The document discusses scientific notation and powers of 10. It explains that in scientific notation, any number x can be written as the product of a number r and a power of 10, r × 10N, where 1 ≤ r < 10 and N is an integer. A positive exponent N shifts the decimal point of r to the right, while a negative N shifts it left. Converting between standard and scientific notation simply moves the decimal in r by the number of places indicated by N.

1. Scientific Notation
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2. Scientific Notation
An important application for exponents is the usage of the
powers of 10 in calculation of very large or very small numbers.

3. 100 = 1
Scientific Notation
An important application for exponents is the usage of the
powers of 10 in calculation of very large or very small numbers.
Powers of 10:
starting with

4. 100 = 1
101 = 10
Scientific Notation
An important application for exponents is the usage of the
powers of 10 in calculation of very large or very small numbers.
Powers of 10:
starting with

5. 100 = 1
101 = 10
102 = 100
Scientific Notation
An important application for exponents is the usage of the
powers of 10 in calculation of very large or very small numbers.
Powers of 10:
starting with

6. 100 = 1
101 = 10
102 = 100
103 = 1000
Scientific Notation
An important application for exponents is the usage of the
powers of 10 in calculation of very large or very small numbers.
Powers of 10:
starting with

7. 100 = 1
101 = 10
102 = 100
103 = 1000
Scientific Notation
An important application for exponents is the usage of the
powers of 10 in calculation of very large or very small numbers.
Powers of 10:
starting with
pack 0’s to the right
for positive exponents
so they get larger

8. 100 = 1
101 = 10
102 = 100
103 = 1000
10–1 = 0.1
Scientific Notation
An important application for exponents is the usage of the
powers of 10 in calculation of very large or very small numbers.
Powers of 10:
starting with
pack 0’s to the right
for positive exponents
so they get larger

9. 100 = 1
101 = 10
102 = 100
103 = 1000
10–1 = 0.1
10–2 = 0.01
Scientific Notation
An important application for exponents is the usage of the
powers of 10 in calculation of very large or very small numbers.
Powers of 10:
starting with
pack 0’s to the right
for positive exponents
so they get larger

10. 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
An important application for exponents is the usage of the
powers of 10 in calculation of very large or very small numbers.
Powers of 10:
starting with
pack 0’s to the right
for positive exponents
so they get larger

11. 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
An important application for exponents is the usage of the
powers of 10 in calculation of very large or very small numbers.
Powers of 10:
starting with
pack 0’s to the right
for positive exponents
so they get larger
pack 0’s to the left for
negative exponents
so they get smaller

12. 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
An important application for exponents is the usage of the
powers of 10 in calculation of very large or very small numbers.
Powers of 10:
starting with
pack 0’s to the right
for positive exponents
so they get larger
pack 0’s to the left for
negative exponents
so they get smaller
If r is a number then r x 10k = shifting the decimal point of r,

13. 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
An important application for exponents is the usage of the
powers of 10 in calculation of very large or very small numbers.
Powers of 10:
starting with
pack 0’s to the right
for positive exponents
so they get larger
pack 0’s to the left for
negative exponents
so they get smaller
If r is a number then r x 10k = shifting the decimal point of r,
if k is positive (+), shift the point right, if k is negative (–), shift left.

14. 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
An important application for exponents is the usage of the
powers of 10 in calculation of very large or very small numbers.
Powers of 10:
starting with
pack 0’s to the right
for positive exponents
so they get larger
pack 0’s to the left for
negative exponents
so they get smaller
If r is a number then r x 10k = shifting the decimal point of r,
if k is positive (+), shift the point right, if k is negative (–), shift left.
In particular, every number x can be written in the form
r x 10K with 1 ≤ r < 10.
This form is called the scientific notation of x.

15. 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
An important application for exponents is the usage of the
powers of 10 in calculation of very large or very small numbers.
Powers of 10:
starting with
pack 0’s to the right
for positive exponents
so they get larger
pack 0’s to the left for
negative exponents
so they get smaller
If r is a number then r x 10k = shifting the decimal point of r,
if k is positive (+), shift the point right, if k is negative (–), shift left.
In particular, every number x can be written in the form
r x 10K with 1 ≤ r < 10.
This form is called the scientific notation of x.
For example, 2500 = 2.5 x 103 and that 0.25 = 2.5 x 10–1.

