1. BMayer@ChabotCollege.edu MTH55_Lec-02_sec_1-6_Exponent_Rules.ppt 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot Mathematics §1.6 Exponent Properties

2. BMayer@ChabotCollege.edu MTH55_Lec-02_sec_1-6_Exponent_Rules.ppt 2 Bruce Mayer, PE Chabot College Mathematics Review §  Any QUESTIONS About §1.5 → (Word) Problem Solving  Any QUESTIONS About HomeWork §1.5 → HW-01 1.5 MTH 55

3. BMayer@ChabotCollege.edu MTH55_Lec-02_sec_1-6_Exponent_Rules.ppt 3 Bruce Mayer, PE Chabot College Mathematics Exponent PRODUCT Rule  For any number a and any positive integers m and n,  In other Words: To MULTIPLY powers with the same base, keep the base and ADD the exponents Exponent Base

4. BMayer@ChabotCollege.edu MTH55_Lec-02_sec_1-6_Exponent_Rules.ppt 4 Bruce Mayer, PE Chabot College Mathematics Quick Test of Product Rule  Test 

5. BMayer@ChabotCollege.edu MTH55_Lec-02_sec_1-6_Exponent_Rules.ppt 5 Bruce Mayer, PE Chabot College Mathematics Example  Product Rule  Multiply and simplify each of the following. (Here “simplify” means express the product as one base to a power whenever possible.) a) x 3  x 5 b) 6 2  6 7  6 3 c) (x + y) 6 (x + y) 9 d) (w 3 z 4 )(w 3 z 7 )

6. BMayer@ChabotCollege.edu MTH55_Lec-02_sec_1-6_Exponent_Rules.ppt 6 Bruce Mayer, PE Chabot College Mathematics Example  Product Rule  Solution a) x 3  x 5 = x 3+5 Adding exponents = x 8  Solution b) 6 2  6 7  6 3 = 6 2+7+3 = 6 12  Solution c) (x + y) 6 (x + y) 9 = (x + y) 6+9 = (x + y) 15  Solution d) (w 3 z 4 )(w 3 z 7 ) = w 3 z 4 w 3 z 7 = w 3 w 3 z 4 z 7 = w 6 z 11 Base is x Base is 6 Base is (x + y) TWO Bases: w & z

7. BMayer@ChabotCollege.edu MTH55_Lec-02_sec_1-6_Exponent_Rules.ppt 7 Bruce Mayer, PE Chabot College Mathematics Exponent QUOTIENT Rule  For any nonzero number a and any positive integers m & n for which m > n,  In other Words: To DIVIDE powers with the same base, SUBTRACT the exponent of the denominator from the exponent of the numerator

8. BMayer@ChabotCollege.edu MTH55_Lec-02_sec_1-6_Exponent_Rules.ppt 8 Bruce Mayer, PE Chabot College Mathematics Quick Test of Quotient Rule  Test 

9. BMayer@ChabotCollege.edu MTH55_Lec-02_sec_1-6_Exponent_Rules.ppt 9 Bruce Mayer, PE Chabot College Mathematics Example  Quotient Rule  Divide and simplify each of the following. (Here “simplify” means express the product as one base to a power whenever possible.) a)b) c)d)

10. BMayer@ChabotCollege.edu MTH55_Lec-02_sec_1-6_Exponent_Rules.ppt 10 Bruce Mayer, PE Chabot College Mathematics Example  Quotient Rule  Solution a)  Solution b)  Solution c)  Solution d) Base is x Base is 8 Base is (6y) TWO Bases: r & t

11. BMayer@ChabotCollege.edu MTH55_Lec-02_sec_1-6_Exponent_Rules.ppt 11 Bruce Mayer, PE Chabot College Mathematics The Exponent Zero  For any number a where a ≠ 0  In other Words: Any nonzero number raised to the 0 power is 1 Remember the base can be ANY Number –0.00073, 19.19, −86, 1000000, anything

