exponents and scientific notation

1. Exponents

2. Exponents
We write the quantity A multiplied to itself N times as AN,

3. Exponents
We write the quantity A multiplied to itself N times as AN, i.e.
A x A x A ….x A = AN

4. base
exponent
Exponents
We write the quantity A multiplied to itself N times as AN, i.e.
A x A x A ….x A = AN

5. Example A.
43
base
exponent
Exponents
We write the quantity A multiplied to itself N times as AN, i.e.
A x A x A ….x A = AN

6. Example A.
43 = (4)(4)(4) = 64
base
exponent
Exponents
We write the quantity A multiplied to itself N times as AN, i.e.
A x A x A ….x A = AN

7. Example A.
43 = (4)(4)(4) = 64
(xy)2
base
exponent
Exponents
We write the quantity A multiplied to itself N times as AN, i.e.
A x A x A ….x A = AN

8. Example A.
43 = (4)(4)(4) = 64
(xy)2= (xy)(xy)
base
exponent
Exponents
We write the quantity A multiplied to itself N times as AN, i.e.
A x A x A ….x A = AN

9. Example A.
43 = (4)(4)(4) = 64
(xy)2= (xy)(xy) = x2y2
base
exponent
Exponents
We write the quantity A multiplied to itself N times as AN, i.e.
A x A x A ….x A = AN

10. Example A.
43 = (4)(4)(4) = 64
(xy)2= (xy)(xy) = x2y2
xy2
base
exponent
Exponents
We write the quantity A multiplied to itself N times as AN, i.e.
A x A x A ….x A = AN

11. Example A.
43 = (4)(4)(4) = 64
(xy)2= (xy)(xy) = x2y2
xy2 = (x)(yy)
base
exponent
Exponents
We write the quantity A multiplied to itself N times as AN, i.e.
A x A x A ….x A = AN

12. Example A.
43 = (4)(4)(4) = 64
(xy)2= (xy)(xy) = x2y2
xy2 = (x)(yy)
–x2 = –(xx)
base
exponent
Exponents
We write the quantity A multiplied to itself N times as AN, i.e.
A x A x A ….x A = AN

13. Example A.
43 = (4)(4)(4) = 64
(xy)2= (xy)(xy) = x2y2
xy2 = (x)(yy)
–x2 = –(xx)
base
exponent
Exponents
We write the quantity A multiplied to itself N times as AN, i.e.
A x A x A ….x A = AN

14. Example A.
43 = (4)(4)(4) = 64
(xy)2= (xy)(xy) = x2y2
xy2 = (x)(yy)
–x2 = –(xx)
base
exponent
Exponents
Rules of Exponents
We write the quantity A multiplied to itself N times as AN, i.e.
A x A x A ….x A = AN

15. Example A.
43 = (4)(4)(4) = 64
(xy)2= (xy)(xy) = x2y2
xy2 = (x)(yy)
–x2 = –(xx)
base
exponent
Exponents
Multiplication Rule: ANAK =AN+K
Rules of Exponents
We write the quantity A multiplied to itself N times as AN, i.e.
A x A x A ….x A = AN

16. Example A.
43 = (4)(4)(4) = 64
(xy)2= (xy)(xy) = x2y2
xy2 = (x)(yy)
–x2 = –(xx)
base
exponent
Exponents
Multiplication Rule: ANAK =AN+K
Example B.
a. 5354
Rules of Exponents
We write the quantity A multiplied to itself N times as AN, i.e.
A x A x A ….x A = AN

17. Example A.
43 = (4)(4)(4) = 64
(xy)2= (xy)(xy) = x2y2
xy2 = (x)(yy)
–x2 = –(xx)
base
exponent
Exponents
Multiplication Rule: ANAK =AN+K
Example B.
a. 5354 = (5*5*5)(5*5*5*5)
Rules of Exponents
We write the quantity A multiplied to itself N times as AN, i.e.
A x A x A ….x A = AN

18. Example A.
43 = (4)(4)(4) = 64
(xy)2= (xy)(xy) = x2y2
xy2 = (x)(yy)
–x2 = –(xx)
base
exponent
Exponents
Multiplication Rule: ANAK =AN+K
Example B.
a. 5354 = (5*5*5)(5*5*5*5) = 53+4 = 57
b. x5y7x4y6
Rules of Exponents
We write the quantity A multiplied to itself N times as AN, i.e.
A x A x A ….x A = AN

19. Example A.
43 = (4)(4)(4) = 64
(xy)2= (xy)(xy) = x2y2
xy2 = (x)(yy)
–x2 = –(xx)
base
exponent
Exponents
Multiplication Rule: ANAK =AN+K
Example B.
a. 5354 = (5*5*5)(5*5*5*5) = 53+4 = 57
b. x5y7x4y6 = x5x4y7y6
Rules of Exponents
We write the quantity A multiplied to itself N times as AN, i.e.
A x A x A ….x A = AN

20. Example A.
43 = (4)(4)(4) = 64
(xy)2= (xy)(xy) = x2y2
xy2 = (x)(yy)
–x2 = –(xx)
base
exponent
Exponents
Multiplication Rule: ANAK =AN+K
Example B.
a. 5354 = (5*5*5)(5*5*5*5) = 53+4 = 57
b. x5y7x4y6 = x5x4y7y6 = x9y13
Rules of Exponents
We write the quantity A multiplied to itself N times as AN, i.e.
A x A x A ….x A = AN

21. Example A.
43 = (4)(4)(4) = 64
(xy)2= (xy)(xy) = x2y2
xy2 = (x)(yy)
–x2 = –(xx)
base
exponent
Exponents
Multiplication Rule: ANAK =AN+K
Example B.
a. 5354 = (5*5*5)(5*5*5*5) = 53+4 = 57
b. x5y7x4y6 = x5x4y7y6 = x9y13
Rules of Exponents
Division Rule:
AN
AK = AN – K
We write the quantity A multiplied to itself N times as AN, i.e.
A x A x A ….x A = AN

22. Example A.
43 = (4)(4)(4) = 64
(xy)2= (xy)(xy) = x2y2
xy2 = (x)(yy)
–x2 = –(xx)
base
exponent
Exponents
Multiplication Rule: ANAK =AN+K
Example B.
a. 5354 = (5*5*5)(5*5*5*5) = 53+4 = 57
b. x5y7x4y6 = x5x4y7y6 = x9y13
Rules of Exponents
Division Rule:
Example C.
AN
AK = AN – K
56
52
We write the quantity A multiplied to itself N times as AN, i.e.
A x A x A ….x A = AN

23. Example A.
43 = (4)(4)(4) = 64
(xy)2= (xy)(xy) = x2y2
xy2 = (x)(yy)
–x2 = –(xx)
base
exponent
Exponents
Multiplication Rule: ANAK =AN+K
Example B.
a. 5354 = (5*5*5)(5*5*5*5) = 53+4 = 57
b. x5y7x4y6 = x5x4y7y6 = x9y13
Rules of Exponents
Division Rule:
Example C.
AN
AK = AN – K
56
52 =
(5)(5)(5)(5)(5)(5)
(5)(5)
We write the quantity A multiplied to itself N times as AN, i.e.
A x A x A ….x A = AN

24. Example A.
43 = (4)(4)(4) = 64
(xy)2= (xy)(xy) = x2y2
xy2 = (x)(yy)
–x2 = –(xx)
base
exponent
Exponents
Multiplication Rule: ANAK =AN+K
Example B.
a. 5354 = (5*5*5)(5*5*5*5) = 53+4 = 57
b. x5y7x4y6 = x5x4y7y6 = x9y13
Rules of Exponents
Division Rule:
Example C.
AN
AK = AN – K
56
52 =
(5)(5)(5)(5)(5)(5)
(5)(5)
We write the quantity A multiplied to itself N times as AN, i.e.
A x A x A ….x A = AN

25. Example A.
43 = (4)(4)(4) = 64
(xy)2= (xy)(xy) = x2y2
xy2 = (x)(yy)
–x2 = –(xx)
base
exponent
Exponents
Multiplication Rule: ANAK =AN+K
Example B.
a. 5354 = (5*5*5)(5*5*5*5) = 53+4 = 57
b. x5y7x4y6 = x5x4y7y6 = x9y13
Rules of Exponents
Division Rule:
Example C.
AN
AK = AN – K
56
52 =
(5)(5)(5)(5)(5)(5)
(5)(5)
= 56 – 2 = 54
We write the quantity A multiplied to itself N times as AN, i.e.
A x A x A ….x A = AN

26. Power Rule: (AN)K = ANK
Exponents

27. Power Rule: (AN)K = ANK
Example D. (34)5
Exponents
A1

28. Power Rule: (AN)K = ANK
Example D. (34)5 = (34)(34)(34)(34)(34)
Exponents

29. Power Rule: (AN)K = ANK
Example D. (34)5 = (34)(34)(34)(34)(34) = 34+4+4+4+4
Exponents

30. Power Rule: (AN)K = ANK
Example D. (34)5 = (34)(34)(34)(34)(34) = 34+4+4+4+4 = 34*5 = 320
Exponents

31. Power Rule: (AN)K = ANK
Example D. (34)5 = (34)(34)(34)(34)(34) = 34+4+4+4+4 = 34*5 = 320
Exponents
Since = 1A1
A1

32. Power Rule: (AN)K = ANK
Example D. (34)5 = (34)(34)(34)(34)(34) = 34+4+4+4+4 = 34*5 = 320
Exponents
Since = 1 = A1 – 1A1
A1

33. Power Rule: (AN)K = ANK
Example D. (34)5 = (34)(34)(34)(34)(34) = 34+4+4+4+4 = 34*5 = 320
Exponents
Since = 1 = A1 – 1 = A0A1
A1

34. Power Rule: (AN)K = ANK
Example D. (34)5 = (34)(34)(34)(34)(34) = 34+4+4+4+4 = 34*5 = 320
Exponents
Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule.A1
A1

35. Power Rule: (AN)K = ANK
Example D. (34)5 = (34)(34)(34)(34)(34) = 34+4+4+4+4 = 34*5 = 320
Exponents
Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule.A1
A1
0-Power Rule: A0 = 1

36. Power Rule: (AN)K = ANK
Example D. (34)5 = (34)(34)(34)(34)(34) = 34+4+4+4+4 = 34*5 = 320
Exponents
Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule.A1
A1
0-Power Rule: A0 = 1
Since =
1
AK
A0
AK

37. Power Rule: (AN)K = ANK
Example D. (34)5 = (34)(34)(34)(34)(34) = 34+4+4+4+4 = 34*5 = 320
Exponents
Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule.A1
A1
0-Power Rule: A0 = 1
Since = = A0 – K1
AK
A0
AK

38. Power Rule: (AN)K = ANK
Example D. (34)5 = (34)(34)(34)(34)(34) = 34+4+4+4+4 = 34*5 = 320
Exponents
Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule.A1
A1
0-Power Rule: A0 = 1
Since = = A0 – K = A–K, we get the negative-power Rule.
1
AK
A0
AK

39. Power Rule: (AN)K = ANK
Example D. (34)5 = (34)(34)(34)(34)(34) = 34+4+4+4+4 = 34*5 = 320
Exponents
Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule.A1
A1
0-Power Rule: A0 = 1
Since = = A0 – K = A–K, we get the negative-power Rule.
1
AK
A0
AK
Negative-Power Rule: A–K = 1
AK

40. Power Rule: (AN)K = ANK
Example D. (34)5 = (34)(34)(34)(34)(34) = 34+4+4+4+4 = 34*5 = 320
Exponents
Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule.A1
A1
0-Power Rule: A0 = 1
Since = = A0 – K = A–K, we get the negative-power Rule.
1
AK
A0
AK
Negative-Power Rule: A–K = 1
AK
Example E. Simplify
a. 30

