Chapter 9
Radicals and Rational Exponents
An exponent has always been a count. Three as an exponent says multiply three copies together, and four says four copies. Then somebody writes a fraction up there, and the count reading collapses: half a copy is not something anyone can multiply. Yet the notation appears on calculators, in science, in growth formulas, and it always returns an ordinary number. Something other than counting is fixing that value, and it ties these exponents to the radical sign you have used on trust for chapters. How that works is the question this chapter opens on.
What You'll Explore
5 lessons.
- Rational Exponents
Zero as an exponent, and negative numbers as exponents, cannot mean repeated multiplication, and yet both have settled values that everybody uses. Some principle other than counting decided those values. This lesson finds that principle and then asks what it forces a fractional exponent to be.
- Simplifying Radical Expressions
The equation saying x squared equals nine has two solutions, positive three and negative three. So which of them does the square root symbol stand for, and can a symbol get away with naming both? The answer decides which rules you have been using on trust actually hold, and what a root in simplest form should look like.
- Operations with Radicals
Adding the square root of two to the square root of three tempts almost everybody into writing the square root of five, and a calculator says otherwise. Some roots really do collect together, though, and some refuse. What separates the two cases is what this lesson is after, along with what happens when roots are multiplied instead.
- Solving Radical Equations
Solving an equation usually means every step preserves the answers: a value that survives to the end really is an answer. That reliability quietly fails when the unknown sits under a root. Candidates can appear that the original equation rejects flatly, and this lesson tracks down where they come from.
- Rationalizing and Radical Conjugates
Dividing by a number with a root in it feels harder than dividing by an ordinary number, and there is a standard way to move the root out of the denominator. You made the same move with complex numbers without being told its name. Here you meet the idea behind it, and the case where the familiar version fails.