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1.5: Algebraic Expressions

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The associative property of multiplication is valid for all numbers.

Associative Property of Multiplication

Let a, b, and c be any numbers. Then: a(bc)=(ab)c

The associative property of multiplication is useful in a number of situations.

Example 1.5.1

Simplify: 3(4y).

Solution

Currently, the grouping 3(4y) demands that we first multiply 4 and y. However, we can use the associative property of multiplication to regroup, first multiplying 3 and 4.

3(4y)=(34)y The associative property of multiplication.=12y Multiply: 34=12

Thus, 3(4y)=12y.

Exercise 1.5.1

Simplify: 2(3x).

Answer

6x

Let’s look at another example.

Example 1.5.2

Simplify: 2(4xy).

Solution

Currently, the grouping 2(4xy) demands that we first multiply 4 and xy. However, we can use the associative property of multiplication to regroup, first multiplying 2 and 4.

2(4xy)=(2(4))xy The associative property of multiplication. =8xy Multiply: 2(4)=8

Thus, 2(4xy)=8xy.

Exercise 1.5.2

Simplify: 3(8u2).

Answer

24u2

In practice, we can move quicker if we perform the regrouping mentally, then simply write down the answer. For example:

2(4t)=8t and 2(5z2)=10z2 and 3(4u3)=12u3

The Distributive Property

We now discuss a property that couples addition and multiplication. Consider the expression 2(3+5). The Rules Guiding Order of Operations require that we first simplify the expression inside the parentheses.

2(3+5)=28 Add: 3+5=8=16 Multiply: 28=16

Alternatively, we can instead distribute the 2 times each term in the parentheses. That is, we will first multiply the 3 by 2, then multiply the 5 by 2. Then we add the results.

2(3+5)=23+25 Distribute the 2. =6+10 Multiply: 23=6 and 25=10=16 Add: 6+10=16

Note that both methods produce the same result, namely 16. This example demonstrates an extremely important property of numbers called the distributive property.

The Distributive Property

Let a, b, and c be any numbers. Then: a(b+c)=ab+ac That is, multiplication is distributive with respect to addition.

Example 1.5.3

Use the distributive property to expand 2(3x+7).

Solution

First distribute the 2 times each term in the parentheses. Then simplify.

2(3x+7)=2(3x)+2(7) Use the distributive property. =6x+14 Multiply: 2(3x)=6x and 2(7)=14

Thus, 2(3x+7)=6x+14.

Exercise 1.5.3

Expand: 5(2y+7).

Answer

10y+35

Multiplication is also distributive with respect to subtraction.

Example 1.5.4

Use the distributive property to expand 2(5y6).

Solution

Change to addition by adding the opposite, then apply the distributive property.

2(5y6)=2(5y+(6)) Add the opposite. =2(5y)+(2)(6) Use the distributive property. =10y+12 Multiply: 2(5y)=10y and (2)(6)=12

Thus, 2(5y6)=10y+12

Exercise 1.5.4

Expand: 3(2z7).

Answer

6z+21

Speeding Things Up a Bit

In Example 1.5.4, we changed the subtraction to addition, applied the distributive property, then several steps later we were finished. However, if you understand that subtraction is really the same as adding the opposite, and if you are willing to do a few steps in your head, you should be able to simply write down the answer immediately following the given problem.

If you look at the expression 2(5y6) from Example 1.5.4 again, only this time think “multiply 2 times 5y, then multiply 2 times 6, then the result is immediate. 2(5y6)=10y+12

Let’s try this “speeding it up” technique in a couple more examples.

Example 1.5.5

Use the distributive property to expand 3(2x+5y12).

Solution

To distribute the 3, we simply think as follows: “3(2x)=6x, 3(5y)=15y, and 3(12)=36.” This sort of thinking allows us to write down the answer immediately without any additional steps. 3(2x+5y12)=6x15y+36

Exercise 1.5.5

Expand: 3(2a+3b7).

Answer

6a9b+21

Example 1.5.6

Use the distributive property to expand 5(2a5b+8).