41. Power Rule: (AN)K = ANK
Example D. (34)5 = (34)(34)(34)(34)(34) = 34+4+4+4+4 = 34*5 = 320
Exponents
Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule.A1
A1
0-Power Rule: A0 = 1
Since = = A0 – K = A–K, we get the negative-power Rule.
1
AK
A0
AK
Negative-Power Rule: A–K = 1
AK
Example E. Simplify
a. 30 = 1

42. Power Rule: (AN)K = ANK
Example D. (34)5 = (34)(34)(34)(34)(34) = 34+4+4+4+4 = 34*5 = 320
Exponents
Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule.A1
A1
0-Power Rule: A0 = 1
Since = = A0 – K = A–K, we get the negative-power Rule.
1
AK
A0
AK
Negative-Power Rule: A–K = 1
AK
Example E. Simplify
b. 3–2
a. 30 = 1

43. Power Rule: (AN)K = ANK
Example D. (34)5 = (34)(34)(34)(34)(34) = 34+4+4+4+4 = 34*5 = 320
Exponents
Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule.A1
A1
0-Power Rule: A0 = 1
Since = = A0 – K = A–K, we get the negative-power Rule.
1
AK
A0
AK
Negative-Power Rule: A–K = 1
AK
Example E. Simplify
1
32b. 3–2 =
a. 30 = 1

44. Power Rule: (AN)K = ANK
Example D. (34)5 = (34)(34)(34)(34)(34) = 34+4+4+4+4 = 34*5 = 320
Exponents
Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule.A1
A1
0-Power Rule: A0 = 1
Since = = A0 – K = A–K, we get the negative-power Rule.
1
AK
A0
AK
Negative-Power Rule: A–K = 1
AK
Example E. Simplify
1
32
1
9b. 3–2 = =
a. 30 = 1

45. Power Rule: (AN)K = ANK
Example D. (34)5 = (34)(34)(34)(34)(34) = 34+4+4+4+4 = 34*5 = 320
Exponents
Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule.A1
A1
0-Power Rule: A0 = 1
Since = = A0 – K = A–K, we get the negative-power Rule.
1
AK
A0
AK
Negative-Power Rule: A–K = 1
AK
Example E. Simplify
1
32
1
9
c. ( )–12
5
b. 3–2 = =
a. 30 = 1

46. Power Rule: (AN)K = ANK
Example D. (34)5 = (34)(34)(34)(34)(34) = 34+4+4+4+4 = 34*5 = 320
Exponents
Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule.A1
A1
0-Power Rule: A0 = 1
Since = = A0 – K = A–K, we get the negative-power Rule.
1
AK
A0
AK
Negative-Power Rule: A–K = 1
AK
Example E. Simplify
1
32
1
9
c. ( )–12
5
=
1
2/5
=
b. 3–2 = =
a. 30 = 1

47. Power Rule: (AN)K = ANK
Example D. (34)5 = (34)(34)(34)(34)(34) = 34+4+4+4+4 = 34*5 = 320
Exponents
Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule.A1
A1
0-Power Rule: A0 = 1
Since = = A0 – K = A–K, we get the negative-power Rule.
1
AK
A0
AK
Negative-Power Rule: A–K = 1
AK
Example E. Simplify
1
32
1
9
c. ( )–12
5
=
1
2/5
= 1*
5
2
=
5
2
b. 3–2 = =
a. 30 = 1

48. Power Rule: (AN)K = ANK
Example D. (34)5 = (34)(34)(34)(34)(34) = 34+4+4+4+4 = 34*5 = 320
Exponents
Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule.A1
A1
0-Power Rule: A0 = 1
Since = = A0 – K = A–K, we get the negative-power Rule.
1
AK
A0
AK
Negative-Power Rule: A–K = 1
AK
Example E. Simplify
1
32
1
9
c. ( )–12
5
=
1
2/5
= 1*
5
2
=
5
2
b. 3–2 = =
a. 30 = 1
In general ( )–Ka
b = ( )K
b
a

49. Power Rule: (AN)K = ANK
Example D. (34)5 = (34)(34)(34)(34)(34) = 34+4+4+4+4 = 34*5 = 320
Exponents
Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule.A1
A1
0-Power Rule: A0 = 1
Since = = A0 – K = A–K, we get the negative-power Rule.
1
AK
A0
AK
Negative-Power Rule: A–K = 1
AK
Example E. Simplify
1
32
1
9
c. ( )–12
5
=
1
2/5
= 1*
5
2
=
5
2
b. 3–2 = =
a. 30 = 1
In general ( )–Ka
b = ( )K
b
a
d. ( )–22
5

50. Power Rule: (AN)K = ANK
Example D. (34)5 = (34)(34)(34)(34)(34) = 34+4+4+4+4 = 34*5 = 320
Exponents
Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule.A1
A1
0-Power Rule: A0 = 1
Since = = A0 – K = A–K, we get the negative-power Rule.
1
AK
A0
AK
Negative-Power Rule: A–K = 1
AK
Example E. Simplify
1
32
1
9
c. ( )–12
5
=
1
2/5
= 1*
5
2
=
5
2
b. 3–2 = =
a. 30 = 1
In general ( )–Ka
b = ( )K
b
a
d. ( )–22
5
= ( )25
2

51. Power Rule: (AN)K = ANK
Example D. (34)5 = (34)(34)(34)(34)(34) = 34+4+4+4+4 = 34*5 = 320
Exponents
Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule.A1
A1
0-Power Rule: A0 = 1
Since = = A0 – K = A–K, we get the negative-power Rule.
1
AK
A0
AK
Negative-Power Rule: A–K = 1
AK
Example E. Simplify
1
32
1
9
c. ( )–12
5
=
1
2/5
= 1*
5
2
=
5
2
b. 3–2 = =
a. 30 = 1
In general ( )–Ka
b = ( )K
b
a
d. ( )–22
5
= ( )2 =
25
4
5
2

52. e. 3–1 – 40 * 2–2 =
Exponents

53. e. 3–1 – 40 * 2–2 =
1
3
Exponents

54. e. 3–1 – 40 * 2–2 =
1
3
– 1*
Exponents

55. e. 3–1 – 40 * 2–2 =
1
3
– 1*
1
22
Exponents

56. e. 3–1 – 40 * 2–2 =
1
3
– 1*
1
22 = 1
3
– 1
4
= 1
12
Exponents

57. e. 3–1 – 40 * 2–2 =
1
3
– 1*
1
22 = 1
3
– 1
4
= 1
12
Exponents
Although the negative power means to reciprocate,
for problems of collecting exponents, we do not reciprocate
the negative exponents.

