Roots and Radicals

Suppose a stone falls from the edge of a cliff. The number of feet the stone has dropped after $t$ seconds can be found by multiplying $16$ times the square of $t$. But to calculate the number of seconds it would take the stone to hit the land below, we need to use a square root. In this chapter, we will introduce and apply the properties of square roots, and extend these concepts to higher-order roots and rational exponents.

Sections

• Simplify and Use Square Roots — simplifying expressions with square roots, estimating square roots, approximating them with a calculator, and simplifying variable expressions with square roots.
• Simplify Square Roots — using the Product Property and the Quotient Property to simplify square roots, including those with variables and coefficients.
• Add and Subtract Square Roots — combining like square roots, and simplifying first so that radicals can be combined.
• Multiply Square Roots — multiplying square roots, using polynomial multiplication patterns with square roots, and finding special products involving conjugates.
• Divide Square Roots — dividing square roots and rationalizing a denominator, including denominators that contain a sum or difference.
• Solve Equations with Square Roots — solving radical equations, checking for extraneous solutions, and using radical equations in applications.
• Higher Roots — simplifying, using, and combining cube roots and other higher-order roots, including expressions with variables.
• Rational Exponents — simplifying expressions with rational exponents of the forms $a^{1/n}$ and $a^{m/n}$, and using the properties of exponents with them.

This chapter overview is adapted from Elementary Algebra 2e, Chapter 9 Introduction by Lynn Marecek, MaryAnne Anthony-Smith, and Andrea Honeycutt Mathis, © OpenStax, licensed under CC BY-NC-SA 4.0. Access the original for free at openstax.org. Changes: condensed the chapter introduction and recast the chapter outline as a section list.