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# Properties Of Exponents

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### Properties Of Exponents

1. 1. Properties of Exponents p. 323
2. 2. Properties of Exponents a&b are real numbers, m&n are integers <ul><li>Product Property : </li></ul><ul><li>Quotient of Powers : </li></ul><ul><li>Power of a Power Property : </li></ul><ul><li>Power of a Product Property : </li></ul><ul><li>Negative Exponent Property : </li></ul><ul><li>Zero Exponent Property : </li></ul><ul><li>Power of Quotient: </li></ul>
3. 3. Example – Product Property <ul><li>(-5) 3 (-5) 2 = </li></ul><ul><li>(-5)(-5)(-5)(-5)(-5)= </li></ul><ul><li>(-5) 5 </li></ul><ul><li>(-5) 3+2 = </li></ul><ul><li>(-5) 5 </li></ul>
4. 4. Example – Product Property <ul><li>x 5 • x 2 = x •x• x •x•x•x•x </li></ul><ul><li>x 5+2 = </li></ul><ul><li>x 7 </li></ul>
5. 5. Product Property a&b are real numbers, m&n are integers <ul><li>Product Property : (a m )(a n )=a m+n </li></ul><ul><li>a 3 • a 5 • a 4 = </li></ul><ul><li>a 3 • a 5 • a 4 = a 3+5+4 </li></ul><ul><li>a 3 • a 5 • a 4 = a 12 </li></ul>
6. 6. Product Property <ul><li>(a 3 b 2 ) (a 4 b 6 ) = </li></ul><ul><li>(a 3 a 4 ) (b 2 b 6 ) = a 3+4 b 2+6 </li></ul><ul><li>a 3+4 b 2+6 = a 7 b 8 </li></ul><ul><li>(x 5 y 2 ) (x 4 y 7 ) = </li></ul><ul><li>(x 5 x 4 ) (y 2 y 7 ) = x 5+4 y 2+7 </li></ul><ul><li>x 5+2 y 2+7 = x 9 y 9 </li></ul>
7. 7. You try <ul><li>(3x 6 y 4 ) (4xy 7 ) = </li></ul><ul><li>(3x 6 y 4 ) (4xy 7 ) = (3 •4) x 6+1 • y 4+7 </li></ul><ul><li>(3 •4) x 6+1 • y 4+7 = 12x 7 y 11 </li></ul><ul><li>(2x 12 y 5 ) (6x 3 y 9 ) = </li></ul><ul><li>(2 • 6) x 12+3 y 5+9 =12x 15 y 14 </li></ul>
8. 8. Do now <ul><li>(2x 4 y 4 ) (5xy 7 ) = </li></ul><ul><li>(2x 4 y 4 ) (5xy 7 ) = (2 •5) x 4+1 • y 4+7 </li></ul><ul><li>(2 •5) x 4+1 • y 4+7 = 10x 5 y 11 </li></ul><ul><li>(3x 14 y 5 ) (9x 3 y) = </li></ul><ul><li>(3 • 9) x 14+3 y 5+1 =27x 17 y 6 </li></ul>
9. 9. Dividing Powers with Like bases <ul><li>-5 3 = -5 • -5 • -5 </li></ul><ul><ul><li>-5 2 -5 • -5 </li></ul></ul><ul><li>-5 • -5 • -5 = -5 </li></ul><ul><li>-5 • -5 </li></ul>
10. 10. Power of a Quotient with like bases <ul><li>x 4 = x • x • x • x </li></ul><ul><li>X 2 x • x </li></ul><ul><li>X 2 </li></ul>
11. 11. Quotient of Powers
12. 12. Quotient of Powers <ul><li>Quotient of Powers : </li></ul><ul><li>a m = a m-n ; a≠0 </li></ul><ul><li>a n </li></ul>
13. 13. You try <ul><li>4 5 x 4 y 7 = </li></ul><ul><li>4 3 x 2 y 6 </li></ul><ul><li>4 5 x 4 y 7 = 4 5-3 x 4-2 y 7-6 </li></ul><ul><li>4 3 x 2 y 6 </li></ul><ul><li>4 5-3 x 4-2 y 7-6 = 4 2 x 2 y = 16x 2 y </li></ul>
14. 14. You try <ul><li>3 7 x 9 y 12 = </li></ul><ul><li>3 4 x 5 y 6 </li></ul><ul><li>3 7 x 9 y 12 = 3 7-4 x 9-5 y 12-6 </li></ul><ul><li>3 4 x 5 y 6 </li></ul><ul><li>3 7-4 x 9-5 y 12-6 = 3 3 x 4 y 8 = 27x 4 y 8 </li></ul>
15. 15. Negative Exponents <ul><li>x 2 = x • x_____ </li></ul><ul><li>x 4 x • x • x • x </li></ul><ul><li>1 = </li></ul><ul><li>x 2 </li></ul><ul><li>x 2 = x 2 -4 = x -2 </li></ul><ul><li>X 4 </li></ul><ul><li>x -2 = 1 </li></ul><ul><li>x 2 </li></ul>
