Day 21 – Multiplication and Division of radical expressions

TEXT: Intermediate Algebra by Miller, O’Neill & Hyde

Videos from a variety of sources, including Khan Academy.

Recall the multiplication property presented in Day 19:

muliplication property of radicals : √(ab)=√a√b

To multiply two radical expressions:

  1. multiply the nonradical parts
  2. “multiply” the radical parts
  3. simplify the new radical
  4. multiply the newly “liberated” part (the root just extracted) with the result from part 1

“multiply” is in quotes. Since you will be simplifying the new radical, you may want to leave the product as factors. Below is a video multiplying 2 monomials with just numbers, presented as an alternative to our procedure. If we follow our procedure with the example from the video, the steps are as follows:

  1. (2√3) (3√6)=6√3 √6
  2. =6√(3·6)
  3. 6√(3·3) √2=6·3 √2
  4. 18 √2

The following video has several examples, the first one being a monomial times monomial, but this time with variables. The other 3 examples are of products of 2 binomials and the presenter uses FOIL:

Here is a more complicated example of the same type:

Division of radical expressions is equivalent to “rationalizing the denominator“. If the denominator is a monomial

  1. Simplify the radical(s)
  2. Multiply numerator and denominator by the radical in the denominator
  3. Simplify

If the denominator is a binomial, you will need to use an object known as the conjugate. The conjugate of a binomial is the same binomial but with the sign of the second term changed. In symbols, ab + cd and abcd are conjugates. One of the radicals b or √d may missing.

To divide by a binomial radical expression:

  1. Simplify the radical(s)
  2. Multiply numerator and denominator by the conjugate of the denominator
  3. Simplify

The following videos have explanation of why the technique works and a large set of examples:

There is a video which you may find amusing about the tyranny of “rationalizing the denominator” (Prof Halleck notes that he disagrees with the point of view of the presenter):

7 Responses to Day 21 – Multiplication and Division of radical expressions

  1. awiltshire says:

    Pro. Halleck,
    This can be very confusing if you don’t take your time and pay attention, while practicing in class, I found myself making simple mistakes, with a little more practice I will have it, Also we didn’t get to do enough division so I hope next class in the other Professor, Pro. Reitz teaches that to us for a few minutes

  2. sbailey says:

    Prof. Halleck
    Today was a little difficult because I was semi confused about how to attack the problems but once I got the hand it became much easier

  3. Mishii says:

    Pro, Reitz
    its was a really good experience, getting the chance to see how Pro, Halleck teaches. but I’m still a bit confused with today’s lesson. i would like it if you can do some more problem in class in general.

  4. Rebecca Kogan
    Professor Halleck

    I found today’s class beneficial. The steps you posted on the board referring to conjugations helped me memorize how to do the problem on my own without looking at the steps

  5. brina92 says:

    It was cool to meet professor reitz, and experience his way of teaching. He takes his time step by step, very clear as if your in 5th grade, so there is no mistake you get it. His teaching style suits me. Wish prof halleck could take his time to be that clear and try to come down to our level to explain things

  6. Zytasia Gaines says:

    Professor Reitz
    Todays class was very interesting i got to meet Mr Halleck and see some new material taught in a different way the class was fairly empty i dont know if it was because it was group day or not but, he had a chance to come to each student individually and help them with what they dont understand so that was cool. the videos on this subject are also very heplful but for me i realized this particular topic is something that has to be praticed alot more it is so easy to forget the steps.

  7. Hibba says:

    Yesterday we gather in Professor Reitz’s class for our group project discussion. For the first hour we studied multiplication and division of radical expressions and for the second hour we got together with our group members for group project discussion.
    Multiplication and Division of radical expressions was little confusing, because we could not spend enough time on it, after watching videos I understand how to multiply radical expressions and how to divide radical expression.

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