58. e. 3–1 – 40 * 2–2 =
1
3
– 1*
1
22 = 1
3
– 1
4
= 1
12
Exponents
Although the negative power means to reciprocate,
for problems of collecting exponents, we do not reciprocate
the negative exponents. Instead we add or subtract them
using the multiplication and division rules first.

59. e. 3–1 – 40 * 2–2 =
Exponents
Although the negative power means to reciprocate,
for problems of collecting exponents, we do not reciprocate
the negative exponents. Instead we add or subtract them
using the multiplication and division rules first.
Example F. Simplify 3–2 x4 y–6 x–8 y 23
1
3
– 1*
1
22 = 1
3
– 1
4
= 1
12

60. e. 3–1 – 40 * 2–2 =
Exponents
Although the negative power means to reciprocate,
for problems of collecting exponents, we do not reciprocate
the negative exponents. Instead we add or subtract them
using the multiplication and division rules first.
Example F. Simplify 3–2 x4 y–6 x–8 y 23
3–2 x4 y–6 x–8 y23
1
3
– 1*
1
22 = 1
3
– 1
4
= 1
12

61. e. 3–1 – 40 * 2–2 =
Exponents
Although the negative power means to reciprocate,
for problems of collecting exponents, we do not reciprocate
the negative exponents. Instead we add or subtract them
using the multiplication and division rules first.
Example F. Simplify 3–2 x4 y–6 x–8 y 23
3–2 x4 y–6 x–8 y23
= 3–2 x4 x–8 y–6 y23
1
3
– 1*
1
22 = 1
3
– 1
4
= 1
12

62. e. 3–1 – 40 * 2–2 =
Exponents
Although the negative power means to reciprocate,
for problems of collecting exponents, we do not reciprocate
the negative exponents. Instead we add or subtract them
using the multiplication and division rules first.
= x4 – 8 y–6+23
Example F. Simplify 3–2 x4 y–6 x–8 y 23
3–2 x4 y–6 x–8 y23
= 3–2 x4 x–8 y–6 y23
1
9
1
3
– 1*
1
22 = 1
3
– 1
4
= 1
12

63. e. 3–1 – 40 * 2–2 =
Exponents
Although the negative power means to reciprocate,
for problems of collecting exponents, we do not reciprocate
the negative exponents. Instead we add or subtract them
using the multiplication and division rules first.
= x4 – 8 y–6+23
= x–4 y17
Example F. Simplify 3–2 x4 y–6 x–8 y 23
3–2 x4 y–6 x–8 y23
= 3–2 x4 x–8 y–6 y23
1
9
1
9
1
3
– 1*
1
22 = 1
3
– 1
4
= 1
12

64. e. 3–1 – 40 * 2–2 =
Exponents
Although the negative power means to reciprocate,
for problems of collecting exponents, we do not reciprocate
the negative exponents. Instead we add or subtract them
using the multiplication and division rules first.
= x4 – 8 y–6+23
= x–4 y17
= y17
Example F. Simplify 3–2 x4 y–6 x–8 y 23
3–2 x4 y–6 x–8 y23
= 3–2 x4 x–8 y–6 y23
1
9
1
9
1
9x4
1
3
– 1*
1
22 = 1
3
– 1
4
= 1
12

65. e. 3–1 – 40 * 2–2 =
Exponents
Although the negative power means to reciprocate,
for problems of collecting exponents, we do not reciprocate
the negative exponents. Instead we add or subtract them
using the multiplication and division rules first.
= x4 – 8 y–6+23
= x–4 y17
= y17
=
Example F. Simplify 3–2 x4 y–6 x–8 y 23
3–2 x4 y–6 x–8 y23
= 3–2 x4 x–8 y–6 y23
1
9
1
9
1
9x4
y17
9x4
1
3
– 1*
1
22 = 1
3
– 1
4
= 1
12

66. Exponents
Example G. Simplify using the rules for exponents.
Leave the answer in positive exponents only.
23x–8
26 x–3

67. Exponents
Example G. Simplify using the rules for exponents.
Leave the answer in positive exponents only.
23x–8
26 x–3
23x–8
26x–3

68. Exponents
Example G. Simplify using the rules for exponents.
Leave the answer in positive exponents only.
23x–8
26 x–3
23x–8
26x–3
= 23 – 6 x–8 – (–3 )

69. Exponents
Example G. Simplify using the rules for exponents.
Leave the answer in positive exponents only.
23x–8
26 x–3
23x–8
26x–3
= 23 – 6 x–8 – (–3 )
= 2–3 x–5

70. Exponents
Example G. Simplify using the rules for exponents.
Leave the answer in positive exponents only.
23x–8
26 x–3
23x–8
26x–3
= 23 – 6 x–8 – (–3 )
= 2–3 x–5
=
23
1
x5
1
* = 8x5
1

71. Exponents
Example G. Simplify using the rules for exponents.
Leave the answer in positive exponents only.
23x–8
26 x–3
23x–8
26x–3
= 23 – 6 x–8 – (–3 )
= 2–3 x–5
=
23
1
x5
1
* = 8x5
1
Example H. Simplify
(3x–2y3)–2 x2
3–5x–3(y–1x2)3

