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

Chabot Mathematics. §1.6 Exponent Properties. Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu. MTH 55. 1.5. Review §. Any QUESTIONS About §1.5 → (Word) Problem Solving Any QUESTIONS About HomeWork §1.5 → HW-01. Exponent PRODUCT Rule.

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

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  1. Chabot Mathematics §1.6 ExponentProperties Bruce Mayer, PE Licensed Electrical & Mechanical EngineerBMayer@ChabotCollege.edu

  2. MTH 55 1.5 Review § • Any QUESTIONS About • §1.5 → (Word) Problem Solving • Any QUESTIONS About HomeWork • §1.5 → HW-01

  3. Exponent PRODUCT Rule • For any number a and any positive integers m and n, Exponent Base • In other Words: To MULTIPLY powers with the same base, keep the base and ADD the exponents

  4. Quick Test of Product Rule • Test 

  5. 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) x3x5 b) 62 67  63 c) (x + y)6(x + y)9 d) (w3z4)(w3z7)

  6. Example  Product Rule • Solution a) x3x5 = x3+5Adding exponents = x8 • Solution b) 62 67  63 = 62+7+3 = 612 • Solution c) (x + y)6(x + y)9 = (x + y)6+9 = (x + y)15 • Solution d) (w3z4)(w3z7) = w3z4w3z7 = w3w3z4z7 = w6z11 Base is x Base is 6 Base is (x + y) TWO Bases: w & z

  7. 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. Quick Test of Quotient Rule • Test 

  9. 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. Example  Quotient Rule • Solution a) Base is x • Solution b) Base is 8 • Solution c) Base is (6y) • Solution d) TWO Bases: r & t

  11. 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. Example  The Exponent Zero • Simplify: a) 12450 b) (−3)0c) (4w)0 d) (−1)80 e) −80 • Solutions • 12450 = 1 • (−3)0 = 1 • (4w)0 = 1, for any w  0. • (−1)80 = (−1)1 = −1 • −80 is read “the opposite of 80” and is equivalent to (−1)80: −80 = (−1)80= (−1)1 = −1

  13. 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. Quick Test of Power Rule • Test 

  15. Example  Power Rule • Simplify: a) (x3)4 b) (42)8 • Solution a) (x3)4= x34 = x12 • Solution b) (42)8= 428 = 416 Base is x Base is 4

  16. 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. Quick Test of Product to Power • Test 

  18. Example  Product to Power • Simplify: a) (3x)4 b) (−2x3)2 c) (a2b3)7(a4b5) • Solutions • (3x)4 = 34x4 = 81x4 • (−2x3)2= (−2)2(x3)2 = (−1)2(2)2(x3)2 = 4x6 • (a2b3)7(a4b5) = (a2)7(b3)7a4b5 = a14b21a4b5Multiplying exponents = a18b26 Adding exponents

  19. 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. Quick Test of Quotient to Power • Test 

  21. Example  Quotient to a Power • Simplify: a) b) c) • Solution a) • Solution b) • Solution c)

  22. Negative Exponents • Integers as Negative Exponents

  23. Negative Exponents • For any real number a that is nonzero and any integer n • The numbers a−n and an are thus RECIPROCALS of each other

  24. Example  Negative Exponents • Express using POSITIVE exponents, and, if possible, simplify. a) m–5b) 5–2 c) (−4)−2 d) xy–1 • SOLUTION a) m–5 = b) 5–2 =

  25. Example  Negative Exponents • Express using POSITIVE exponents, and, if possible, simplify. a) m–5 b) 5–2c) (−4)−2d) xy−1 • SOLUTION c) (−4)−2= d) xy–1 = • Remember PEMDAS

  26. More Examples • Simplify. Do NOT use NEGATIVE exponents in the answer.a) b) (x4)3 c) (3a2b4)3d) e) f) • Solution a)

  27. More Examples • Solution b) (x−4)−3 = x(−4)(−3) = x12 c) (3a2b−4)3 = 33(a2)3(b−4)3 = 27 a6b−12 = d) e) f)

  28. 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. 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. 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. Examples  Flippers • Simplify • SOLUTION

  32. Summary – Exponent Properties This summary assumes that no denominators are 0 and that 00 is not considered. For any integers m and n

  33. WhiteBoard Work • Problems From §1.6 Exercise Set • 14, 24, 52, 70, 84, 92, 112, 130 • Base & Exponent →Which is Which?

  34. All Done for Today AstronomicalUnit (AU)

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

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