Day 19 – Square roots and radical expressions

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

Videos from Khan Academy.

We introduce square roots and radical expressions. Keep in mind that the exposition in the text book is with roots in general, in other words cube roots, fourth roots, etc. While we will not spend much time in class with these other roots, you will see these in 1275, the next course in the sequence. Hence, it will do no harm to familiarize yourself with them now, especially if you are already familiar with square roots.

What is a square root?

For example, if a is 49, then the square roots (b) are 7 and -7, because


Now normally, when someone says “the square root”, they are referring to the nonnegative or prinicipal square root:

The part under the square root sign is the radicand. To simplify square roots, you must use the muliplication property of radicals [√(ab)=√a√b] and  extract as factors, either all at once or in stages, perfect squares from the radicand.

The book has provided a definition for when a radical is simplified:

Since we are  only  concerned with square roots, we can simplify condition 1 and get:

DEFINITION Simplified Form of a Radical

Consider any radical expression where the radicand is written as a product of prime factors. The expression is in simplified form if all the following conditions are met:

  1. The radicand has no repeated factor
  2. The radicand does not contain a fraction.
  3. There are no radicals in the denominator of a fraction.

For the purpose of simplifying radicals, it is a good idea to memorize the perfect squares up to 15 squared:

Here is a video with many examples just with numbers (no variables):

Finally, here is an example with variables:


10 Responses to Day 19 – Square roots and radical expressions

  1. Todays class was refreshed most of the information that i previously learned

  2. Mishii says:

    Pro Reitz,
    today i refreshed my memory on stuff from high school. the video’s helped too.

  3. lalayudi says:

    the radicals are getting harder and harder.

  4. sbailey says:

    class about square roots was a recap for me but i did learn different methods other than mine.

  5. This lesson was intresting it was a good review of the subject radicals which i learned in highschool so when i got home i did the homework on radicals and just added the knowledge from class and the videos and i was able to finish the homework.

  6. awiltshire says:

    okay so I got it but I’m about to watch this video and see if it helps me understand it more

  7. Professor Halleck
    Rebecca Kogan

    It’s difficult remembering certain aspects of math when I haven’t taken any math since my third year of high school but after doing problems in the textbook (recommended hw) i understand everything much better. The videos on here help a lot as well.

  8. Hibba says:

    Today’s lesson was easy. I just had problem with solving radicals with variables., but after watching this video I am satisfied.

  9. Hibba says:

    Today’s lesson was easy. I just had problem solving radicals with variables, but after watching this video I am satisfied.

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