Every math student must learn how to work with numbers inside of a radical (√) at some point. While working with perfect squares inside of radicals can be quite easy, the task becomes a bit trickier when non-perfect squares are involved and you have to simplify.
While learning how to simplify a radical of a non-perfect square may seem challenging, it’s actually a relatively easy math skill to learn as long as you have a strong understanding of perfect squares and factoring (we will review both of these topics in this guide).
This free How to Simplify Radicals Step-by-Step Guide will teach you how to simplify a radical when the number inside of it is not a perfect square using a simple 3-step strategy that you can use to solve any problem involving simplifying radicals.
The following topics and examples will be covered in this guide:
You can use the quick-links above to jump to any section of this guide. However, we highly recommend that you work through each section in order, including the following review section where we will quickly recap some important features of perfect squares, non-perfect squares, factors, and the properties of radicals. This review will recap some important vocabulary terms and prerequisite math skills that you will need to be successful with this new math skill.
But, before you learn how to add fractions, let’s do a quick review of some key characteristics and vocabulary terms related to fractions before we move onto a few step-by-step examples of how to add fractions.
Let’s get started!
Before you learn how to simplify a radical, it’s important that you understand what a radical is and what the difference between a perfect square and a non-perfect square is.
Definition: In math, a radical (√) is a symbol that is associated with the operation of finding the square root of a number.
Finding the square root of a number means finding an answer to the question: “What number times itself results in the original number?” For example, the square root of 25 (or √25) is equal to 5 because 5 times 5 equals 25.
Definition: In math, a perfect square is a number that is the square of any integer. This means that a perfect square is a number that can be represented as the product of an integer squared (or an integer multiplied by itself). For example, 36 is a perfect square because 6 times 6 equals 36.
Note that perfect squares are unique and there are not that many of them. In fact, most numbers are considered non-perfect squares because they can not be represented as an integer times itself.
Definition: In math, a non-perfect square is a number that is not the square of any integer. This means that a non-perfect square is a number that can’t be represented as the product of an integer squared (or an integer multiplied by itself). For example, 20 is not a non-perfect square because it is impossible to take an integer and multiply by itself to get a result of 20.
Again, most numbers are non-perfect squares. When we see these types of numbers inside of a radical, they can be simplified or expressed as decimal numbers, but never as an integer.
The chart in Figure 02 below shows all of the perfect squares and non-perfect squares up to 144. Note that the numbers highlighted in orange are all perfect squares. For example:
For quick reference, here is a list of the perfect squares up to 144:
Now that you are familiar with radicals, perfect squares, and non-perfect squares, you are ready to learn how to simplify radicals.
Now you are ready to learn how to simplify a radical using our easy 3-step radical.
Let’s start by considering a problem where you are asked to simplify c. You should notice that 16 is a perfect square, so you can easily conclude that √16 = 4 (because 4 x 4 =16).
This kind of problem is pretty easy when the number inside of the radical is a perfect square, but what happens when it is not a perfect square? For example, what if you were dealing with a non-perfect square like 12 inside of the radical? How can you simplify √12? This guide will teach you how to do just that (i.e. how to simplify a radical containing a non-perfect square).
To simplify radicals like √12, we will use the following 3-step strategy:
Does this sound confusing? The process gets easier with practice. Let’s go ahead and apply these three steps to √12 as follows:
Step One: List all of the factors of the number inside of the radical
First, we will list all of the factors of 12:
Step Two: Determine if any of the factors are perfect squares (not including 1). If there are multiple perfect squares, choose the largest one.
From the list of factors above, we can see that only 4 is a perfect square and it is the only perfect square that is a factor of 12. This is important because we will need to find at least one factor that is a perfect square in order to simplify a radical of a non-perfect square.
