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Course: Algebra III / Intermediate Algebra
Topic: Radical Expressions
Subtopic: Fractional Exponents

Overview

You already know about positive whole number powers like x3 and negative integer powers like x-2 (recall x-2 means 1/x2). In this lesson we learn about fractional powers like x½ and decimal powers like x1.2. Variables to fractional powers can be rewritten as radicals. Working with fractional powers is actually more efficient than radicals. In fact some radical expressions cannot even be simplified without using fractional exponents.

To convert from fractional exponents to radicals the denominator of the fraction in the power becomes the index of the radical. The numerator of the fraction remains as the power on the variable. For example, x2/3=cubert(x2) and x1/2=sqrt(x).

You will find that sometimes expressions are written using radicals, sometimes using fractional exponents. All the rules of exponents that you learned with integer powers hold for fractional powers. E.g., the power rule of exponents says that (x2/3)1/2=x2/3·1/2=x1/3. Takes practice, so try a good variety of problems!

Objectives

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

• 10.2.1 Convert fractional exponent expressions radicals
• 10.2.2 Evaluate fractional exponent expressions (algebraically for exact answer, electronically for approximation)
• 10.2.3 Extend rules of exponents (product rule, quotient rule, power rule) to fractional exponents
• 10.2.4 Simplify fractional exponent expressions including those requiring FOIL
• 10.2.5 Perform operations on fractional exponent expressions that do not contain variables (e.g., 91/3*31/2)
• 10.2.6 Simplify radicals via fractional exponents (e.g., cubert(9)*sqrt(3))

Terminology

Define: fractional exponent (a.k.a. rational exponent), fractional exponent expression

Text Notes

The "laws/properties of fractional/rational exponents" are important not just to memorize but to know how and when to use each!