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This lessons looks at how to simplify radical expressions which are not perfect squares or cubes. I include an example of square root, cube root, and 4th root. I show a technique to simplify n-root expressions. Also I look at how to simplify variables in radical expressions.

Welcome to Your Tutor Video lessons. Today we’ll look at how to simplify radicals.

A radical is anything with a radical symbol which looks like this. It can be square root, cube root, fourth root, or any numbered root. If there is no number in the symbol then its a square root, otherwise there will be a tiny number here to show you what root it is. In this example it is cube root. These expressions need to be simplified like anything else in math.

Lets look at this example: radical 54 or square root of 54. The easiest way I have found to simplify radicals is to do a factor tree for the terms inside the radical. Now this only works if everything inside the radical is multiplied together or you only have one term.

First draw two lines out from the number or term inside the radical. And on this line we are going to write any two numbers that multiply together to give us our originial. I’m going to choose 9 and 6. We are going to repeat the process until we have nothing left except for prime numbers. 3 times 3 is equal to 9 and 3 times 2 is equal to 6. Now we are left with only prime numbers here at the bottom, 2s and 3s.

Now we want to look and see which root we are dealing with. Here it is the square root or 2nd root because there is no number in the symbol itself. Since its the second root, we are looking for groups of two of the same numbers. Here I see two-3s. When you get a group, you can write the group number outside of the radical, in this case its 3. Then we’ll go ahead and write our radical and write everything thats left over inside, multiplied together: 3 times 2. Go ahead and multiply those together: 3 radical 6, and thats our simplified expression.

You can look for shortcuts as you do your factor tree. As you get more practice with this type of problem, you will know that 9 is a perfect square and so it’s square root can automatically come out of the radical. For cube roots you are looking for perfect cubes and so on according to whatever radical you happen to be dealing with. Now its going to get harder as you get into higher radicals and you probably won’t have high powers memorized, but thats okay you can just do your factor tree when in doubt.

Lets look at how this works for a cube root. We’ll use the example cube root 135. We are still going to go the factor tree just like we did in the previous example. 5 times 27 is equal to 135. 5 is a prime number so we will just leave that alone. 27 is 3 times 9 and 9 is 3 times 3. Notice we could of had a shortcut there if you knew that 27 was a perfect cube.

This time we are going to look for groups of three because we are dealing with the cube root. Here you can see three-3s. So that means a 3 is allowed to come out of the radical. So I write 3, the radical symbol with its cube root and whatever is left over that didn’t find a group of three, which is 5. So this radical simplified is 3 cube roots of 5.

Lets look at how to do radicals with variables in them. For a coefficient we are still going to do a factor tree. 48 is 12 times 4. 12 is 3 times 4. 3 is a prime number. 4 is 2 times 2. And this 4 is 2 times 2. This time we are going to be looking for groups of four because we are dealing with the fourth root. So here, I found four-2s. So the two is going to come out and a 3 will stay inside.

Now lets look at our variables. This part is a little tricky. We are going to look at each of our exponents for our variables. We are going to take our root number and divide it into each of the exponents. The number of times that the root goes into the exponent, will be our new exponent on the outside of the radical and the remainder will be the new exponent that stays behind inside the radical.

Now we can get started on our answer. We know a 2 comes out so I’m going to write that. I’ll leave some space, draw my radical symbol with is the fourth root. I know a 3 is stuck inside. Now I can start dealing with the variables.

4 goes into 6 one time so x to the first power is on the outside. 4 goes into 6 once and there are 2 left over, so its x squared left on the inside. 4 does not go into 2 any so a y does not come out of the radical. 4 goes into 2 zero times with 2 left over so we have a y squared left on the inside. 4 goes into 8 two times so we have z squared on the outside. 4 goes into 8 twice with no remaineder so there is no z left on the inside. And this is our final answer: 2 x z squared times the fourth root of 3 x squared y squared.

Thanks for watching Your Tutor video lessons, now I know today was a little bit of a tricky topic so if you have any questions head over to the blog and leave a question at www.YourTutorOnline.com And also, if you haven’t been there yet be sure to check out the quizzes I’m going to put up after each lesson and see how much you understand. Thanks for watching, class dismissed.