72. Exponents
Example G. Simplify using the rules for exponents.
Leave the answer in positive exponents only.
23x–8
26 x–3
23x–8
26x–3
= 23 – 6 x–8 – (–3 )
= 2–3 x–5
=
23
1
x5
1
* = 8x5
1
Example H. Simplify
(3x–2y3)–2 x2
3–5x–3(y–1x2)3
(3x–2y3)–2 x2
3–5x–3(y–1x2)3

73. Exponents
Example G. Simplify using the rules for exponents.
Leave the answer in positive exponents only.
23x–8
26 x–3
23x–8
26x–3
= 23 – 6 x–8 – (–3 )
= 2–3 x–5
=
23
1
x5
1
* = 8x5
1
Example H. Simplify
(3x–2y3)–2 x2
3–5x–3(y–1x2)3
=
3–2x4y–6x2
3–5x–3y–3 x6
(3x–2y3)–2 x2
3–5x–3(y–1x2)3

74. Exponents
Example G. Simplify using the rules for exponents.
Leave the answer in positive exponents only.
23x–8
26 x–3
23x–8
26x–3
= 23 – 6 x–8 – (–3 )
= 2–3 x–5
=
23
1
x5
1
* = 8x5
1
Example H. Simplify
(3x–2y3)–2 x2
3–5x–3(y–1x2)3
=
3–2x4y–6x2
3–5x–3y–3 x6 =
(3x–2y3)–2 x2
3–5x–3(y–1x2)3 3–5x–3x6y–3
3–2x4x2y–6

75. Exponents
Example G. Simplify using the rules for exponents.
Leave the answer in positive exponents only.
23x–8
26 x–3
23x–8
26x–3
= 23 – 6 x–8 – (–3 )
= 2–3 x–5
=
23
1
x5
1
* = 8x5
1
Example H. Simplify
(3x–2y3)–2 x2
3–5x–3(y–1x2)3
=
3–2x4y–6x2
3–5x–3y–3 x6 =
=
(3x–2y3)–2 x2
3–5x–3(y–1x2)3 3–5x–3x6y–3
3–2x4x2y–6
3–2x6y–6
3–5x3y–3

76. Exponents
Example G. Simplify using the rules for exponents.
Leave the answer in positive exponents only.
23x–8
26 x–3
23x–8
26x–3
= 23 – 6 x–8 – (–3 )
= 2–3 x–5
=
23
1
x5
1
* = 8x5
1
Example H. Simplify
(3x–2y3)–2 x2
3–5x–3(y–1x2)3
=
3–2x4y–6x2
3–5x–3y–3 x6 =
= = 3–2 – (–5) x6 – 3 y–6 – (–3)
(3x–2y3)–2 x2
3–5x–3(y–1x2)3 3–5x–3x6y–3
3–2x4x2y–6
3–2x6y–6
3–5x3y–3

77. Exponents
Example G. Simplify using the rules for exponents.
Leave the answer in positive exponents only.
23x–8
26 x–3
23x–8
26x–3
= 23 – 6 x–8 – (–3 )
= 2–3 x–5
=
23
1
x5
1
* = 8x5
1
Example H. Simplify
(3x–2y3)–2 x2
3–5x–3(y–1x2)3
=
3–2x4y–6x2
3–5x–3y–3 x6 =
= = 3–2 – (–5) x6 – 3 y–6 – (–3)
= 33 x3 y–3=
(3x–2y3)–2 x2
3–5x–3(y–1x2)3 3–5x–3x6y–3
3–2x4x2y–6
3–2x6y–6
3–5x3y–3

78. Exponents
Example G. Simplify using the rules for exponents.
Leave the answer in positive exponents only.
23x–8
26 x–3
23x–8
26x–3
= 23 – 6 x–8 – (–3 )
= 2–3 x–5
=
23
1
x5
1
* = 8x5
1
Example H. Simplify
(3x–2y3)–2 x2
3–5x–3(y–1x2)3
=
3–2x4y–6x2
3–5x–3y–3 x6 =
= = 3–2 – (–5) x6 – 3 y–6 – (–3)
= 33 x3 y–3=
27 x3
(3x–2y3)–2 x2
3–5x–3(y–1x2)3 3–5x–3x6y–3
3–2x4x2y–6
3–2x6y–6
3–5x3y–3
y3

79. Exponents
Example G. Simplify using the rules for exponents.
Leave the answer in positive exponents only.
23x–8
26 x–3
23x–8
26x–3
= 23 – 6 x–8 – (–3 )
= 2–3 x–5
=
23
1
x5
1
* = 8x5
1
Example H. Simplify
(3x–2y3)–2 x2
3–5x–3(y–1x2)3
=
3–2x4y–6x2
3–5x–3y–3 x6 =
= = 3–2 – (–5) x6 – 3 y–6 – (–3)
= 33 x3 y–3=
27 x3
(3x–2y3)–2 x2
3–5x–3(y–1x2)3 3–5x–3x6y–3
3–2x4x2y–6
3–2x6y–6
3–5x3y–3
y3
An important application of exponents is the scientific notation.

