Using Order of Operations in Algebra

Introduction

This Webquest will reintroduce students who may have already learned the Order of Operations once before, and tack on some things you may not have known apply when it comes to algebra. For example, when should a square root be calculated, or when does sine and cosine apply? Don't worry if you don't remember old stuff, we'll review that too!

Task

Let's begin with a quick exercise: grab a calculator and plug in this problem: 76 - 48 x 168 + 2 x (53+4) = ?

What did you come up with? If you answered 588,254 the calculator did not take order of operations into account. This is a common problem for everyday calculators, and it can be a problem for students too! The true answer is -7,730. So now let's get into why this is the case and how to get it right every time!

Process

Remember the acronym PEMDAS? Maybe you remember the mnemonic Please Excuse My Dear Aunt Sally. This gives you a hint into the Order of Operations. Let's go over them once:

Parenthesis

Exponents

Multiplication

Division

Addition

Subraction

1. Parentheses

The equation inside of a set of parentheses should always be solved first. Let's look at another example:

Solving the parentheses first will give you 3*(9) which equals 27. If the equation was solved simply left to right, the 3 probably would have been multiplied by 7, then adding the 2 would result in the wrong answer. Parentheses can come in a few forms such as () or [], but most commonly you'll just see ().

2. Exponents

After the parentheses have been solved, we can start solving exponents! Solving an exponent gives a nice easy-to-work-with number that can be used in a later step. Now let's add to the previous example:

Parentheses still come first and our equation becomes 3 squared times 9. Before multiplying 3 to 9 we have to take care of that exponent! Squaring 3 gives us 9, and the equation becomes 9*(9)=81. If we had multiplied 3 and 9 and THEN squared it we would have ended up with 729, and that's just not right at all!

But what if we had an exponent INSIDE the parentheses? Nothing really changes, simply solve the equation inside the parentheses first, as if it was all by itself, and once that is taken care of continue as you normally would. Let's see it in action:

Take the parentheses by itself and square 7 to get 49, then add 2. Now the equation becomes 3 squared times 51. Follow the rules we learned so far and you'll get 459.

3. Multiplication and Division

Multiplication and division are interchangeable, and either may be solved first. As a general rule of thumb, it is better to solve these left to right to make it easy to keep track of what has been done. Going back to our example:

Here everything begins the same. Parentheses, exponents, and now we have some extra multiplication. But remember: just because the 4 comes before everything else does NOT mean we multiply first! Start with parentheses, then the exponent, THEN we multiply with the 4 and divide by the 2. Since we already know part of the equation we can skip ahead a little bit. The equation becomes:

Now we can simply multiply left to right and get our answer of 918! If we changed the equation a little bit and moved the 2 next to the 4 we would get the same answer. Just be sure to keep the division sign with the 2! Here is what it would look like:

4. Addition and Subtraction

Like multiplication and division, addition and subraction also hold the same weight as each other and be solved out of order. Keep in mind that it usually is easier to keep it simple and solve these left to right. Let's add one final piece to our example equation:

Again, don't just start solving left to right! Instead simply follow the order of operations. Previously we had solved most of this already, so we can skip ahead again. The equation becomes:

With everything else out of the way we can finally solve the equation to get 90. Remember: addition and subtraction can be solved in any order, but always make sure the addition and subtraction signs stay the same!

5. More complicated functions

Sometimes our equations do not always use nice whole numbers. Sometimes, they don't use numbers at all! So when should those other functions be solved? First let's look at some examples of what we're up against.

Roots:

Logarithms:

Trigonometry:

Each of these examples is a function in and of itself, and should be treated as if they are in parentheses. That means solve what you can on the inside, and if the rest cannot be solved without a calculator then you probably don't have to worry about it. When you leave things unsolved in an equation it is called the exact value. Whether you solve a function or leave it in an exact value format should be discussed with your instructor. If calculators are allowed instructors will often ask you to solve the equation with no more than a number for the final answer, but this is not always the case. Here's some examples:

______________________________________________________

Evaluation

To put this knowledge into practice we'll have three problems to complete. Each one increases in difficulty. Follow the Order of Operations and complete the following problems. Solutions are on the Conclusion page.

Conclusion

After reviewing this short webquest you should now have the tools required to solve both current problems and many future problems! Keep in mind that problems do not always work out nicely here, and especially in algebra things can look a little different when functions and variables come into play. There's no need to worry though, just keep things simple and work out those problems as you would if they were actual numbers.

Solutions: