There is a specific order for performing computations when solving a mathematical expression with more than one mathematical operation. The order of these steps can be remembered through the acronym PEMDAS:
Let's explore this process using the following expression:
Step 1: Resolve any parentheses
In our expression, we have 5 * 3 inside a set of parentheses, so we need to find that product first. We can then substitute that back into the expression and remove the parentheses since they are now resolved.
Step 2: Complete exponents
Next, we need to resolve any exponents by performing the appropriate calculation. Remember, an exponent tells us how many times the base number is multiplied by itself. In this instance, that means we will rethink 4 squared as 4 x 4. We will again find that product and plug it back into the expression.
Step 3: Resolve multiplication and division from left to right
We can now resolve any multiplication or division that remains in the expression. We always solve these from left to right. In our expression, we only have division. So, we will take 16 divided by 8 and plug that quotient back into the expression.
Step 4: Resolve addition and subtraction from left to right
In our last step, we will perform any adding or subtracting that remains. Again, this is done from left to right. Reading across our expression, we have 2 - 15 first. We can plug this difference into the expression before performing the addition.
All that's left to do is add the last two numbers!
Sometimes when trying to combine like terms, we need to move terms from one side of the equation to the other side. Thankfully, we can do this by applying the inverse or opposite function. Addition and subtraction are inverse functions, and multiplication and division are inverse functions. Sometimes this is called reverse PEMDAS.
Just remember: to keep the equation balanced, what we do on one side, we must do on the other side.
Another important inverse to know is exponents and radicals. That is, the opposite of squared is the square root. The opposite of cubed is a cube root.
Let's apply the order of operations to equations that involve an unknown. Oftentimes the unknown is written as x in the equation, but it can be any letter. To reinforce this idea, these guides will use an assortment of letters when working through problems with unknown values.
Before we look at an equation, let's talk about the idea that a whole is the sum of its parts. That is, if I have a string that is 10 inches long with a knot on it, and I know that the length of the string to the right of that knot is 7 inches, then I can find the length of the string to the left of the knot. Here's how I might work through this logic:
Whole string length (W) = 10 inches
Right length (r) = 7 inches
Left length (l) = ?
I can write an equation to represent this scenario:
With this equation, I can plug in the values that I do know from above for the appropriate variables in the equation. I keep the unknown the same. That would look like this:
Now that I only have one unknown variable, I can move the numbers to one side of the equation to isolate the unknown, or get it by itself. To do that, I consider what is currently on the same side as the unknown. +7 is on the same side of the equation, so I need to move it to the other side by using the inverse operation. That is, since it is being added on one side, I can move it by subtracting it from both sides. The last part is important. I must do it on both sides of the equation to keep it balanced. That step would look like this:
Based on this work, we now know that the length of the left side must be equal to 3 inches. We can check our work by plugging that in for l. Since 7 + 3 = 10, we know our answer is correct!
The above example demonstrates how we can use given information in an equation to solve for the value of an unknown variable. For this to work, we must have an equation, meaning there must be an = in the problem. Let's consider another example:
In this equation, we still have one unknown, x. That unknown is divided by 3. To move the 3 to the other side of the equation, we must multiply both sides by 3 (because multiplication is the inverse of division):
It's always a good habit to plug the answer back into the original equation to check your work. On simpler problems, it may seem silly, but when you get to more complex problems, this step can help you avoid silly mistakes. If we plug 27 in for x, we see 27 divided by 3. If we reduce that, we get 9 = 9. Since the numbers match, our answer is correct!