16. 16. Negative exponets <ul><li>x 3 = x • x_ • x___ </li></ul><ul><li>x 5 x • x • x • x • x </li></ul><ul><li>1 = </li></ul><ul><li>x 3 </li></ul><ul><li>x 3 = x 3 -5 = x -3 </li></ul><ul><li>x 5 </li></ul><ul><li>x -3 = 1 </li></ul><ul><li>x 3 </li></ul>
17. 17. Example – Quotient of Powers
18. 18. You try <ul><li>x -2 = </li></ul><ul><li>1 </li></ul><ul><li>x 2 </li></ul><ul><li>2x -2 y = </li></ul><ul><li>2x -2 y = 2y </li></ul><ul><li>x 2 </li></ul>
19. 19. You try <ul><li>(-5) -6 (-5) 4 = </li></ul><ul><li>(-5) -6+4 = </li></ul><ul><li>(-5) -2 = </li></ul>
20. 20. Properties of Exponents a&b are real numbers, m&n are integers <ul><li>Negative Exponent Property : </li></ul><ul><li>a -m = ; a ≠0 </li></ul>
21. 21. Zero Exponent Property <ul><li>x 0 </li></ul><ul><li>x 2 = x 2-2 </li></ul><ul><li>x 2 </li></ul><ul><li>x 2-2 = x 0 </li></ul><ul><li>x 2 = 1 </li></ul><ul><li>x 2 </li></ul><ul><li>x 0 = 1 </li></ul>
22. 22. You try <ul><li>(x -2 ) (x 2 ) = </li></ul><ul><li>(x -2 ) (x 2 ) = x -2+2 </li></ul><ul><li>x -2+2 = x 0 </li></ul><ul><li>x 0 = 1 </li></ul>
23. 23. Properties of Exponents a&b are real numbers, m&n are integers ets Review <ul><li>Zero Exponent Property : a 0 =1; a≠0 </li></ul>
24. 24. Properties of Exponents a&b are real numbers, m&n are integers <ul><li>Product Property : a m * a n =a m+n </li></ul><ul><li>Quotient of Powers : a m = a m-n ; a≠0 a n </li></ul><ul><li>Negative Exponent Property : a -m = ; a ≠0 </li></ul><ul><li>Zero Exponent Property : a 0 =1; a≠0 </li></ul>
25. 25. Journal Entry: Describe the rules for the follwoing <ul><li>Product Property : </li></ul><ul><li>Quotient of Powers : </li></ul><ul><li>Negative Exponent Property : </li></ul><ul><li>Zero Exponent Property : </li></ul>
26. 26. Example – Power of a Power <ul><li>(2 3 ) 4 = (2 3 ) (2 3 ) (2 3 ) (2 3 ) 4 </li></ul><ul><li>2 3+3+3+3 = </li></ul><ul><li>(2 3 ) 4 = 2 12 </li></ul>
27. 27. Example - Power of a Power <ul><li>(3 4 ) 3 = (3 4 ) (3 4 ) (3 4 ) </li></ul><ul><li>(3 4 ) (3 4 ) (3 4 ) = 3 4+4+4 </li></ul><ul><li>(3 4 ) 3 = 3 12 </li></ul>
28. 28. Raising a Power to a Power <ul><li>(X 5 ) 2 = (X 5 ) (X 5 ) </li></ul><ul><li>(X 5 ) (X 5 )= x 5+5 </li></ul><ul><li>(X 5 ) 2 = x 10 </li></ul>
29. 29. Power of a Power Property a&b are real numbers, m&n are integers <ul><li>Power of a Power Property : (a m ) n =a mn </li></ul><ul><li>(x 5 ) 3 = x 5 • 3 </li></ul><ul><li>x 5 • 3 = x 15 </li></ul>
30. 30. You try <ul><li>(y 4 ) 8 = </li></ul><ul><li>(y 4 ) 8 = y 4 •8 = y 24 </li></ul><ul><li>(s 3 ) 4 = </li></ul><ul><li>(s 3 ) 4 = s 3 •4 = s 12 </li></ul>
31. 31. Power of a Product Property <ul><li>(-2x 7 ) 2 = (-2x 7 ) (-2x 7 ) </li></ul><ul><li>(-2x 7 ) (-2x 7 ) = (-2 • - 2) (x 7 • x 7 ) =4x 14 </li></ul><ul><li>(-2x 7 ) 2 = (-2) 2 (x 7 ) 2 = -2 1 •2 x 7 • 2 </li></ul><ul><li>= 4x 14 </li></ul>
32. 32. <ul><li>Power of a Product Property : (ab) m =a m b m </li></ul><ul><li>(a 3 b 2 ) 4 = (a 3 ) 4 (b 2 ) 4 </li></ul><ul><li>(a 3 ) 4 (b 2 ) 4 = a 3 •4 b 2 •4 =a 12 b 8 </li></ul>
33. 33. You try <ul><li>(-2x 4 ) 3 = (-2) 1 • 3 x 5 • 3 </li></ul><ul><li>(-2x 4 ) 3 = (-2) 1 • 3 x 4 • 3 </li></ul><ul><li>(-2) 1 • 3 x 4 • 3 = (-2) 3 x 12 = -16x 12 </li></ul><ul><li>(4x 4 y 5 ) 2 </li></ul><ul><li>(4x 4 y 5 ) 2 = 4 1 • 2 x 4 • 2 y 5 • 2 </li></ul><ul><li>(4x 4 y 5 ) 2 = 16x 8 y 10 </li></ul>