Solution

To distribute the 5, we simply think as follows: “5(2a)=10a, 5(5b)=25b, and 5(8)=40.” This sort of thinking allows us to write down the answer immediately without any additional steps. 5(2a5b+8)=10a+25b40

Exercise 1.5.6

Expand: 4(x2y7).

Answer

4x+8y+28

Distributing a Negative Sign

Recall that negating a number is equivalent to multiplying the number by 1.

Multiplicative Property of Minus One

If a is any number, then: (1)a=a

This means that if we negate an expression, it is equivalent to multiplying the expression by 1.

Example 1.5.7

Expand (7x8y10).

Solution

First, negating is equivalent to multiplying by 1. Then we can change subtraction to addition by “adding the opposite” and use the distributive property to finish the expansion.

(7x8y10)=1(7x8y10) Negating is equivalent to multiplying by 1=1(7x+(8y)+(10)) Add the opposite. =1(7x)+(1)(8y)+(1)(10) Distribute the 1=7x+8y+10 Multiply.

Thus, (7x8y10)=7x+8y+10

Exercise 1.5.7

Expand: (a2b+11).

Answer

a+2b11

While being mathematically precise, the technique of Example 1.5.7 can be simplified by noting that negating an expression surrounded by parentheses simply changes the sign of each term inside the parentheses to the opposite sign.

Once we understand this, we can simply “distribute the minus sign” and write:

(7x8y10)=7x+8y+10

In similar fashion,

(3a+5bc)=3a5b+c

and,

(3x8y+11)=3x+8y11

Combining Like Terms

We can use the distributive property to distribute a number times a sum. a(b+c)=ab+ac

However, the distributive property can also be used in reverse, to “unmultiply” or factor an expression. Thus, we can start with the expression ab+ac and “factor out” the common factor a as follows:

ab+ac=a(b+c)

You can also factor out the common factor on the right.

ac+bc=(a+b)c

We can use this latter technique to combine like terms.

Example 1.5.8

Simplify: 7x+5x.

Solution

Use the distributive property to factor out the common factor x from each term, then simplify the result.

7x+5x=(7+5)x Factor out an x using the distributive property. =12x Simplify: 7+5=12

Thus, 7x+5x=12x.

Exercise 1.5.8

Simplify: 3y+8y.

Answer

11y

Example 1.5.9

Simplify: 8a2+5a2.

Solution

Use the distributive property to factor out the common factor a2 from each term, then simplify the result.

8a2+5a2=(8+5)a2 Factor out an a2 using the distributive property. =3a2 Simplify: 8+5=3

Thus, 8a2+5a2=3a2.

Exercise 1.5.9

Simplify: 5z3+9z3.

Answer

4z3

Examples 1.5.8 and 1.5.9 combine what are known as “like terms.” Examples 1.5.8 and 1.5.9 also suggest a possible shortcut for combining like terms.

Like Terms

Two terms are called like terms if they have identical variable parts, which means that the terms must contain the same variables raised to the same exponents.

For example, 2x2y and 11x2y are like terms because they contain identical variables raised to the same exponents. On the other hand, 3st2 and 4s2t are not like terms. They contain the same variables, but the variables are not raised to the same exponents.

Consider the like terms 2x2y and 11x2y. The numbers 2 and 11 are called the coefficients of the like terms. We can use the distributive property to combine these like terms as we did in Examples 1.5.8 and 1.5.9, factoring out the common factor x2y.

2x2y+11x2y=(2+11)x2y=13x2y

However, a much quicker approach is simply to add the coefficients of the like terms, keeping the same variable part. That is, 2+11=13, so:

2x2y+11x2y=13x2y

This is the procedure we will follow from now on.

Example 1.5.10

Simplify: 8w2+17w2.

Solution

These are like terms. If we add the coefficients 8 and 17, we get 9. Thus:

8w2+17w2=9w2Add the coefficients and repeat the variable part.

Exercise 1.5.10

Simplify: 4ab15ab.

Answer

11ab

Example 1.5.11

Simplify: 4uv9uv.

Solution

These are like terms. If we add 4 and 9, we get 13. Thus:

4uv9uv=13uvAdd the coefficients and repeat the variable part.

Exercise 1.5.11

Simplify: 3xy8xy.