16. Scientific Notation
Let's change a number in scientific notation r x 10K back to the
standard form by moving the decimal point of r according to N.

17. Scientific Notation
Let's change a number in scientific notation r x 10K back to the
standard form by moving the decimal point of r according to N.
i. If N is positive, move the decimal point of r to the right,
i.e. make r into a larger number.

18. Scientific Notation
Example B. Write the following numbers in the standard form.
a. 1. 23 x 10 +4
Let's change a number in scientific notation r x 10K back to the
standard form by moving the decimal point of r according to N.
i. If N is positive, move the decimal point of r to the right,
i.e. make r into a larger number.

19. Scientific Notation
Move right 4 places,
Example B. Write the following numbers in the standard form.
a. 1. 23 x 10 +4 = 1 2300 . = 12300.
Let's change a number in scientific notation r x 10K back to the
standard form by moving the decimal point of r according to N.
i. If N is positive, move the decimal point of r to the right,
i.e. make r into a larger number.

20. Scientific Notation
Move right 4 places,
Example B. Write the following numbers in the standard form.
a. 1. 23 x 10 +4 = 1 2300 . = 12300.
Let's change a number in scientific notation r x 10K back to the
standard form by moving the decimal point of r according to N.
i. If N is positive, move the decimal point of r to the right,
i.e. make r into a larger number.
ii. If N is negative, move the decimal point of r to the left,
i.e. make r into a smaller number.

21. Scientific Notation
Move right 4 places,
Example B. Write the following numbers in the standard form.
a. 1. 23 x 10 +4 = 1 2300 . = 12300.
b. 1. 23 x 10 –3
Let's change a number in scientific notation r x 10K back to the
standard form by moving the decimal point of r according to N.
i. If N is positive, move the decimal point of r to the right,
i.e. make r into a larger number.
ii. If N is negative, move the decimal point of r to the left,
i.e. make r into a smaller number.

22. Scientific Notation
Move right 4 places,
Move left 3 places
Example B. 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
Let's change a number in scientific notation r x 10K back to the
standard form by moving the decimal point of r according to N.
i. If N is positive, move the decimal point of r to the right,
i.e. make r into a larger number.
ii. If N is negative, move the decimal point of r to the left,
i.e. make r into a smaller number.

23. Scientific Notation
Move right 4 places,
Move left 3 places
Example B. 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
Let's change a number in scientific notation r x 10K back to the
standard form by moving the decimal point of r according to N.
i. If N is positive, move the decimal point of r to the right,
i.e. make r into a larger number.
ii. If N is negative, move the decimal point of r to the left,
i.e. make r into a smaller number.

24. Scientific Notation
Move right 4 places,
Move left 3 places
Example B. 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
To represent a number x with scientific notation as r x 10N,
first identify the r using x,
Let's change a number in scientific notation r x 10K back to the
standard form by moving the decimal point of r according to N.
i. If N is positive, move the decimal point of r to the right,
i.e. make r into a larger number.
ii. If N is negative, move the decimal point of r to the left,
i.e. make r into a smaller number.

25. Scientific Notation
Let's change a number in scientific notation r x 10K back to the
standard form by moving the decimal point of r according to N.
i. If N is positive, move the decimal point of r to the right,
i.e. make r into a larger number.
ii. If N is negative, move the decimal point of r to the left,
i.e. make r into a smaller number.
Move right 4 places,
Move left 3 places
Example B. 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
To represent a number x with scientific notation as r x 10N,
first identify the r using x, then multiply r by 10N to adjust the
decimal point of r to get back the x.

26. Scientific Notation
Let's change a number in scientific notation r x 10K back to the
standard form by moving the decimal point of r according to N.
i. If N is positive, move the decimal point of r to the right,
i.e. make r into a larger number.
ii. If N is negative, move the decimal point of r to the left,
i.e. make r into a smaller number.
Move right 4 places,
Move left 3 places
Example B. 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
To represent a number x with scientific notation as r x 10N,
first identify the r using x, then multiply r by 10N to adjust the
decimal point of r to get back the x. To find N, we count.

27. Scientific Notation
To express a given number x with scientific notation as r x 10N,
move the decimal point of x to the back of the first nonzero digit,
this is r.

28. Scientific Notation
To express a given number x with scientific notation as r x 10N,
move the decimal point of x to the back of the first nonzero digit,
this is r.
Example B. Write the following numbers in scientific notation.
a. 12300. .