12. BMayer@ChabotCollege.edu MTH55_Lec-02_sec_1-6_Exponent_Rules.ppt 12 Bruce Mayer, PE Chabot College Mathematics Example  The Exponent Zero  Simplify: a) 1245 0 b) (−3) 0 c) (4w) 0 d) (−1)8 0 e) −8 0  Solutions a)1245 0 = 1 b)(−3) 0 = 1 c)(4w) 0 = 1, for any w  0. d)(−1)8 0 = (−1)1 = −1 e)−8 0 is read “the opposite of 8 0 ” and is equivalent to (−1)8 0 : −8 0 = (−1)8 0 = (−1)1 = −1

13. BMayer@ChabotCollege.edu MTH55_Lec-02_sec_1-6_Exponent_Rules.ppt 13 Bruce Mayer, PE Chabot College Mathematics The POWER Rule  For any number a and any whole numbers m and n  In other Words: To RAISE a POWER to a POWER, MULTIPLY the exponents and leave the base unchanged

14. BMayer@ChabotCollege.edu MTH55_Lec-02_sec_1-6_Exponent_Rules.ppt 14 Bruce Mayer, PE Chabot College Mathematics Quick Test of Power Rule  Test 

15. BMayer@ChabotCollege.edu MTH55_Lec-02_sec_1-6_Exponent_Rules.ppt 15 Bruce Mayer, PE Chabot College Mathematics Example  Power Rule  Simplify: a) (x 3 ) 4 b) (4 2 ) 8  Solution a) (x 3 ) 4 = x 3  4 = x 12  Solution b) (4 2 ) 8 = 4 2  8 = 4 16 Base is x Base is 4

16. BMayer@ChabotCollege.edu MTH55_Lec-02_sec_1-6_Exponent_Rules.ppt 16 Bruce Mayer, PE Chabot College Mathematics Raising a Product to a Power  For any numbers a and b and any whole number n,  In other Words: To RAISE A PRODUCT to a POWER, RAISE Each Factor to that POWER

17. BMayer@ChabotCollege.edu MTH55_Lec-02_sec_1-6_Exponent_Rules.ppt 17 Bruce Mayer, PE Chabot College Mathematics Quick Test of Product to Power  Test 

18. BMayer@ChabotCollege.edu MTH55_Lec-02_sec_1-6_Exponent_Rules.ppt 18 Bruce Mayer, PE Chabot College Mathematics Example  Product to Power  Simplify: a) (3x) 4 b) (−2x 3 ) 2 c) (a 2 b 3 ) 7 (a 4 b 5 )  Solutions a)(3x) 4 = 3 4 x 4 = 81x 4 b)(−2x 3 ) 2 = (−2) 2 (x 3 ) 2 = (−1) 2 (2) 2 (x 3 ) 2 = 4x 6 c)(a 2 b 3 ) 7 (a 4 b 5 ) = (a 2 ) 7 (b 3 ) 7 a 4 b 5 = a 14 b 21 a 4 b 5 Multiplying exponents = a 18 b 26 Adding exponents

19. BMayer@ChabotCollege.edu MTH55_Lec-02_sec_1-6_Exponent_Rules.ppt 19 Bruce Mayer, PE Chabot College Mathematics Raising a Quotient to a Power  For any real numbers a and b, b ≠ 0, and any whole number n  In other Words: To Raise a Quotient to a power, raise BOTH the numerator & denominator to the power

20. BMayer@ChabotCollege.edu MTH55_Lec-02_sec_1-6_Exponent_Rules.ppt 20 Bruce Mayer, PE Chabot College Mathematics Quick Test of Quotient to Power  Test 

21. BMayer@ChabotCollege.edu MTH55_Lec-02_sec_1-6_Exponent_Rules.ppt 21 Bruce Mayer, PE Chabot College Mathematics Example  Quotient to a Power  Simplify: a) b) c)  Solution a)  Solution b)  Solution c)

22. BMayer@ChabotCollege.edu MTH55_Lec-02_sec_1-6_Exponent_Rules.ppt 22 Bruce Mayer, PE Chabot College Mathematics Negative Exponents  Integers as Negative Exponents

23. BMayer@ChabotCollege.edu MTH55_Lec-02_sec_1-6_Exponent_Rules.ppt 23 Bruce Mayer, PE Chabot College Mathematics Negative Exponents  For any real number a that is nonzero and any integer n  The numbers a −n and a n are thus RECIPROCALS of each other