80. Scientific Notation

81. Scientific Notation
Scientific notation simplifies the tracking and calculation of very
large or very small numbers. We note the relation between
the exponents and
the base-10 numbers:

82. 100 = 1
Scientific Notation
Scientific notation simplifies the tracking and calculation of very
large or very small numbers. We note the relation between
the exponents and
the base-10 numbers:

83. 100 = 1
101 = 10
Scientific Notation
Scientific notation simplifies the tracking and calculation of very
large or very small numbers. We note the relation between
the exponents and
the base-10 numbers:

84. 100 = 1
101 = 10
102 = 100
Scientific Notation
Scientific notation simplifies the tracking and calculation of very
large or very small numbers. We note the relation between
the exponents and
the base-10 numbers:

85. 100 = 1
101 = 10
102 = 100
103 = 1000
Scientific Notation
Scientific notation simplifies the tracking and calculation of very
large or very small numbers. We note the relation between
the exponents and
the base-10 numbers:

86. 100 = 1
101 = 10
102 = 100
103 = 1000
10–1 = 0.1
Scientific Notation
Scientific notation simplifies the tracking and calculation of very
large or very small numbers. We note the relation between
the exponents and
the base-10 numbers:

87. 100 = 1
101 = 10
102 = 100
103 = 1000
10–1 = 0.1
10–2 = 0.01
Scientific Notation
Scientific notation simplifies the tracking and calculation of very
large or very small numbers. We note the relation between
the exponents and
the base-10 numbers:

88. 100 = 1
101 = 10
102 = 100
103 = 1000
10–1 = 0.1
10–2 = 0.01
10–3 = 0.001
10–4 = 0.0001
Scientific Notation
Scientific notation simplifies the tracking and calculation of very
large or very small numbers. We note the relation between
the exponents and
the base-10 numbers:

89. 100 = 1
101 = 10
102 = 100
103 = 1000
10–1 = 0.1
10–2 = 0.01
10–3 = 0.001
10–4 = 0.0001
Scientific Notation
Scientific Notation
Scientific notation simplifies the tracking and calculation of very
large or very small numbers. We note the relation between
the exponents and
the base-10 numbers:

90. 100 = 1
101 = 10
102 = 100
103 = 1000
10–1 = 0.1
10–2 = 0.01
10–3 = 0.001
10–4 = 0.0001
Scientific Notation
Scientific Notation
Any number can be written in the form
A x 10N
where 1 < A < 10.
Scientific notation simplifies the tracking and calculation of very
large or very small numbers. We note the relation between
the exponents and
the base-10 numbers:

91. 100 = 1
101 = 10
102 = 100
103 = 1000
10–1 = 0.1
10–2 = 0.01
10–3 = 0.001
10–4 = 0.0001
Scientific Notation
Scientific Notation
Any number can be written in the form
A x 10N
where 1 < A < 10. This form is called the scientific notation of
the number.
Scientific notation simplifies the tracking and calculation of very
large or very small numbers. We note the relation between
the exponents and
the base-10 numbers:

92. Scientific Notation
To write a number in scientific notation, we move the decimal
point behind the first nonzero digit.

93. Scientific Notation
To write a number in scientific notation, we move the decimal
point behind the first nonzero digit.
i. If the decimal point moves to the left N spaces, then the
exponent over 10 is positive N,

94. Scientific Notation
To write a number in scientific notation, we move the decimal
point behind the first nonzero digit.
i. If the decimal point moves to the left N spaces, then the
exponent over 10 is positive N, i.e. if after moving the
decimal point we get a smaller number A, then N is positive.

95. Scientific Notation
To write a number in scientific notation, we move the decimal
point behind the first nonzero digit.
i. If the decimal point moves to the left N spaces, then the
exponent over 10 is positive N, i.e. if after moving the
decimal point we get a smaller number A, then N is positive.
Example I. Write the following numbers in scientific
notation.
a. 12300.

96. Scientific Notation
To write a number in scientific notation, we move the decimal
point behind the first nonzero digit.
i. If the decimal point moves to the left N spaces, then the
exponent over 10 is positive N, i.e. if after moving the
decimal point we get a smaller number A, then N is positive.
Move left 4 places.
Example I. Write the following numbers in scientific
notation.
a. 12300. = 1 2300 .

97. Scientific Notation
To write a number in scientific notation, we move the decimal
point behind the first nonzero digit.
i. If the decimal point moves to the left N spaces, then the
exponent over 10 is positive N, i.e. if after moving the
decimal point we get a smaller number A, then N is positive.
Move left 4 places.
Example I. Write the following numbers in scientific
notation.
a. 12300. = 1 2300 . = 1. 23 x 10 +4

98. Scientific Notation
To write a number in scientific notation, we move the decimal
point behind the first nonzero digit.
i. If the decimal point moves to the left N spaces, then the
exponent over 10 is positive N, i.e. if after moving the
decimal point we get a smaller number A, then N is positive.
ii. If the decimal point moves to the right N spaces, then the
exponent over 10 is negative,
Move left 4 places.
Example I. Write the following numbers in scientific
notation.
a. 12300. = 1 2300 . = 1. 23 x 10 +4

99. Scientific Notation
To write a number in scientific notation, we move the decimal
point behind the first nonzero digit.
i. If the decimal point moves to the left N spaces, then the
exponent over 10 is positive N, i.e. if after moving the
decimal point we get a smaller number A, then N is positive.
ii. If the decimal point moves to the right N spaces, then the
exponent over 10 is negative, i.e. if after moving the decimal
point we get a larger number A, then N is negative.
Move left 4 places.
Example I. Write the following numbers in scientific
notation.
a. 12300. = 1 2300 . = 1. 23 x 10 +4

100. Scientific Notation
To write a number in scientific notation, we move the decimal
point behind the first nonzero digit.
i. If the decimal point moves to the left N spaces, then the
exponent over 10 is positive N, i.e. if after moving the
decimal point we get a smaller number A, then N is positive.
ii. If the decimal point moves to the right N spaces, then the
exponent over 10 is negative, i.e. if after moving the decimal
point we get a larger number A, then N is negative.
Move left 4 places.
Example I. Write the following numbers in scientific
notation.
a. 12300. = 1 2300 . = 1. 23 x 10 +4
b. 0.00123

101. Scientific Notation
To write a number in scientific notation, we move the decimal
point behind the first nonzero digit.
i. If the decimal point moves to the left N spaces, then the
exponent over 10 is positive N, i.e. if after moving the
decimal point we get a smaller number A, then N is positive.
ii. If the decimal point moves to the right N spaces, then the
exponent over 10 is negative, i.e. if after moving the decimal
point we get a larger number A, then N is negative.
Move left 4 places.
Move right 3 places
Example I. Write the following numbers in scientific
notation.
a. 12300. = 1 2300 . = 1. 23 x 10 +4
b. 0.00123 = 0. 001 23