29. Scientific Notation
To express a given number x with scientific notation as r x 10N,
move the decimal point of x to the back of the first nonzero digit,
this is r.
Move left 4 places.
Example B. Write the following numbers in scientific notation.
a. 12300. = 1 2300 . = 1. 23 x

30. Scientific Notation
To express a given number x with scientific notation as r x 10N,
move the decimal point of x to the back of the first nonzero digit,
this is r.
Move left 4 places.
Example B. Write the following numbers in scientific notation.
a. 12300. = 1 2300 . = 1. 23 x
r

31. Scientific Notation
To express a given number x with scientific notation as r x 10N,
move the decimal point of x to the back of the first nonzero digit,
this is r.
i. If the point moved left N spaces so the r is smaller than x,
Move left 4 places.
Example B. Write the following numbers in scientific notation.
a. 12300. = 1 2300 . = 1. 23 x
r

32. Scientific Notation
To express a given number x with scientific notation as r x 10N,
move the decimal point of x to the back of the first nonzero digit,
this is r.
i. If the point moved left N spaces so the r is smaller than x,
then use positive exponent N to compensate for the change.
Move left 4 places.
Example B. Write the following numbers in scientific notation.
a. 12300. = 1 2300 . = 1. 23 x
r

33. Scientific Notation
To express a given number x with scientific notation as r x 10N,
move the decimal point of x to the back of the first nonzero digit,
this is r.
i. If the point moved left N spaces so the r is smaller than x,
then use positive exponent N to compensate for the change.
Move left 4 places.
Example B. Write the following numbers in scientific notation.
a. 12300. = 1 2300 . = 1. 23 x 10 +4
r

34. Scientific Notation
To express a given number x with scientific notation as r x 10N,
move the decimal point of x to the back of the first nonzero digit,
this is r.
i. If the point moved left N spaces so the r is smaller than x,
then use positive exponent N to compensate for the change.
Move left 4 places.
Example B. Write the following numbers in scientific notation.
a. 12300. = 1 2300 . = 1. 23 x 10 +4
b. 0.00123.
r

35. Scientific Notation
To express a given number x with scientific notation as r x 10N,
move the decimal point of x to the back of the first nonzero digit,
this is r.
i. If the point moved left N spaces so the r is smaller than x,
then use positive exponent N to compensate for the change.
ii. If the point moved to the right N spaces so r is more than x,
Move left 4 places.
Example B. Write the following numbers in scientific notation.
a. 12300. = 1 2300 . = 1. 23 x 10 +4
b. 0.00123.
r

36. Scientific Notation
To express a given number x with scientific notation as r x 10N,
move the decimal point of x to the back of the first nonzero digit,
this is r.
i. If the point moved left N spaces so the r is smaller than x,
then use positive exponent N to compensate for the change.
ii. If the point moved to the right N spaces so r is more than x,
Move left 4 places.
Move right 3 places
Example B. 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
r
r

37. Scientific Notation
To express a given number x with scientific notation as r x 10N,
move the decimal point of x to the back of the first nonzero digit,
this is r.
i. If the point moved left N spaces so the r is smaller than x,
then use positive exponent N to compensate for the change.
ii. If the point moved to the right N spaces so r is more than x,
then use negative exponent N to compensate for the change.
Move left 4 places.
Move right 3 places
Example B. 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
r
r

38. Scientific Notation
To express a given number x with scientific notation as r x 10N,
move the decimal point of x to the back of the first nonzero digit,
this is r.
i. If the point moved left N spaces so the r is smaller than x,
then use positive exponent N to compensate for the change.
ii. If the point moved to the right N spaces so r is more than x,
then use negative exponent N to compensate for the change.
Move left 4 places.
Move right 3 places
Example B. 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
Scientific notation simplifies complicated calculation of
very large and very small numbers.
r
r

39. Example C. 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

40. Example C. 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

41. Example C. 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

42. Example C. 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

43. Example C. 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

44. Example C. 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

45. Example C. 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

46. Example C. 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

47. Example C. 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

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

49. Example D. 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

50. Example D. 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

51. Example D. 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

52. Example D. 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

53. Example D. 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

54. Example D. 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

55. Example D. 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

56. Example D. 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
For calculators, the 10N portion in scientific notation is displayed
as E+N or E–N where E means exponents.
Hence 2.5 x10–6 is displayed as 2.5 E–6 on the calculators.