Answer

11xy

Example 1.5.12

Simplify: 3x2y+2xy2

Solution

These are not like terms. They do not have the same variable parts. They do have the same variables, but the variables are not raised to the same exponents. Consequently, this expression is already simplified as much as possible.

3x2y+2xy2Unlike terms. Already simplified.

Exercise 1.5.12

Simplify: 5ab+11bc.

Answer

5ab+11bc

Sometimes we have more than just a single pair of like terms. In that case, we want to group together the like terms and combine them.

Example 1.5.13

Simplify: 8u4v12u+9v.

Solution

Use the associative and commutative property of addition to change the order and regroup, then combine line terms.

8u4v12u+9v=(8u12u)+(4v+9v) Reorder and regroup. =20u+5v Combine like terms. 

Note that 8u12u=20u and 4v+9v=5v.

Alternate solution

You may skip the reordering and regrouping step if you wish, simply combining like terms mentally. That is, it is entirely possible to order your work as follows:

8u4v12u+9v=20u+5vCombine like terms.

Exercise 1.5.13

Simplify: 3z2+4z8z29z.

Answer

11z25z

In Example 1.5.13, the “Alternate solution” allows us to move more quickly and will be the technique we follow from here on, grouping and combining terms mentally.

Order of Operations

Now that we know how to combine like terms, let’s tackle some more complicated expressions that require the Rules Guiding Order of Operations.

Rules Guiding Order of Operations

When evaluating expressions, proceed in the following order.

  1. Evaluate expressions contained in grouping symbols first. If grouping symbols are nested, evaluate the expression in the innermost pair of grouping symbols first.
  2. Evaluate all exponents that appear in the expression.
  3. Perform all multiplications and divisions in the order that they appear in the expression, moving left to right.
  4. Perform all additions and subtractions in the order that they appear in the expression, moving left to right.

Example 1.5.14

Simplify: 4(3a+2b)3(4a5b).

Solution

Use the distributive property to distribute the 4 and the 3, then combine like terms.

4(3a+2b)3(4a5b)=12a+8b12a+15b Distribute. =24a+23b Combine like terms. 

Note that 12a12a=24a and 8b+15b=23b

Exercise 1.5.14

Simplify: 2x3(52x).

Answer

4x15

Example 1.5.15

Simplify: 2(3x4y)(5x2y).

Solution

Use the distributive property to multiply 2 times 3x4y, then distribute the minus sign times each term of the expression 5x2y. After that, combine like terms.

2(3x4y)(5x2y)=6x+8y5x+2y Distribute. =11x+10y Combine like terms. 

Note that 6x5x=11x and 8y+2y=10y.

Exercise 1.5.15

Simplify: 3(u+v)(u5v).

Answer

4u+2v

Example 1.5.16

Simplify: 2(x2y3xy2)4(x2y+3xy2).

Solution

Use the distributive property to multiply 2 times x2y3xy2 and 4 times x2y+3xy2. After that, combine like terms.

2(x2y3xy2)4(x2y+3xy2)=2x2y+6xy2+4x2y12xy2=2x2y6xy2

Note that 2x2y+4x2y=2x2y and 6xy212xy2=6xy2.

Exercise 1.5.16

Simplify: 8u2v3(u2v+4uv2).

Answer

5u2v12uv2

When grouping symbols are nested, evaluate the expression inside the innermost pair of grouping symbols first.

Example 1.5.17

Simplify: 2x2(2x2[2x2]).

Solution

Inside the parentheses, we have the expression 2x2[2x2]. The Rules Guiding Order of Operations dictate that we should multiply first, expanding 2[2x2] and combining like terms.

2x2(2x2[2x2])=2x2(2x+4x+2)=2x2(2x+2)

In the remaining expression, we again multiply first, expanding 2(2x+2) and combining like terms.

=2x4x4=6x4

Exercise 1.5.17

Simplify: x2[x+4(x+1)].

Answer

5x8

Contributors


This page titled 1.5: Algebraic Expressions is shared under a CC BY-NC-ND 3.0 license and was authored, remixed, and/or curated by David Arnold via source content that was edited to the style and standards of the LibreTexts platform.

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