Topic: Radical Expressions
Subtopic: Operations II - Add, Subtract, & Divide
Overview
Continue to expand your knowledge of radical expressions by adding like radicals, subtracting like radicals, and dividing two radicals. When dividing radicals remember that you may not leave a radical in the denominator of a fraction nor a fraction in the radicand. In either of these circumstances you must rationalize the denominator to eliminate the radical in the denominator.
Rule: When taking a root of a fraction, you can split it up into the root of the numerator over the root of the denominator. It's a good idea to reduce the fraction first though!
Rule: When dividing roots that have the same index, you can rewrite them as one big root with the radicands divided as one fraction underneath, and then simplify completely.
So sometimes you want to take the radical of a fraction and write it as a fraction of radicals and sometimes you want to take the fraction of radicals and write it as a radical of a fraction!
Rule: Never leave a fraction under a radical nor a radical in the denominator of a fraction. It is illegal! You must rationalize the denominator. This is an important process. Concentrate on rationalizing the denominator when the fraction has a square root in the bottom, but try some with higher roots too.
Rule: You can only add or subtract "like" radical expressions, meaning that they must have the same index and identical radicands.
Objectives
By the end of this topic you should know and be prepared to be tested on:
- 10.4.1 Recognize "like" radicals
- 10.4.2 Know that it is only "like" radicals that can be added or subtracted
- 10.4.3 Add or subtract radical expressions simplifying as needed
- 10.4.4 Divide two radical expressions
- 10.4.5 Rewrite a radical of a fraction as a fraction of radicals and visa versa
- 10.4.6 Rationalize the denominator of an expression to eliminate the radical in the denominator
- 10.4.7 Rationalize the denominator when the radical in the denominator is a square root, cube root, fourth root, etc.
Terminology
Define: "like" radicals, the process of rationalizing the denominator
Text Notes
Text:
Intro & Inter Algebra for CS 3ed by Blitzer, sect. 10.4
- 10.3-10.5 contain many important rules and processes. Spend ample time working through these to get them down well!
- This text is unusual in that it covers rationalizing the denominator in section 10.5 not in the section where dividing radicals is covered, so check those lecture notes for text notes regarding rationalizing.