Elementary Algebra
Introduction to Algebra Linear Equations and Inequalities Functions and Graphs I Lines and thier Graphs Linear Systems Exponents & Polynomials
Intermediate Algebra
Factoring Rational Expressions Rational Equations and Applications Radical Expressions Nonlinear Equations and Applications Functions and Graphs II Exponential and Logarithmic Functions
Precalculus I / College Algebra
Equations and Inequalitites Functions and Graphs Polynomial and Rational Functions Exponential and Logarithmic Functions Systems and Matrices Geometry Basics Conic Sections Sequences and Series
Precalculus II / Trigonometry
The Six Trigonometric Functions Right Triangle Trigonometry Circular Functions Graphs of Trigonometric Functions Trigonometric Identities Trigonometric Equations Oblique Triangles and the Laws Vectors Complex, Parametric, and Polar Forms
Calculus I
Limits and Continuity Derivatives Analysis of Curves Antiderivatives
Calculus II
Transcendental Functions
Geometry Physics Integration Techniques Calculus of Infinity Parametric, Polar, and Conic Curves
Calculus III

Course: Intermediate Algebra
Subtopic: Operations III - Divide & Rationalize

Overview

Continue to expand your knowledge of radical expressions by dividing two radicals. When dividing radicals note 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.

Objectives

By the end of this topic you should know and be prepared to be tested on:

• 10.5.1 Divide two radical expressions
• 10.5.2 Rewrite a radical of a fraction as a fraction of radicals and visa versa
• 10.5.3 Rationalize the denominator of an expression to eliminate the radical in the denominator
• 10.5.4 Rationalize the denominator when the radical in the denominator is a square root, cube root, fourth root, etc.

Terminology

Terms you should be able to define: the process of rationalizing the denominator

Text Notes

"Rationalizing the denominator" is an important concept, but concentrate on those that have single terms in the denominator (so you can SKIP the "conjugates" type). Watch the index carefully -- the higher it is the more complicated the rationalization process may become. SKIP "rationalizing the numerator".