1. Slide 7- 1 Copyright © 2012 Pearson Education, Inc.

2. 7.1 Radical Expressions, Functions, and Models ■ Square Roots and Square-Root Functions ■ Expressions of the form ■ Cube roots ■ Odd and Even nth Roots ■ Radical Functions and Models

3. Slide 7- 3 Copyright © 2012 Pearson Education, Inc. Square Roots and Square- Root Functions When a number is multiplied by itself, we say that the number is squared. Often we need to know what number was squared in order to produce some value a. If such a number can be found, we call that number a square root of a.

4. Slide 7- 4 Copyright © 2012 Pearson Education, Inc. Square Root The number c is a square root of a if c 2 = a.

5. Slide 7- 5 Copyright © 2012 Pearson Education, Inc. For example, 16 has –4 and 4 as square roots because (–4) 2 = 16 and 4 2 = 16. –9 does not have a real-number square root because there is no real number c for which c 2 = –9.

6. Slide 7- 6 Copyright © 2012 Pearson Education, Inc. Example Solution Find the two square roots of 49. The square roots are 7 and –7, because 7 2 =49 and (–7) 2 = 49.

7. Slide 7- 7 Copyright © 2012 Pearson Education, Inc. Whenever we refer to the square root of a number, we mean the nonnegative square root of that number. This is often referred to as the principal square root of the number.

8. Slide 7- 8 Copyright © 2012 Pearson Education, Inc. Principal Square Root The principal square root of a nonnegative number is its nonnegative square root. The symbol is called a radical sign and is used to indicate the principal square root of the number over which it appears.

9. Slide 7- 9 Copyright © 2012 Pearson Education, Inc. Example Solution Simplify each of the following.

10. Slide 7- 10 Copyright © 2012 Pearson Education, Inc. is also read as “the square root of a,” “root a,” or “radical a.” Any expression in which a radical sign appears is called a radical expression. The following are examples of radical expressions: The expression under the radical sign is called the radicand.

11. Slide 7- 11 Copyright © 2012 Pearson Education, Inc. Expressions of the Form It is tempting to write but the next example shows that, as a rule, this is untrue. Example

12. Slide 7- 12 Copyright © 2012 Pearson Education, Inc. Simplifying For any real number a, (The principal square root of a 2 is the absolute value of a.)

13. Slide 7- 13 Copyright © 2012 Pearson Education, Inc. Example Solution Simplify each expression. Assume that the variable can represent any real number. Since y + 3 might be negative, absolute-value notation is necessary.

14. Slide 7- 14 Copyright © 2012 Pearson Education, Inc. Solution continued b) Note that (m 6 ) 2 = m 12 and that m 6 is never negative. Thus, c) Note that (x 5 ) 2 = x 10 and that x 5 might be negative. Thus,

15. Slide 7- 15 Copyright © 2012 Pearson Education, Inc. Cube Roots We often need to know what number was cubed in order to produce a certain value. When such a number is found, we say that we have found the cube root. For example, 3 is the cube root of 27 because 3 3 =27.

16. Slide 7- 16 Copyright © 2012 Pearson Education, Inc. Cube Root The number c is a cube root of a if c 3 = a. In symbols, we write to denote the cube root of a.

17. Slide 7- 17 Copyright © 2012 Pearson Education, Inc. Example Solution Simplify Since ( – 3x)( – 3x)( – 3x) = –27x 3

18. Slide 7- 18 Copyright © 2012 Pearson Education, Inc. Odd and Even nth Roots The fourth root of a number a is the number c for which c 4 = a. We write for the nth root. The number n is called the index (plural, indices). When the index is 2, we do not write it.

19. Slide 7- 19 Copyright © 2012 Pearson Education, Inc. Example Solution Find each of the following. Since 3 5 = 243 Since (–3) 5 = – 243

20. Slide 7- 20 Copyright © 2012 Pearson Education, Inc. Example Solution Find each of the following. Since 3 4 = 81 Not a real number Use absolute-value notation since m could represent a negative number

21. Slide 7- 21 Copyright © 2012 Pearson Education, Inc. na Even Positive NegativeNot a real number Odd Positive a Negative Simplifying nth Roots

22. Slide 7- 22 Copyright © 2012 Pearson Education, Inc. Radical Functions and Models A radical function is a function that can be described by a radical expression. If a function is given by a radical expression with an odd index, the domain is the set of all real numbers. If a function is given by a radical expression with an even index, the domain is the set of replacements for which the radicand is nonnegative.

23. Slide 7- 23 Copyright © 2012 Pearson Education, Inc. Example Find the domain of the function given by the equation Check by graphing the function. Then, from the graph, estimate the range of the function. Solution The radicand of is x 2 + 4. We must have

24. Slide 7- 24 Copyright © 2012 Pearson Education, Inc. continued The domain is (–∞,∞). The range appears to be [2,∞). Domain Range