Order of Operations

LESSON OBJECTIVES

When you complete this lesson, you will be able to:

  • Use the standard order of operations to simplify and evaluate algebraic expressions.
  • Simplify and evaluate numerical and algebraic expressions that contain parentheses.

LESSON SUMMARY

Adding, subtracting, multiplying and dividing are examples of operations. When we do any of
these things, or when we raise a number to a power or find its square root (or any other root), we
are performing an operation.

Sometimes, we’re asked to evaluate an algebraic expression like this:

\(2+5\times3\;+\;4^2\;-8\;\div\;2\)

In such cases the answer may depend on which operation we do first. To handle this difficulty,
we follow the standard order of operations:

  • First, we figure out any expressions with exponents or radicals. In the above, we have one such operation: 42

    Of course, 42 = 16. The expression becomes:

    \(2+5\times3\;+\;16\;-8\;\div\;2\)

  • Second, we figure out any multiplication or division. If multiplication or division appears
    more than once, we work from left to right. In the above example, this means our next step is to
    figure out five times three first and then eight divided by two.

    We obtain:

    \(2+15+16-4\)

  • Finally, we figure out any addition or subtraction. If these operations appear more than once,
    we again work from left to right. We see that:

    2 + 15 + 16 – 4
    17 + 16 – 4
    33 – 4
    29

However, if you are working with a fraction, and the numerator or denominator (or both) require
simplifying, you begin right there. For example:

\(\frac{14-3^2}7=\frac{14-9}7=\frac57\)

If we want to change the order of operations in an expression, we use parentheses. The parentheses indicate that the contents are a package. We can’t break it up. We must do it first. Hence the most common meaning of parentheses in mathematics is “do this first.” We perform the operations inside parentheses before going on with the rest of the problem. For example, (10+4)x2 means that first we add ten and four. Then we multiply the answer by two.

So:

(10+4)x2
14×2
28

Notice that without the parentheses, the expression becomes:

10+4×2
10+8
18

(since multiplication precedes addition in the standard order of operations.)

Parentheses also have other meanings in mathematics. Here are some examples:

  • (3+4)(8−5) . This means first add 3 and 4. Next subtract 5 from 8. Finally, multiply the
    two results. (3+4)(8−5) = (7)(3) = 21
  • 3−(−5). Here the parentheses are used to separate the two signs. In general, we never
    write two signs alongside each other. We always separate them with parentheses.

Examples

These examples can help you master the new ideas in this lesson.

Example 1

Evaluate each of the following algebraic expressions using these values: m=0, p=3, r=2

  1. (p-1)(mr)
  2. \(r+\frac{12}p=2+\frac{12}3\)
  3. \(\frac{4m+p}{p-r}\)

Solution

  1. Our first step is to replace the variables with the numbers they stand for.

    \((p-1)(m-r)=(3-1)(0-2)\)

    We must now evaluate the expressions within the parentheses. There are two pairs and they have
    equal priority, so we start at the left.

    That gives us:

    \((3-1)=2\;and\;(0-2)=-2\)
    \((2)(-2)=-4\)

  2. Again, we begin by replacing the variables with the numbers they stand for. We obtain:

    \(r+\frac{12}p=2+\frac{12}3\)

    Division comes before addition in the standard order of operations, so we begin here. Since
    twelve divided by three equals four, we get:

    \(2+4=6\)

  3. As before, we begin by replacing the variables with the numbers they stand for.

    \(\frac{4m+p}{p-r}=\frac{4(0)+3}{3-2}\)

    When we work with a fraction, we figure out the numerator and denominator separately before we divide. Within each part, we follow the standard order of operations.

    numerator: 4(0)-3=0-3=-3
    denominator: 3-2=1
    So the fraction is \(\frac31=3\)

Example 2

Evaluate each of the following algebraic expressions using these values: K=2,L=5,N=-3

  1. K3+(4L)2
  2. -2(5-(N+K2))

Solution

  1. Substituting, we find that: \(K^3+{(4L)}^2=2^3+{(4\times5)}^2\)

Since 4 x 5 is in parentheses, we multiply four times five before squaring.

23+(4×5)2
23+(20)2
8+400
408

  • In this example, we have one pair of parentheses within another. We begin with the inner pair.

    (N+K2)
    (-3+22)
    (-3+4)
    1

    So far, we have:

    -2(5-1)

    Again, the parentheses tell us to “do this first”.

    (5-1)
    4

    So the expression becomes: -2(4)=-8

    Self-Test

    Use this test to check your understanding.

    1. Evaluate each of the following algebraic expressions using values of j=3,m=4

      a) (j-7)(m-2)

      b) j-(-m)
      c) \(\frac{2j+m}5\)

    2. Evaluate each of the following algebraic expressions using values of N=-2,P=5,R=0

      a) \(3N^2+\frac{2R}P\)
      b) \(N^3-\sqrt{5P}\)
      c) \(\frac{4P}{2(N+(R-3))}\)

    3. Evaluate each of the following algebraic expressions using values of f=2,g=1,h=3

      a) \({(fh)}^2+8g^3\)
      b) \(\frac{f-6g^2}{2h}\)

    Compare your answers with those below.

    Answers to Self-Test

    1. a) -8
      b) 7
      c) 2
    2. a) 12
      b) -13
      c) -2
    3. a) 44
      b) \(-\frac23\)

    Practice Problems

    1. Evaluate each of the following algebraic expressions using these values: g=2,h=3
      a) (g+1)(h+1)
      b) 6-g+h
      c) 6-(g+h)
    2. Evaluate each of the following algebraic expressions using these values: R=1,T=4
      a) R2-2T+7
      b) \(\frac{T+5R}3\)
      c) (TR)2
    3. Evaluate each of the following algebraic expressions using these values: P=5,W=-1,Z=-2
      a) \(\frac P{5W+Z}\)
      b) (WP)(Z-2)
      c) 3Z2+PW
    4. Evaluate each of the following algebraic expressions using these values: k=2,z=3,w=1
      a) (kz)(w-4)
      b) k+(z(w+2))
      c) \(\frac{z^2\sqrt{3K+10}}3\)

    Answers

    1. a) 12
      b) 7
      c) 1
    2. a) 0
      b) 3
      c) 16
    3. a) \(-\frac57\)
      b) 24
      c) 7
    4. a) 3
      b) 11
      c) \(\frac{13}3\)
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