1. Precalculus Essentials
Fifth Edition
Chapter P
Prerequisites: Fundamental Concepts of Algebra 1
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2. P.3 Radicals and Rational Exponents

3. Objectives Evaluate square roots.
Use the product rule to simplify square roots.
Use the quotient rule to simplify square roots.
Add and subtract square roots.
Rationalize denominators.
Evaluate and perform operations with higher roots.
Understand and use rational exponents.

4. Definition of the Principal Square Root

5. Example: Evaluating Square Roots

6. Simplifying Expressions of the Form Square Root of a Squared
In words, the principal square root of a² is the absolute value of a.

7. Example: Simplifying Expressions of the Form Square Root of a Squared

8. The Product Rule for Square Roots
If a and b represent nonnegative real numbers, then

9. Example: Using the Product Rule to Simplify Square Roots

10. The Quotient Rule for Square Roots
If a and b represent nonnegative real numbers and b ≠ 0, then .

11. Example: Using the Quotient Rule to Simplify Square Roots

12. Adding and Subtracting Square Roots
Two or more square roots can be combined using the distributive property provided that they have the same radicand. Such radicals are called like radicals.

13. Example: Adding and Subtracting Like Radicals

14. Rationalizing Denominators (1 of 2)
The process of rewriting a radical expression as an equivalent expression in which the denominator no longer contains any radicals is called rationalizing the denominator. If the denominator consists of the square root of a natural number that is not a perfect square, multiply the numerator and the denominator by the smallest number that produces the square root of a perfect square in the denominator.

15. Rationalizing Denominators (2 of 2)
Radical expressions that involve the sum and difference of the same two terms are called conjugates. Thus,
are conjugates. If the denominator contains two terms with one or more square roots, multiply the numerator and denominator by the conjugate of the denominator.

16. Example: Rationalizing Denominators
Rationalize the denominator:
Rationalize the denominator:

17. Definition of the Principal nth Root of a Real Number

18. Finding nth Roots of Perfect nth Powers

19. The Product and Quotient Rules for nth Roots
For all real numbers a and b, where the indicated roots represent real numbers, .

20. Example: Simplifying, Multiplying, and Dividing Higher Roots

21. The Definition of a to the m over n Power

22. Example: Using the definition of a to the m over n Power
Simplify:

23. Example: Simplifying Expressions with Rational Exponents
Simplify using properties of exponents: