Use of Parentheses and
Other Grouping Symbols

Already in this unit you have seen the use of parentheses. In the section on exponents we used parentheses to illustrate how entire terms were raised to a power. This was an example of using parentheses for grouping. Using grouping symbols helps us to identify the order in which we should apply mathematical operations.

When we have more complex expressions, which may combine several terms, and use multiple operations, we may need to group terms to help stay organized. Parentheses, ( ), are most commonly used in grouping but you may also see brackets, [ ], or braces, { }. When a term or expression is inside one of these grouping symbols it means that any operation indicated to be done on the group is done to the entire term or expression. For example, in the section on exponents, when a term inside parentheses is raised to a power, it means we raise the entire term to that power. With terms, this means we raise each numerical coefficient and variable in the term to that power.

(2x)2 = (2x)(2x) = 22x2 = 4x2

Also, when we raise an expression with parentheses to a power, it means to multiply the entire expression by itself the number of times indicated by the exponent.

(5x2 + y)4 = (5x2 + y) (5x2 + y) (5x2 + y) (5x2 + y)

When there are multiple grouping symbols in a single expression, the preferred way to write them is {[( )]} with the parentheses on the inside, the square-shaped brackets used next, and braces used outermost. An example of an expression that uses multiple grouping symbols is:

3{[2(2x + 6) + 3x] – 12z}= 21x + 36 – 36z

Grouping Algebraic Terms

When simplifying an expression, we always work from within the grouping symbols first. When we have multiple grouping symbols, we work from the innermost set towards the outside. Below is an example of this.

  • We first begin simplifying within the parentheses. In the example, this means adding the terms inside the parentheses.
  • Then multiply the term inside the parentheses with the one outside the parentheses.

3 (5x + 2x) = 3 (7x)

3 (7x) = 21x

The example above only has one set of grouping symbols, the parentheses. In the example below we have two sets of grouping symbols, parentheses and brackets.

  • We begin by performing the operation in the innermost grouping symbols, the parentheses.

15x [(3xy + 4xy) + 10xy] = 15x [7xy + 10xy]
  • We then perform the operation in the brackets.

15x [7xy + 10xy] = 15x [17xy]
  • Finally we perform the multiplication of the term outside the brackets with the one inside.

15x [17xy] = 255x2y

Order of Operations

The mathematical tools we've reviewed in this unit will help you to evaluate expressions. Remember, that means reducing an expression to its simplified form. The final tool you will need to evaluate complex expressions is the order of operations. The order in which you perform these steps makes a difference. There is a systematic way to evaluate expressions to ensure you do it correctly.

Order Of Operations

The order of operations tells us a step-by-step method for evaluating expressions:

  1. Evaluate all expressions within parentheses and other grouping symbols.
    (See page 20 for a discussion of this step.)
  2. Evaluate all expressions involving exponents. (See page 12 for a discussion
    of this step.)
  3. Do the remaining multiplication and division, as you come to them, when working from left to right in the expression. (See below for a discussion of this step.)
  4. Do the remaining addition and subtraction, as you come to them, when working from left to right in the expression. (See below for a discussion of this step.)

One trick to help remember the correct order of operations is to think of the acronym PE[MD][AS]. This stands for Parentheses, Exponents, [Multiplication and Division], [Addition and Subtraction].

For example, suppose you want to evaluate a simple expression. Look at the example shown below. If we follow the order of operations, we get a result of 16.

6 + 2 • 5 = 6 + 10 = 16

When dealing with algebraic expressions, or terms which contain variables, we follow the same order of operations. It is important to begin within the parentheses (or other grouping symbols) and follow the rest of the steps from there.

Example

Evaluate the following expressions.

  1. [(10 • 2 + 8) ÷ 14]2
     
  2. 12x2y ÷ 4x + (6x – x)2
     
  3. (x + z) + (5x • 2)
     
  4. {4 [2xz4 (2c3v ÷ 2cv2)3 ÷ 2]} ÷ c6x • v3

Answers

Evaluate the following expressions.

  1. [(10 • 2 + 8) ÷ 14]2 = 4
     
  2. 12x2y ÷ 4x + (6x – x)2 = 3xy + 25x2
     
  3. (x + z) + (5x • 2) = 11x + z
     
  4. {4 [2xz4 (2c3v ÷ 2cv2)3 ÷ 2]} ÷ c6x • v3 = 4z4

Detailed Answers

When evaluating expressions we first must consider the order of operations, PE[MD][AS].

