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?
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:
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:
- The radicand has no repeated factor
- The radicand does not contain a fraction.
- 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: