1. BMayer@ChabotCollege.edu MTH55_Lec-39_sec_7-2a_Rational_Exponents.ppt 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot Mathematics §7.2 Radical Functions

2. BMayer@ChabotCollege.edu MTH55_Lec-39_sec_7-2a_Rational_Exponents.ppt 2 Bruce Mayer, PE Chabot College Mathematics Review §  Any QUESTIONS About §7.1 → Cube & n th Roots  Any QUESTIONS About HomeWork §7.1 → HW-30 7.1 MTH 55

3. BMayer@ChabotCollege.edu MTH55_Lec-39_sec_7-2a_Rational_Exponents.ppt 3 Bruce Mayer, PE Chabot College Mathematics 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 c 3 = a, then

4. BMayer@ChabotCollege.edu MTH55_Lec-39_sec_7-2a_Rational_Exponents.ppt 4 Bruce Mayer, PE Chabot College Mathematics 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 c) As 3 3 = 27 and 4 3 = 64, so (3/4) 3 = 27/64

5. BMayer@ChabotCollege.edu MTH55_Lec-39_sec_7-2a_Rational_Exponents.ppt 5 Bruce Mayer, PE Chabot College Mathematics Rational Exponents  Consider a 1/2 a 1/2. If we still want to add exponents when multiplying, it must follow from the Exponent PRODUCT RULE that a 1/2 a 1/2 = a 1/2 + 1/2, or a 1  Recall  [SomeThing]·[SomeThing] = [SomeThing] 2  This suggests that a 1/2 is a square root of a.

6. BMayer@ChabotCollege.edu MTH55_Lec-39_sec_7-2a_Rational_Exponents.ppt 6 Bruce Mayer, PE Chabot College Mathematics Definition of a 1/n  When a is NONnegative, n can be any natural number greater than 1. When a is negative, n must be odd.  Note that the denominator of the exponent becomes the index and the BASE becomes the RADICAND.

7. BMayer@ChabotCollege.edu MTH55_Lec-39_sec_7-2a_Rational_Exponents.ppt 7 Bruce Mayer, PE Chabot College Mathematics n th Roots  n th root: The number c is an n th root of a number a if c n = a.  The fourth root of a number a is the number c for which c 4 = a. We write for the n th root. The number n is called the index (plural, indices). When the index is 2 (for a Square Root), the Index is omitted.

8. BMayer@ChabotCollege.edu MTH55_Lec-39_sec_7-2a_Rational_Exponents.ppt 8 Bruce Mayer, PE Chabot College Mathematics Evaluating a 1/n  Evaluate Each Expression (a) 27 1/3 27 3 = 3 (b) 64 1/2 64 = 8 = = (c) –625 1/4 625 4 = –5= – (d) (–625) 1/4 –625 4 is not a real number because the radicand, –625, is negative and the index is even. =

9. BMayer@ChabotCollege.edu MTH55_Lec-39_sec_7-2a_Rational_Exponents.ppt 9 Bruce Mayer, PE Chabot College Mathematics Caveat on Roots  CAUTION  CAUTION: Notice the difference between parts (c) and (d) in the last Example.  The radical in part (c) is the negative fourth root of a positive number, while the radical in part (d) is the principal fourth root of a negative number, which is NOT a real no. (c) –625 1/4 625 4 = –5= – (d) (–625) 1/4 –625 4 is not a real number because the radicand, –625, is negative and the index is even. =

10. BMayer@ChabotCollege.edu MTH55_Lec-39_sec_7-2a_Rational_Exponents.ppt 10 Bruce Mayer, PE Chabot College Mathematics Radical Functions  Given PolyNomial, P, a RADICAL FUNCTION Takes this form:  Example  Given f(x) = Then find f(3).  SOLUTION  To find f(3), substitute 3 for x and simplify.

11. BMayer@ChabotCollege.edu MTH55_Lec-39_sec_7-2a_Rational_Exponents.ppt 11 Bruce Mayer, PE Chabot College Mathematics Example  Exponent to Radical  Write an equivalent expression using RADICAL notation a)b)c)  SOLUTION a) b) c)

12. BMayer@ChabotCollege.edu MTH55_Lec-39_sec_7-2a_Rational_Exponents.ppt 12 Bruce Mayer, PE Chabot College Mathematics Example  Radical to Exponent  Write an equivalent expression using EXPONENT notation a)b)  SOLUTION a)b)

13. BMayer@ChabotCollege.edu MTH55_Lec-39_sec_7-2a_Rational_Exponents.ppt 13 Bruce Mayer, PE Chabot College Mathematics Exponent ↔ Index Base ↔ Radicand  From the Previous Examples Notice: The denominator of the exponent becomes the index. The base becomes the radicand. The index becomes the denominator of the exponent. The radicand becomes the base.

14. BMayer@ChabotCollege.edu MTH55_Lec-39_sec_7-2a_Rational_Exponents.ppt 14 Bruce Mayer, PE Chabot College Mathematics Definition of a m/n  For any natural numbers m and n (n not 1) and any real number a for which the radical exists,

15. BMayer@ChabotCollege.edu MTH55_Lec-39_sec_7-2a_Rational_Exponents.ppt 15 Bruce Mayer, PE Chabot College Mathematics Example  a m/n Radicals  Rewrite as radicals, then simplify a. 27 2/3 b. 243 3/4 c. 9 5/2  SOLUTION a. b. c.

16. BMayer@ChabotCollege.edu MTH55_Lec-39_sec_7-2a_Rational_Exponents.ppt 16 Bruce Mayer, PE Chabot College Mathematics Example  a m/n Exponents  Rewrite with rational exponents  SOLUTION

17. BMayer@ChabotCollege.edu MTH55_Lec-39_sec_7-2a_Rational_Exponents.ppt 17 Bruce Mayer, PE Chabot College Mathematics Definition of a −m/n  For any rational number m/n and any positive real number a the NEGATIVE rational exponent:  That is, a m/n and a −m/n are reciprocals

18. BMayer@ChabotCollege.edu MTH55_Lec-39_sec_7-2a_Rational_Exponents.ppt 18 Bruce Mayer, PE Chabot College Mathematics Caveat on Negative Exponents  A negative exponent does not indicate that the expression in which it appears is negative; i.e.;

19. BMayer@ChabotCollege.edu MTH55_Lec-39_sec_7-2a_Rational_Exponents.ppt 19 Bruce Mayer, PE Chabot College Mathematics Example  Negative Exponents  Rewrite with positive exponents, & simplify a. 8 −2/3 b. 9 −3/2 x 1/5 c.  SOLUTION

20. BMayer@ChabotCollege.edu MTH55_Lec-39_sec_7-2a_Rational_Exponents.ppt 20 Bruce Mayer, PE Chabot College Mathematics Example  Speed of Sound  Many applications translate to radical equations.  For example, at a temperature of t degrees Fahrenheit, sound travels S feet per second According to the Formula

21. BMayer@ChabotCollege.edu MTH55_Lec-39_sec_7-2a_Rational_Exponents.ppt 21 Bruce Mayer, PE Chabot College Mathematics Example  Speed of Sound  During orchestra practice, the temperature of a room was 74 °F. How fast was the sound of the orchestra traveling through the room?  SOLUTION: Substitute 74 for t in the Formula and find an approximation using a calculator.

22. BMayer@ChabotCollege.edu MTH55_Lec-39_sec_7-2a_Rational_Exponents.ppt 22 Bruce Mayer, PE Chabot College Mathematics WhiteBoard Work  Problems From §7.2 Exercise Set 4, 10, 18, 32, 48, 54, 130  The MACH No. M

23. BMayer@ChabotCollege.edu MTH55_Lec-39_sec_7-2a_Rational_Exponents.ppt 23 Bruce Mayer, PE Chabot College Mathematics All Done for Today Ernst Mach Fluid Dynamicist  Born 8Feb1838 in Brno, Austria  Died 19Feb1916 (aged 78) in Munich, Germany

24. BMayer@ChabotCollege.edu MTH55_Lec-39_sec_7-2a_Rational_Exponents.ppt 24 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot Mathematics Appendix –

25. BMayer@ChabotCollege.edu MTH55_Lec-39_sec_7-2a_Rational_Exponents.ppt 25 Bruce Mayer, PE Chabot College Mathematics

26. BMayer@ChabotCollege.edu MTH55_Lec-39_sec_7-2a_Rational_Exponents.ppt 26 Bruce Mayer, PE Chabot College Mathematics Graph y = |x|  Make T-table

27. BMayer@ChabotCollege.edu MTH55_Lec-39_sec_7-2a_Rational_Exponents.ppt 27 Bruce Mayer, PE Chabot College Mathematics