Lesson 138 – Reducing Higher Order Roots


Today your child continues working with radicals and learning to reduce them.

As a bit of preparatory work today, it will probably help your child for you to sit down with him and review two of the rules of powers that were summarized in lesson 137.

First review the concept of



x^1/2 is the same as the square root of x

Write the the two concepts down on a piece of paper for your child and explain the concepts, or review then as you write. Then show your child that you can use these two concepts to get to the concept he will need today.

First: if x^1/2 is the same as the square root of x, then x^1/3 is the same as the cube root of x and x^4 is the same as the fourth root of x.

Then you can reduce a root order using the first concept. This means that if you have the fourth root of x, then that is the same as x^1/4 which is the same as x^1/2*x^1/2 because 1/2*1/2=1/4, or you can think of it as square root of the square root is the fourth root.

The examples on the worksheet also need your child to understand that the third root of the third root is the 9th root (cause 3*3=9, or thinking of it differently 1/3*1/3=1/9) and the third root of the second root is the sixth root (cause 3*2=6, or 1/3*1/2=1/6).

Please feel free to comment below if you have any questions. If your child doesn’t really understand I can help you figure out a different way of explaining the concept.

I always approach homeschooling my own kids in terms of finding the right explanation or method for them to understand a subject. I tell them that they are smart and able and if they have trouble with fundamentally understanding a subject, we just have to work together to find a way of explaining things so that they understand. It is not at all related to how smart they are, only dependent on how they learn. This is one of the real benefits of homeschooling – your child doesn’t ever have to feel like he is not smart. If he works at it, he will absolutely be able to learn anything and if a subject is hard it is most likely due to a mismatch between how he learns and how the material is presented to him. In that regard, the best thing I ever did was to use a learning style test on both my kids and on me when we began homeschooling. This gave me insight into my kids, who are dramatically different in spite of being twins. I also took the test so that I could see the difference between how I learn and how they do individually. This was very, very useful as I can translate material for each of them from the form I would most naturally understand into a form that suits each of them best. I highly recommend you do something like this with your own children.

Download Lesson 138 of Doodles Do Algebra HERE


1. The sixth root of 4a^2 is the same as the sixth root of 4 times the sixth root of a^2. The sixth root of 4 is the same as the cube root of the square root of 4 (I am sorry I can’t write this out right now in math symbols – I am working on getting the write scripting language to be able to display these kinds of answers on the web site, just bear with me for now please) and that is the cube root of 2 (cause the square root of 4 is 2. Then the sixth root of a^2 is the cube root of the square root of a^2 which is the cube root of a. So the answer is the cube root of 2 times the cube root of a, or the cube root of 2a.

2. Here you go throught the same process, only there is one more component in the problem. The answer is the cube root of [math]