34. 34. You try <ul><li>(-3x 5 y 3 ) 4 = </li></ul><ul><li>(-3) 1 • 4 x 5 • 4 y 4 • 4 = (-3) 4 x 20 y 16 </li></ul><ul><li>(7x 3 y -5 ) 2 </li></ul><ul><li>7 1 • 2 x 3 • 2 y -5 • 2 </li></ul><ul><li>16x 8 y -10 = 16x 8 </li></ul><ul><li>y 10 </li></ul>
35. 35. Example – Power of Quotient
36. 36. Properties of Exponents a&b are real numbers, m&n are integers <ul><li>Power of Quotient: </li></ul><ul><li>b≠0 </li></ul>
37. 37. Properties of Exponents a&b are real numbers, m&n are integers <ul><li>Product Property : a m * a n =a m+n </li></ul><ul><li>Quotient of Powers : a m = a m-n ; a≠0 a n </li></ul><ul><li>Power of a Power Property : (a m ) n =a mn </li></ul><ul><li>Power of a Product Property : (ab) m =a m b m </li></ul><ul><li>Negative Exponent Property : a -m = ; a ≠0 </li></ul><ul><li>Zero Exponent Property : a 0 =1; a≠0 </li></ul><ul><li>Power of Quotient: b≠0 </li></ul>
38. 38. Multiplying and Dividing Monomials <ul><li>Monomial – an expression that is either a numeral, a variable or a product of numerals and variables with whole number exponents. </li></ul><ul><li>Constant – Monomial that is a numeral. Example - 2 </li></ul>
39. 39. Journal Entry: Describe the rules for the follwoing <ul><li>Power of a Power Property : </li></ul><ul><li>Power of a Product Property : </li></ul><ul><li>Power of Quotient: </li></ul>
40. 40. Multiplying Monomials <ul><li>(-2x 4 y 2 ) (-3xy 2 z 3 ) = </li></ul><ul><li>(-2)(-3)(x 4 x )(y 2 y 2 ) z 3 </li></ul><ul><li>6x 5 y 4 z 3 </li></ul><ul><li>(-2x 3 y 4 ) 2 (-3xy 2 ) </li></ul><ul><li>(-2 ) 2 x 3∙2 y 4∙2 ) (-3xy 2 ) </li></ul><ul><li>(4)(-3)(x 6 x) (y 8 y 2 ) </li></ul><ul><li>-12x 7 y 10 </li></ul>
41. 41. Scientific Notation <ul><li>A number is expressed in scientific notation when it is written as the product of a factor and a power of 10. The factor must be greater than or equal to 1 and less then 10 </li></ul><ul><li>a x 10ⁿ, where 1 ≤ a < 10 </li></ul>
42. 42. Scientific Notation <ul><li>131,400,000,000= </li></ul><ul><li>1.314 x 10 11 </li></ul>Move the decimal behind the 1 st number How many places did you have to move the decimal? Put that number here!
43. 43. Write using scientific notation <ul><li>12,300= </li></ul><ul><li>1.23 x 10 4 </li></ul><ul><li>Write using standard notation </li></ul><ul><li>1.76 x 10 3 </li></ul><ul><li>1,760 </li></ul>
44. 44. Example – Scientific Notation <ul><li>131,400,000,000 = </li></ul><ul><li>5,284,000 </li></ul><ul><li>1.314 x 10 11 = </li></ul><ul><li>5.284 x 10 6 </li></ul>
45. 45. Example – Scientific Notation <ul><li>(5.2 x 10 9 )(3.0 x 10 -3 )= </li></ul><ul><li>(5.2 x 3.0) (10 9 x 10 -3 )= </li></ul><ul><li>15.6 x 10 6 </li></ul><ul><li>1.56 x 10 7 </li></ul><ul><li>2.45 x 10 -3 = </li></ul><ul><li>0.00245 </li></ul>
46. 46. Properties of Exponents a&b are real numbers, m&n are integers <ul><li>Product Property : a m * a n =a m+n </li></ul><ul><li>Quotient of Powers : a m = a m-n ; a≠0 a n </li></ul><ul><li>Power of a Power Property : (a m ) n =a mn </li></ul><ul><li>Power of a Product Property : (ab) m =a m b m </li></ul><ul><li>Negative Exponent Property : a -m = ; a ≠0 </li></ul><ul><li>Zero Exponent Property : a 0 =1; a≠0 </li></ul><ul><li>Power of Quotient: b≠0 </li></ul>