1. [(10 • 2 + 8) ÷ 14]2 = 4

In this expression we have two sets of parentheses.

We begin by evaluating the expression in the innermost parentheses, following the order of operations within that inner set of parentheses. This is shown on the right.

[(10 • 2 + 8) ÷ 14]2 =
[(20 + 8) ÷ 14]2 =
[28 ÷ 14]2

We then look at the second expression in brackets and follow the order of operations within this set of brackets. This is shown on the right.

 [28 ÷ 14]2 =[2]2

Now we move on to exponents outside the brackets.

 [2]2 = 4

2. 12x2y ÷ 4x + (6x – x)2 = 3xy + 25x2

For this expression we work through the order of operations: P-Parentheses, E-Exponents, [M-Multiplication and D-Division], [A-Addition and S-Subtraction].

We must begin with the parentheses, (6x – x), in this expression. We first do the subtraction here. Because the terms are like terms, x can be subtracted from 6x. Once this is done, we perform the operation indicated by the exponent associated with the parentheses.

12x2y ÷ 4x + (6x – x)2 =
12x2y ÷ 4x + (5x) 2 =
12x2y ÷ 4x + 25x 2

Now we need to perform the multiplication and division in the expression. Notice that there is only division, no multiplication. When we divide 12x2y by 4x, we get 3xy.

The next step is to perform any addition or subtraction. Since the two terms left in the expression are not like terms, they cannot be subtracted. The expression has been simplified.

12x2y ÷ 4x + 25x 2 =
3xy + 25x 2

3. (x + z) + (5x • 2) = 11x + z

We begin with both sets of parentheses and perform the indicated operations within each set of parentheses. Note that x and z in the first set are not like terms so they cannot be added.

(x + z) + (5x • 2) =
(x + z) + 10x

If we refer to the commutative property:

a + b = b + a,

and associative property:

a + (b + c) = (a + b) + c

we reorder and regroup the terms.

(x + z) + 10x = 10x + (x + z)
10x + (x + z) = (10x + x) + z
(10x + x) + z = 11x + z

4. {4 [2xz4 (2c3v ÷ 2cv2)3 ÷ 2]} ÷ c6x • v3 = 4z4

Here we have a number of sets of parentheses and brackets that group parts of the expression. We perform the order of operations from the inside out.

We begin with the inside set of parentheses, which contains division.

{4 [2xz4 (2c3v ÷ 2cv2)3 ÷ 2]} ÷ c6x • v3 ={4 [2xz4 (c2/v)3 ÷ 2]} ÷ c6x • v3

Then the exponent outside of the parentheses.

{4 [2xz4 (c2/v)3 ÷ 2]} ÷ c6x • v3 ={4 [2xz4 (c6/v3) ÷ 2]} ÷ c6x • v3

Now we look at the inner set of brackets and follow the order of operations within this set of brackets. First we multiply 2xz4 by c6/v3.

{4 [2xz4 (c6/v3) ÷ 2]} ÷ c6x • v3 = {4 [2xz4c6/v3 ÷ 2]} ÷ c6x • v3

Then we divide the result of this multiplication, 2xz4c6/v3, by 2.

{4 [2xz4c6/v3 ÷ 2]} ÷ c6x • v3 = {4 [xz4c6/v3]} ÷ c6x • v3

If we then look at the remaining braces, there is just a multiplication by 4 and
division by v3.

{4 [xz4c6/v3]} ÷ c6x • v3 = 4xz4c6/v3 ÷ c6x • v3

We are now left only with an expression with no grouping symbols, so follow the remainder of the order of operations. The first term in this expression is the
fraction 4xz4c6/v3. We can rewrite the exponents of the variable in the denominator of this expression so it can be rewritten as 4xz4c6v–3. We begin with the division of 4xz4c6v–3 by c6x.

4xz4c6v–3 ÷ c6x • v3 = 4x1–1z4c6–6v–3 • v3 = 4z4v–3 • v3

Then we need to multiply 4z4v–3 by v3.

4z4v–3 • v3 = 4z4v–3 + 3 = 4z4

For each set of grouping symbols, the order of operations holds.

[introduction] [exponents] [multiply/divide] [add/subtract] [grouping]