Amy has a master's degree in secondary education and has been teaching math for over 9 years. Amy has worked with students at all levels from those with special needs to those that are gifted.
If you see an equation like the square root of (x + 2) = 0, you can say, 'That's so radical!' Do you see the symbol that the square root uses? In math, we call that the radical symbol. Besides the square root, we can also have third roots, fourth roots, and so on.
What do these mean? While the square root tells you to look for something that if you multiply it together twice you will get the number inside the square root symbol, a third root tells you to look for something that if you multiply it together three times, you will get the number inside the radical. Can you guess how the fourth root works? Yes, a fourth root is a number where if you multiply it together four times, you will get your number inside the radical.
So, how do these look? Well, you know what the square root symbol looks like. The third root and the fourth root also use the same symbol except that now you will see a little number to the upper left of the symbol like this.
Do you see the little number right where the symbol makes a dip? That number tells you how many times to multiply your answer to get the number inside the radical. If you don't see a number, then you are looking at a square root that has an invisible little two there.
How to Remove Radicals
How would you solve something like this? What do you think? We would want to remove the radical somehow so that we could get the x so we can solve for it. So how do we do that? Let me show you.
What we need to do first is to look for the little number if it is there. If there is no number, then the number is 2. It is this little number that holds the key to removing the radical. If you think of the radical symbol as a lock, then this little number tells you the password that will unlock it.
Okay, so how do we unlock it? We unlock it by multiplying the side of the equation with the radical by itself as many times as that little number tells us. In other words, we would take that side of the equation to the power that the little number tells us to.
For a square root, we square it; for a third root, we cube it; for a fourth root, we take it to the fourth power, and so on and so forth. We also remember that if we do one operation on one side of the equation, we have to do it to the other side as well.
Also, if you have an equation such as sqrt(x + 2) - 1 = 0, you would want to move the -1 over to the other side of the equation first so that your radical is by itself before you unlock the radical. Let's take a look at a couple of examples.
Change to a Linear Equation
First, if inside our radical, we have an x, we will end up with a linear equation after removing the radical. Let's solve this equation.
We see that our radical is not by itself. We need to move the 4 over to the other side first. We go ahead and subtract 4 from both sides. Now we can use the little number to help us unlock our radical. We see a little 3, so we need to take both sides of our equation to the third power. This is what we get.
Now we have a linear equation, which we know very well how to solve. We see that we have one final step left which is to move the 5 over to the other side. Doing this, we get an answer of -69.
Change to a Quadratic Equation
If we have an x squared inside our radical, then we will end up with a quadratic equation after removing the radical. This means that once we remove the radical, we will use what we know about quadratic equations to help us finish solving. Let's take a look by solving this problem.
We see our radical by itself on its side of the equation, so we can go ahead and unlock it. We look for our password. We don't see a number, so that means our password is 2. A 2 means we will square both sides of our equation to remove the radical.
Doing that, we end up with x^2 - 1 = 0. Now we can move the -1 over to get x^2 = 1. Next we take the square root of our x^2 to get x by itself. So our answer is -1 and 1.
Now, let's recap. We learned that the radical symbol is the symbol that is part of the square root. But we can also have third roots, fourth roots, etc. These are all radicals where the type is determined by the little number located where the symbol makes a dip.
Before we can solve our equation, we need to remove the radical. We do this by looking to the little number. To remove the radical, we first make sure it is by itself on its side of the equation. Then we take both sides of our equation to the power of the little number. We then finish solving by using what we know of linear and quadratic equations.
By the end of this lesson, you should be able to identify a radical and remove it to create a linear or quadratic equation.
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