Simplifying Algebraic Expressions
A Beginner’s Guide
What are Algebraic Expressions?
The first question you might have is “What exactly are algebraic expressions?” As the name implies you have expressions involving algebra, but what does that mean?
An expression is a grouping that can include any combination of numbers, symbols, and operators (addition, subtraction, multiplication, division). The one thing that expressions don’t have are equal signs. The presence of an equal sign makes an equation, not an expression, although an expression can be part of an equation.
Since the expressions we’ll be dealing with are algebraic we know that they will include algebraic variables, making them a tad bit more abstract. You’ll often be asked to “simplify algebraic expressions”, that means you want to rewrite the expression as shortly as possible, so combine all the things you can!
Watch my video tutorials for fully worked-out examples and read on to learn all the notation and exponent rules you need to know 😁
Level 1: Simplifying Algebraic Expressions
Level 2: Challenging Algebraic Expressions
Yes, there are always rules.
Like everything you have ever done in mathematics, there are some rules you must obey. But don’t worry the rules are pretty straightforward and once you have the hang of it you’ll hardly notice them. In fact, these rules build on your prior knowledge of mathematics, so they’ll be a breeze for you!
Rule #1: When Multiplying Like Bases, Add the Exponents
Basically, when you multiply two of the same type of things together (same base), you add the exponents. This is nothing new. You’ve been doing this for a while. For example, when you multiply 2³ x 2⁵ you get 2⁸, since 3 + 5 = 8.
The same applies when you are working with variables. For instance, a³ x a⁵ = a⁸. Now you must be careful not to accidentally combine two things that are different together. You can’t simplify a³ x b⁵ because they are different types of things.
Rule #2: When Dividing Like Bases, Subtract the Exponents (or use the Canceling technique)
Whenever you stumble across two like things being divided, you have two options for handling it. They are essentially the same idea, but depending on how your brain works you’ll probably prefer one way over the other.
The first way you can handle a division of like things is to subtract the exponents. So for example, if I have a⁸/a⁵ I can handle the operation by subtracting the exponent of 5 from 8 to get an answer of a³.
Again, I must have like things. So I can’t do any simplification to the expression a⁸/b⁵ since they have different bases.
The second way you can handle a division of like things is to cancel out like terms. You’ll be using the canceling technique lots in simplifying algebraic expressions, so this is a great opportunity to learn about it early on!
Remember that a⁸ = a ∙ a ∙ a ∙ a ∙ a ∙ a ∙ a ∙ a, and a⁵ = a ∙ a ∙ a ∙ a ∙ a, and recall that a / a = 1. That means that if I’m dividing the product of eight a’s by the product of five a’s, I have five a/a’s. Those in turn divide or cancel to 1, leaving only a ∙ a ∙ a, or a³.
Remember that you can use the canceling technique whenever you have a number or variable in the numerator and denominator that will divide to 1. In the video tutorial, we’ll walk through lots of instances where you can cancel out like terms to rapidly simplify your expressions.
Rule #3: When Raising a Power to a Power, Multiply Powers
You’ll often run into the scenario where you have a value with an exponent in parenthesis and then a power on the parenthesis. It’ll look something like this:
In this case, you want to multiply the powers together. So (a³)⁵ is equivalent to a¹⁵. Let’s take a look at why this is true.
Applying a power of 5 on the outside of the parenthesis means that we have whatever is inside the parenthesis multiplied with itself five times. In mathematical terms we have:
Using Rule #1 from above, we can see how (a³)⁵ equals a¹⁵ since a³ ∙ a³ ∙ a³ ∙ a³ ∙ a³ = a¹⁵.
Rule #4: When You See Negative Exponents, Flip It
The last thing that typically shows up in simplifying algebraic expressions problems is negative exponents. Whenever you see a number or variable with a negative exponent, move it to the opposite side of the fraction bar and remove the negative symbol.
So if you have a negative exponent in the numerator, such as:
Move it to the denominator, remove the negative, and simplify from there.
If you have a negative exponent in the denominator, move it to the top, remove the negative and simplify it.
And if you don’t have a fraction bar, you can always write your expression over 1 to create a fraction and simplify from there.
Putting It All Together
Of course, the difficult part is putting all of this together to simplify more complicated algebraic expressions. For this part, I think it’s best to see real problems worked out step-by-step. Check out the following tutorials for more help with algebraic expressions of all kinds!
Need More Math Help?
- Check out Math Hacks on YouTube for more hands-on math tutorials covering popular topics from Algebra through College Math.
- You can find a number of interesting math topics and problems right here on Medium, just click that follow button!
Thanks for joining me!
❤ Brett
