Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege

1. Chabot Mathematics §7.1 Cube& nth Roots Bruce Mayer, PE Licensed Electrical & Mechanical EngineerBMayer@ChabotCollege.edu

2. MTH 55 7.1 Review § • Any QUESTIONS About • §7.1 → Square-Roots and Radical Expessions • Any QUESTIONS About HomeWork • §7.1 → HW-30

3. Cube Root • The CUBE root, c, of a Number a is written as: • The number c is the cube root of a, if the third power of c is a; that is; if c3 = a, then

4. Example  Cube Root of No.s • Find Cube Roots a) b) c) • SOLUTION • a) As 0.2·0.2·0.2 = 0.008 • b) As (−13)(−13)(−13) = −2197 As 33 = 27 and 43 = 64,so (3/4)3 = 27/64 • c)

5. Cube Root Functions • Since EVERY Real Number has a Cube Root Define the Cube Root Function: • The GraphReveals • Domain ={all Real numbers} • Range ={all Real numbers}

6. Evaluate Cube Root Functions • Evaluate Cube Root Functions • SOLUTION (using calculator)

7. Simplify Cube Roots • For any Real Number, a • Use this property to simplify Cube Root Expressions. • For EXAMPLE  Simplify • SOLUTION because (–3x)(–3x)(–3x) = –27x3

8. nth Roots • nth root: The number c is an nth root of a number a if cn = a. • The fourth root of a number a is the number c for which c4 = a. We write for the nth root. The number n is called the index (plural, indices). When the index is 2 (for a Square Root), the Index is ommitted.

9. Odd & Even nth Roots → • When the index number, n, is ODD the root itself is also called ODD • A Cube-Root (n = 3) is Odd. Other Odd roots share the properties of Cube-Roots • the most important property of ODD roots is that we can take the ODD-Root of any Real Number – positive or NEGATIVE • Domain of Odd Roots = (−, +) • Range of Odd Roots =(−, +)

10. Example  nth Roots of No.s • Find ODD Roots a) b) c) • SOLUTION • a) Since 35 = 243 • b) As (−3)(−3)(−3)(−3)(−3) = −243 When the index equals the exponent under the radical we recover the Base • c)

11. Odd & Even nth Roots → • When the index number, n, is EVEN the root itself is also called EVEN • A Sq-Root (n = 2) is Even. Other Even roots share the properties of Sq-Roots • The most important property of EVEN roots is that we canNOT take the EVEN-Root of a NEGATIVE number. • Domain of Even Roots = {x|x ≥ 0} • Range of Even Roots = {y|y ≥ 0}

12. Example  nth Roots of No.s • Find EVEN Roots a) b) c) • SOLUTION • a) Since 34 = 81 • b) Even Root is Not a Real No. Use absolute-value notation since m could represent a negative number • c)

13. Simplifying nth Roots

14. Example  Radical Expressions • Find nth Roots a) b) c) • SOLUTION • a) • b) • c)

15. WhiteBoard Work • Problems From §7.1 Exercise Set • 50, 74, 84, 88, 98, 102 • Principalnth Root

16. All Done for Today SkidMarkAnalysis Skid Distances

17. Chabot Mathematics Appendix Bruce Mayer, PE Licensed Electrical & Mechanical EngineerBMayer@ChabotCollege.edu –

18. Graph y = |x| • Make T-table