24. BMayer@ChabotCollege.edu MTH55_Lec-02_sec_1-6_Exponent_Rules.ppt 24 Bruce Mayer, PE Chabot College Mathematics Example  Negative Exponents  Express using POSITIVE exponents, and, if possible, simplify. a) m –5 b) 5 –2 c) (−4) −2 d) xy –1  SOLUTION a) m –5 = b) 5 –2 =

25. BMayer@ChabotCollege.edu MTH55_Lec-02_sec_1-6_Exponent_Rules.ppt 25 Bruce Mayer, PE Chabot College Mathematics Example  Negative Exponents  Express using POSITIVE exponents, and, if possible, simplify. a) m –5 b) 5 –2 c) (−4) −2 d) xy −1  SOLUTION c) (−4) −2 = d) xy –1 = Remember PEMDAS

26. BMayer@ChabotCollege.edu MTH55_Lec-02_sec_1-6_Exponent_Rules.ppt 26 Bruce Mayer, PE Chabot College Mathematics More Examples  Simplify. Do NOT use NEGATIVE exponents in the answer. a) b) (x  4 )  3 c) (3a 2 b  4 ) 3 d)e) f)  Solution a)

27. BMayer@ChabotCollege.edu MTH55_Lec-02_sec_1-6_Exponent_Rules.ppt 27 Bruce Mayer, PE Chabot College Mathematics More Examples  Solution b) (x −4 ) −3 = x (−4)(−3) = x 12 c) (3a 2 b −4 ) 3 = 3 3 (a 2 ) 3 (b −4 ) 3 = 27 a 6 b −12 = d) e) f)

28. BMayer@ChabotCollege.edu MTH55_Lec-02_sec_1-6_Exponent_Rules.ppt 28 Bruce Mayer, PE Chabot College Mathematics Factors & Negative Exponents  For any nonzero real numbers a and b and any integers m and n  A factor can be moved to the other side of the fraction bar if the sign of the exponent is changed

29. BMayer@ChabotCollege.edu MTH55_Lec-02_sec_1-6_Exponent_Rules.ppt 29 Bruce Mayer, PE Chabot College Mathematics Examples  Flippers  Simplify  SOLUTION  We can move the negative factors to the other side of the fraction bar if we change the sign of each exponent.

30. BMayer@ChabotCollege.edu MTH55_Lec-02_sec_1-6_Exponent_Rules.ppt 30 Bruce Mayer, PE Chabot College Mathematics Reciprocals & Negative Exponents  For any nonzero real numbers a and b and any integer n  Any base to a power is equal to the reciprocal of the base raised to the opposite power

31. BMayer@ChabotCollege.edu MTH55_Lec-02_sec_1-6_Exponent_Rules.ppt 31 Bruce Mayer, PE Chabot College Mathematics Examples  Flippers  Simplify  SOLUTION

32. BMayer@ChabotCollege.edu MTH55_Lec-02_sec_1-6_Exponent_Rules.ppt 32 Bruce Mayer, PE Chabot College Mathematics Summary – Exponent Properties 1 as an exponenta 1 = a 0 as an exponenta 0 = 1 Negative Exponents (flippers) The Product Rule The Quotient Rule The Power Rule(a m ) n = a mn The Product to a Power Rule (ab) n = a n b n The Quotient to a Power Rule This summary assumes that no denominators are 0 and that 00 00 is not considered. For any integers m and n

33. BMayer@ChabotCollege.edu MTH55_Lec-02_sec_1-6_Exponent_Rules.ppt 33 Bruce Mayer, PE Chabot College Mathematics WhiteBoard Work  Problems From §1.6 Exercise Set 14, 24, 52, 70, 84, 92, 112, 130  Base & Exponent → Which is Which?

34. BMayer@ChabotCollege.edu MTH55_Lec-02_sec_1-6_Exponent_Rules.ppt 34 Bruce Mayer, PE Chabot College Mathematics All Done for Today Astronomical Unit (AU)

35. BMayer@ChabotCollege.edu MTH55_Lec-02_sec_1-6_Exponent_Rules.ppt 35 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot Mathematics Appendix –