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Study Guides > Mathematics for the Liberal Arts

B1.05: Section 4

Evaluate variable expressions when given the value of each variable, using the correct order of operations. Recall these rules about order of operations. Do them in this order.

  1. All operations inside symbols of grouping (parentheses) from the inside out.
  2. All operations of exponents or roots.
  3. All multiplications and divisions, in order from left to right.
  4. All additions and subtractions, in order from left to right.
The best method I know is to write the expression first with the variable, then with parentheses in place of the variable, and then with the values inserted into the parentheses. Then you can remove any of those parentheses that aren’t needed to keep the negative numbers clear and to keep the products of two numbers clear. Then, begin to evaluate the expression according to the order of operations, doing one operation per step. For problems in this course, the most important of these rules are to do exponents first and then to do multiplications/divisions before additions/subtractions.

Example 1

Evaluate y=6+2xy=6+2x when x=7x=7.

Answer:

Solution:   \begin{align}&y=6+2x\\&y=6+2\cdot(\,\,\,)\\&y=6+2\cdot(7)\\&y=6+2\cdot7\\&y=6+14\\&y=20\\\end{align} Notice that the 7 here is in parentheses, but it is not an OPERATION in parentheses, so there is no need to think about that first rule in the order of operations.
Discussion: Later in the course, some students frequently do problems like this incorrectly because they do the addition before the multiplication. It is important to learn this rule well.

Example 2

Evaluate y=ux4y=u{{x}^{4}} when x=2   and   u=9x=2\,\,\,and\,\,\,u=9.

Answer: y=ux4y=(  )⋅(  )4y=( 9)⋅( 2)4y=9⋅24y=9⋅16y=144\begin{align}&y=u{{x}^{4}}\\&y=(\,\,)\cdot{{(\,\,)}^{4}}\\&y=(\,9)\cdot{{(\,2)}^{4}}\\&y=9\cdot{{2}^{4}}\\&y=9\cdot16\\&y=144\\\end{align} Discussion: Later in the course, some students have trouble remembering not to do the multiplication of 9 times 2 here first. Many of our formulas will include exponents such as this, so it is important to learn this rule well.

Checking your work

When evaluating expressions, the only clear method to check your work is to simply re-work it. That’s not completely satisfactory because if you have a mistake in understanding, you’re likely to make it both times. However, in this course, most of the times when we will evaluate an expression, it is in the context of a larger problem for which there is a good method of checking available.

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