7.1 Radical Expressions

1. 7.1 Radical Expressions

2. Objective 1: Find principal square roots of numbers A square root of a number a is a number c such that Examples: • 25 has a square root of 5 because • 25 has a square root of -5 because • -16 does not have a real-number square root because there is no real number c such that

3. Theorem 7-1 -Every positive real number has two real-number square roots. -The number 0 has just one square root, 0 itself. -Negative numbers do not have real-number square roots. Ex. Find the two square roots of 64. The square roots are 8 and -8.

4. Try This Find the square roots of each number. • 9 • 36 • 121 • 0 • -49

5. Definition The principal square root of a nonnegative number is its nonnegative square root. The symbol represents the principal square root of a. the negative square root of a is written . Ex. Simplify. 1. 2. 3. 4.

6. Try This Simplify. 6. 9. 7. 10. 8. 11.

7. Definition The symbol is a radical sign. An expression written with a radical sign is a radical expression. The expression written under the radical sign is the radicand.

8. Theorem 7-2 For any real number a, . The principal (nonnegative) square root of is the absolute value of a. Ex. 1. 3. 2. 4.

9. Try This 12. 13. 14. 15.

10. Objective 2: Find odd and even kth roots The number c is the cube root of a if . • 2 is the cube root of 8 because . • -5 is the cube root of -125 because Ex. Simplify. 1. 2. 3. 4.

11. Try This Simplify. 16. 17. 18.

12. Rewrite using exponential notation 1. 2. 3.

13. Try This 19. 20. 21.

14. The number k in is called the index. If k is an odd number, we say that we are finding an odd root. Examples. Find the following. 1. 2. 3. 4.

15. Try This Find the following. 22. 23. 24. 25.

16. Theorem 7-3 For any real number a, the following statements are true. A. When k is even. B. When k is odd.

17. Try This Find the following. 26. 29. 27. 30. 28. 31.