102. Scientific Notation
To write a number in scientific notation, we move the decimal
point behind the first nonzero digit.
i. If the decimal point moves to the left N spaces, then the
exponent over 10 is positive N, i.e. if after moving the
decimal point we get a smaller number A, then N is positive.
ii. If the decimal point moves to the right N spaces, then the
exponent over 10 is negative, i.e. if after moving the decimal
point we get a larger number A, then N is negative.
Move left 4 places.
Move right 3 places
Example I. Write the following numbers in scientific
notation.
a. 12300. = 1 2300 . = 1. 23 x 10 +4
b. 0.00123 = 0. 001 23 = 1. 23 x 10 –3

103. Scientific Notation
To change a number in scientific notation back to the standard
form, we move the decimal point according to N.

104. Scientific Notation
To change a number in scientific notation back to the standard
form, we move the decimal point according to N.
i. If N is positive, move the decimal point in A to the right,

105. Scientific Notation
To change a number in scientific notation back to the standard
form, we move the decimal point according to N.
i. If N is positive, move the decimal point in A to the right,
i.e. make A into a larger number.

106. Scientific Notation
To change a number in scientific notation back to the standard
form, we move the decimal point according to N.
i. If N is positive, move the decimal point in A to the right,
i.e. make A into a larger number.
Example J. Write the following numbers in the standard form.
a. 1. 23 x 10 +4

107. Scientific Notation
To change a number in scientific notation back to the standard
form, we move the decimal point according to N.
i. If N is positive, move the decimal point in A to the right,
i.e. make A into a larger number.
Move right 4 places,
Example J. Write the following numbers in the standard form.
a. 1. 23 x 10 +4 = 1 2300 . = 12300.

108. Scientific Notation
To change a number in scientific notation back to the standard
form, we move the decimal point according to N.
i. If N is positive, move the decimal point in A to the right,
i.e. make A into a larger number.
ii. If N is negative, move the decimal point in A to the left,
Move right 4 places,
Example J. Write the following numbers in the standard form.
a. 1. 23 x 10 +4 = 1 2300 . = 12300.

109. Scientific Notation
To change a number in scientific notation back to the standard
form, we move the decimal point according to N.
i. If N is positive, move the decimal point in A to the right,
i.e. make A into a larger number.
ii. If N is negative, move the decimal point in A to the left,
i.e. make A into a smaller number.
Move right 4 places,
Example J. Write the following numbers in the standard form.
a. 1. 23 x 10 +4 = 1 2300 . = 12300.

110. Scientific Notation
To change a number in scientific notation back to the standard
form, we move the decimal point according to N.
i. If N is positive, move the decimal point in A to the right,
i.e. make A into a larger number.
ii. If N is negative, move the decimal point in A to the left,
i.e. make A into a smaller number.
Move right 4 places,
Example J. Write the following numbers in the standard form.
a. 1. 23 x 10 +4 = 1 2300 . = 12300.
b. 1. 23 x 10 –3

111. Scientific Notation
To change a number in scientific notation back to the standard
form, we move the decimal point according to N.
i. If N is positive, move the decimal point in A to the right,
i.e. make A into a larger number.
ii. If N is negative, move the decimal point in A to the left,
i.e. make A into a smaller number.
Move right 4 places,
Move left 3 places
Example J. Write the following numbers in the standard form.
a. 1. 23 x 10 +4 = 1 2300 . = 12300.
b. 1. 23 x 10 –3 = 0. 001 23 = 0.00123

112. Scientific Notation
To change a number in scientific notation back to the standard
form, we move the decimal point according to N.
i. If N is positive, move the decimal point in A to the right,
i.e. make A into a larger number.
ii. If N is negative, move the decimal point in A to the left,
i.e. make A into a smaller number.
Move right 4 places,
Move left 3 places
Example J. Write the following numbers in the standard form.
a. 1. 23 x 10 +4 = 1 2300 . = 12300.
b. 1. 23 x 10 –3 = 0. 001 23 = 0.00123
Scientific notation simplifies multiplication and division of
very large and very small numbers.

113. Example K. Calculate. Give the answer in both scientific
notation and the standard notation.
a. (1.2 x 108) x (1.3 x 10–12)
Scientific Notation

114. Example K. Calculate. Give the answer in both scientific
notation and the standard notation.
a. (1.2 x 108) x (1.3 x 10–12)
= 1.2 x 1.3 x 108 x 10 –12
Scientific Notation

115. Example K. Calculate. Give the answer in both scientific
notation and the standard notation.
a. (1.2 x 108) x (1.3 x 10–12)
= 1.2 x 1.3 x 108 x 10 –12
= 1.56 x 108 –12
Scientific Notation

116. Example K. Calculate. Give the answer in both scientific
notation and the standard notation.
a. (1.2 x 108) x (1.3 x 10–12)
= 1.2 x 1.3 x 108 x 10 –12
= 1.56 x 108 –12
= 1.56 x 10 –4
Scientific Notation

117. Example K. Calculate. Give the answer in both scientific
notation and the standard notation.
a. (1.2 x 108) x (1.3 x 10–12)
= 1.2 x 1.3 x 108 x 10 –12
= 1.56 x 108 –12
= 1.56 x 10 –4
= 0.000156
Scientific Notation

118. Example K. Calculate. Give the answer in both scientific
notation and the standard notation.
a. (1.2 x 108) x (1.3 x 10–12)
= 1.2 x 1.3 x 108 x 10 –12
= 1.56 x 108 –12
= 1.56 x 10 –4
= 0.000156
b.
6.3 x 10-2
2.1 x 10-10
Scientific Notation

