Algebra 1 - Exponents and the Order of Operations 1.2

6/28/2010 By David Pilarski

Simplifying and Evaluating Expressions and Formulas

When you simplify numerical expression you perform all the operations so the new expression is in its simplest form. For example: 2*3 + 2*4 will simplify to 14. To simplify or evaluate numerical expressions, you must follow the order of operations.

Let’s simplify the following 4 + 6 *5. 

Order of Operations Comparison

Without a defined order to follow, different people could get different answers when simplifying the same numerical expression. That would be bad news. That is why there is the Order of Operations, so we all should get the same solution. Below is the order of operations. You will find it written like this or very similar to this in any math text book.

The Order of OperationsThe Order of Operations

Exponent, Base and PowerBefore we simplify full blown numerical expressions, we should review the parts of a power. To the right is the power, 3 to the 4th power, which means we need to multiply 3 by itself four times. In other words, 3 is used as a factor 4 times. In this power, 3 is the base and 4 is the exponent. It is common to call the 4 the “power”, but it is really the exponent and the base and the exponent together are called a power.

Example 1 Evaluating Numerical Expressions

Example 1 Evaluating a Numerical ExpressionIn this example, there are 5 operations that need to be completed. Referring the order from above, there are no parentheses in the expression. There is an exponent that should be evaluated first, but I did the multiplication and the division in the same step. How can I do that? Well, if you remember, an exponent represents repeated multiplication and therefore can be completed with the multiplication/division step of the order. So, to transform (1) into (2), I multiplied 5 and 8 to get 40, I divided 12 and 6 to get 2 and I raised 4 to second power to get 16. Notice how I brought down the remaining signs and where I placed them. From line (2) to line (3) I subtracted 2 from 40 to get 38 and I brought down the plus 16 Finally, I added 38 to 16 in line (3) to obtain 54.

I caution you and a problem like this because of line (2). Many students make the mistake of adding 2 to 16 first, because the last final step of the order of operations is said, “Addition and Subtraction”. Well it does not just say “Addition and Subtraction”, is SAYS, “Addition and Subtraction - Complete all addition and subtraction from left to right as it appears in the problem.”  

Example 2 Evaluating an Algebraic Expression

When you are asked to evaluate an algebraic expression you need to remember a math tool called substitution. Many math students and math teachers use words such as plug it in for the letter or replace the letter with the number. What you need to do is “substitute the given values in for the corresponding variable.” Of course the other ways are easier to remember, but it is important to be able to use the language of mathematics. So, in this example we will need to substitute the values in for the respective variables.

Evaluate 4x – y3 ÷ z if x = 8, y = 2 and z = 4.

4x – y3 ÷ z    Copy down the original expression

4(8) – 23 ÷ 4   Substitute 8 for x, 2 for y and 4 for z

32 – 8 ÷ 4   Evaluate 2 to the third and multiply 4 and 8

32 – 2  Divide 4 into 8

30 Final Answer

The main purpose of this example was to review using substitution.

Grouping Symbols and the Order of Operations

It is good to know that a fraction bar acts as a grouping symbol. It groups whatever is in the numerator together and whatever is in the denominator together.

  is the same as (25 – 3) ÷ (6 + 5)

Normally we would complete the division before any addition or subtraction, but in the above problem, the fraction bar or the parenthesis change the order. Remember, you must complete any operations within the parenthesis first. If there is more than one set of grouping symbols, then work from the inner most set first.

Example 3 Evaluating Expressions with Grouping Symbols

Example 3 Order of OperationsIn this problem, you first have to substitute 4 in for x and 2 in for y and this is how the first line changes into the second line.

Following the order of operations, I evaluate all of the exponents and that changes the problem’s second line into the problem’s third line.

Next, I perform two operations, the addition in the numerator and the subtraction in the denominator. This produces the fourth line in the problem.

Finally, I do the division. Remember, when completing the division indicated by a fraction bar, you divide the top number by the bottom number.

Example 4 Evaluating Expressions with Parentheses

Example 4 Order of Operations In this example, there will actually be parentheses. The original problem is  and it is worked out to the right. The most common mistake for students to make with this problem occurs from the third to the fourth line. The mistake that is made is similar to the mistake made in example 1 with addition and subtraction. In this problem, you are left with 27 ÷ 3(4). The mistake occurs by multiplying before dividing. I believe this mistake occurs because how the order of operations is written. To review: the third step of the order of operations reads, “Multiplication and Division – Complete all multiplication and division from left to right as it appears in the problem.” The bold part is the part that is forgotten when this mistake is made.

By no means does this Algebra 1 blog article encompass all of the possible examples that could be made up using the order of operations. If you understand the order and how to apply the steps, you should do fine with other order of operation problems.