doesn't change, as in \(x^2+x^2=2x^2\text{.}\)
When we multiply expressions with the same variable, its exponent will change, as in \((x^2)(x^2)=x^4\text{.}\)
We cannot combine "unlike terms," as \(x^2+x\) will stay like it is.
We can multiply expressions with "unlike terms," as in \((x^2)(x)=x^3\text{.}\)
The next few examples test your understanding of these concepts.
Example2.8.24
Simplify the following expressions using the rules of exponents and the distributive property.
\(3x^2+2x+x^2\)
\((3x^2)(2x)(x^2)\)
\(2x(3x+4)\)
\(3x^2(5x-2)\)
\(x^3-3x^2(5x-2)\)
Solution
-
We will combine like terms \(3x^2\) and \(x^2\text{:}\)
\begin{align*} 3x^2+2x+x^2\amp=4x^2+2x \end{align*} -
We will apply the Product Rule:
\begin{align*} (3x^2)(2x)(x^2)\amp=6x^5 \end{align*} -
To simplify \(2x(3x+4)\text{,}\) we want to first distribute \(2x\text{,}\) and then we can apply the Product Rule:
\begin{align*} 2x(3x+4)\amp=2x(3x)+2x(4)\\ \amp=6x^2+8x \end{align*} -
We will use the distributive property first, and then apply the Product Rule:
\begin{align*} 3x^2(5x-2)\amp=3x^2(5x)-3x^2(2)\\ \amp=15x^3-6x^2 \end{align*} -
We will use the distributive property first, apply the Product Rule, and combine like terms:
\begin{align*} x^3-3x^2(5x-2)\amp=x^3-3x^2(5x)-(-3x^2)(2)\\ \amp=x^3-15x^3+6x^2\\ \amp=-14x^3+6x^2 \end{align*}
Subsection2.8.7Exercises
These exercises involve the concepts of like terms and the commutative, associative, and distributive properties.
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These exercises involve the rules of exponents.
These exercises involve rules of exponents and combining like terms.
These exercises involve the distributive property and rules of exponents.