119. Example K. Calculate. Give the answer in both scientific
notation and the standard notation.
a. (1.2 x 108) x (1.3 x 10–12)
= 1.2 x 1.3 x 108 x 10 –12
= 1.56 x 108 –12
= 1.56 x 10 –4
= 0.000156
b.
6.3 x 10-2
2.1 x 10-10
=
6.3
2.1
x 10 – 2 – ( – 10)
Scientific Notation

120. Example K. Calculate. Give the answer in both scientific
notation and the standard notation.
a. (1.2 x 108) x (1.3 x 10–12)
= 1.2 x 1.3 x 108 x 10 –12
= 1.56 x 108 –12
= 1.56 x 10 –4
= 0.000156
b.
6.3 x 10-2
2.1 x 10-10
=
6.3
2.1
x 10 – 2 – ( – 10)
= 3 x 108
Scientific Notation

121. Example K. Calculate. Give the answer in both scientific
notation and the standard notation.
a. (1.2 x 108) x (1.3 x 10–12)
= 1.2 x 1.3 x 108 x 10 –12
= 1.56 x 108 –12
= 1.56 x 10 –4
= 0.000156
b.
6.3 x 10-2
2.1 x 10-10
=
6.3
2.1
x 10 – 2 – ( – 10)
= 3 x 108
= 300,000,000
Scientific Notation

122. Example L. Convert each numbers into scientific notation.
Calculate the result. Give the answer in both scientific
notation and the standard notation.
Scientific Notation

123. Example L. Convert each numbers into scientific notation.
Calculate the result. Give the answer in both scientific
notation and the standard notation.
240,000,000 x 0.0000025
0.00015
Scientific Notation

124. Example L. Convert each numbers into scientific notation.
Calculate the result. Give the answer in both scientific
notation and the standard notation.
240,000,000 x 0.0000025
=
0.00015
2.4 x 108
Scientific Notation

125. Example L. Convert each numbers into scientific notation.
Calculate the result. Give the answer in both scientific
notation and the standard notation.
240,000,000 x 0.0000025
=
0.00015
2.4 x 108 x 2.5 x 10–6
Scientific Notation

126. Example L. Convert each numbers into scientific notation.
Calculate the result. Give the answer in both scientific
notation and the standard notation.
240,000,000 x 0.0000025
=
0.00015
2.4 x 108 x 2.5 x 10–6
1.5 x 10–4
Scientific Notation

127. Example L. Convert each numbers into scientific notation.
Calculate the result. Give the answer in both scientific
notation and the standard notation.
240,000,000 x 0.0000025
=
0.00015
2.4 x 108 x 2.5 x 10–6
1.5 x 10–4
=
2.4 x 2.5 x 108 x 10–6
1.5 x 10–4
Scientific Notation

128. Example L. Convert each numbers into scientific notation.
Calculate the result. Give the answer in both scientific
notation and the standard notation.
240,000,000 x 0.0000025
=
0.00015
2.4 x 108 x 2.5 x 10–6
1.5 x 10–4
=
2.4 x 2.5
1.5
x 10 8 + (–6) – ( – 4)
=
2.4 x 2.5 x 108 x 10–6
1.5 x 10–4
Scientific Notation

129. Example L. Convert each numbers into scientific notation.
Calculate the result. Give the answer in both scientific
notation and the standard notation.
240,000,000 x 0.0000025
=
0.00015
2.4 x 108 x 2.5 x 10–6
1.5 x 10–4
=
2.4 x 2.5
1.5
x 10 8 + (–6) – ( – 4)
=
2.4 x 2.5 x 108 x 10–6
1.5 x 10–4
= 4 x 108 – 6 + 4
Scientific Notation

130. Example L. Convert each numbers into scientific notation.
Calculate the result. Give the answer in both scientific
notation and the standard notation.
240,000,000 x 0.0000025
=
0.00015
2.4 x 108 x 2.5 x 10–6
1.5 x 10–4
=
2.4 x 2.5
1.5
x 10 8 + (–6) – ( – 4)
=
2.4 x 2.5 x 108 x 10–6
1.5 x 10–4
= 4 x 108 – 6 + 4
= 4 x 106 = 4,000,000
Scientific Notation

131. Ex. A. Write the numbers without the negative exponents and
compute the answers.
1. 2–1 2. –2–2 3. 2–3 4. (–3)–2 5. 3–3
6. 5–2 7. 4–3 8. 1
2
( )
–3
9. 2
3
( )
–1
10. 3
2
( )
–2
11. 2–1* 3–2 12. 2–2+ 3–1 13. 2* 4–1– 50 * 3–1
14. 32 * 6–1– 6 * 2–3 15. 2–2* 3–1 + 80 * 2–1
Ex. B. Combine the exponents. Leave the answers in positive
exponents–but do not reciprocate the negative exponents until
the final step.
16. x3x5 17. x–3x5 18. x3x–5 19. x–3x–5
20. x4y2x3y–4 21. y–3x–2 y–4x4 22. 22x–3xy2x32–5
23. 32y–152–2x5y2x–9 24. 42x252–3y–34 x–41y–11
25. x2(x3)5 26. (x–3)–5x –6 27. x4(x3y–5) –3y–8
Exponents

132. x–8
x–3
B. Combine the exponents. Leave the answers in positive
exponents–but do not reciprocate the negative exponents until
the final step.
28. x8
x–329.
x–8
x330. y6x–8
x–2y331.
x6x–2y–8
y–3x–5y232.
2–3x6y–8
2–5y–5x233.
3–2y2x4
2–3x3y–234.
4–1(x3y–2)–2
2–3(y–5x2)–135.
6–2 y2(x4y–3)–1
9–1(x3y–2)–4y236.
C. Combine the exponents as much as possible.
38. 232x 39. 3x+23x 40. ax–3ax+5
41. (b2)x+1b–x+3 42. e3e2x+1e–x
43. e3e2x+1e–x
44. How would you make sense of 23 ?
2