Scientific Notation

7.2. SCIENTIFIC NOTATION
425
The negative exponent means invert, so we can replace y −20 with 1/y 20 , then
multiply numerators and denominators.
1
1
·
32 y 20
1
=
32y 20
=
7.2
Scientific Notation
1. When the power of ten is a negative integer, it dictates the total number of
decimal places to use in expressing the number in decimal form. In the case of
10−4 , the exponent −4 tells us to use 4 decimal places. This requires that we
write a decimal point, 3 zeros, then the number 1.
10−4 = 0.0001
3. When the power of ten is a negative integer, it dictates the total number of
decimal places to use in expressing the number in decimal form. In the case of
10−8 , the exponent −8 tells us to use 8 decimal places. This requires that we
write a decimal point, 7 zeros, then the number 1.
10−8 = 0.00000001
5. When the power of ten is a whole number, it dictates the number of zeros
that you should add after the number 1. In the case of 108 , the exponent 8
tells us to write 8 zeros after the number 1.
108 = 100000000
Next, we delimit our answer with commas in the appropriate places.
108 = 100, 000, 000
7. When the power of ten is a whole number, it dictates the number of zeros
that you should add after the number 1. In the case of 107 , the exponent 7
tells us to write 7 zeros after the number 1.
107 = 10000000
Next, we delimit our answer with commas in the appropriate places.
107 = 10, 000, 000
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9. When multiplying by a power of ten, such as 10n , the exponent tells us how
many places to move the decimal point. If n is greater than or equal to zero
(nonegative), then we move the decimal point n places to the right. If n is less
than zero (negative), then we move the decimal point n places to the left. In
the case of 6506399.9 × 10−4, the exponent is negative, so we move the decimal
point 4 places to the left. Hence:
6506399.9 × 10−4 = 650.63999
11. When multiplying by a power of ten, such as 10n , the exponent tells us
how many places to move the decimal point. If n is greater than or equal to
zero (nonegative), then we move the decimal point n places to the right. If n is
less than zero (negative), then we move the decimal point n places to the left.
In the case of 3959.430928 × 102 , the exponent is nonnegative, so we move the
decimal point 2 places to the right. Hence:
3959.430928 × 102 = 395943.0928
13. When multiplying by a power of ten, such as 10n , the exponent tells us
how many places to move the decimal point. If n is greater than or equal to
zero (nonegative), then we move the decimal point n places to the right. If n
is less than zero (negative), then we move the decimal point n places to the
left. In the case of 440906.28 × 10−4 , the exponent is negative, so we move the
decimal point 4 places to the left. Hence:
440906.28 × 10−4 = 44.090628
15. When multiplying by a power of ten, such as 10n , the exponent tells us
how many places to move the decimal point. If n is greater than or equal to
zero (nonegative), then we move the decimal point n places to the right. If n is
less than zero (negative), then we move the decimal point n places to the left.
In the case of 849.855115 × 104 , the exponent is nonnegative, so we move the
decimal point 4 places to the right. Hence:
849.855115 × 104 = 8498551.15
17. Converting a number into scientifc notation requires that we convert the
given number into the form a × 10k , where k is an integer and 1 ≤ |a| < 10.
The requirement 1 ≤ |a| < 10 says that the magnitude of the number a must
be greater than or equal to 1, but strictly less than 10. This means that there
must be a single nonzero digit to the left of the decimal point. In the case of
the number 390000, we must move the decimal point in the number 390000 five
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7.2. SCIENTIFIC NOTATION
427
places to the left, then compensate by multiplying by multiplying by a power
of ten so that the result is still identical to the original number. That is:
390000 = 3.9 × 105
Check: To check the solution, recall that multiplying by 105 moves the decimal
point five places to the right. Hence:
3.9 × 105 = 390000
Thus, the solution checks.
19. Converting a number into scientifc notation requires that we convert the
given number into the form a × 10k , where k is an integer and 1 ≤ |a| < 10.
The requirement 1 ≤ |a| < 10 says that the magnitude of the number a must
be greater than or equal to 1, but strictly less than 10. This means that there
must be a single nonzero digit to the left of the decimal point. In the case of
the number 0.202, we must move the decimal point in the number 0.202 one
place to the right, then compensate by multiplying by multiplying by a power
of ten so that the result is still identical to the original number. That is:
0.202 = 2.02 × 10−1
Check: To check the solution, recall that multiplying by 10−1 moves the
decimal point one place to the left. Hence:
2.02 × 10−1 = 0.202
Thus, the solution checks.
21. Converting a number into scientifc notation requires that we convert the
given number into the form a × 10k , where k is an integer and 1 ≤ |a| < 10.
The requirement 1 ≤ |a| < 10 says that the magnitude of the number a must
be greater than or equal to 1, but strictly less than 10. This means that there
must be a single nonzero digit to the left of the decimal point. In the case
of the number 0.81, we must move the decimal point in the number 0.81 one
place to the right, then compensate by multiplying by multiplying by a power
of ten so that the result is still identical to the original number. That is:
0.81 = 8.1 × 10−1
Check: To check the solution, recall that multiplying by 10−1 moves the
decimal point one place to the left. Hence:
8.1 × 10−1 = 0.81
Thus, the solution checks.
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23. Converting a number into scientifc notation requires that we convert the
given number into the form a × 10k , where k is an integer and 1 ≤ |a| < 10.
The requirement 1 ≤ |a| < 10 says that the magnitude of the number a must
be greater than or equal to 1, but strictly less than 10. This means that there
must be a single nonzero digit to the left of the decimal point. In the case of the
number 0.0007264, we must move the decimal point in the number 0.0007264
four places to the right, then compensate by multiplying by multiplying by a
power of ten so that the result is still identical to the original number. That is:
0.0007264 = 7.264 × 10−4
Check: To check the solution, recall that multiplying by 10−4 moves the
decimal point four places to the left. Hence:
7.264 × 10−4 = 0.0007264
Thus, the solution checks.
25. Converting a number into scientifc notation requires that we convert the
given number into the form a × 10k , where k is an integer and 1 ≤ |a| < 10.
The requirement 1 ≤ |a| < 10 says that the magnitude of the number a must
be greater than or equal to 1, but strictly less than 10. This means that
there must be a single nonzero digit to the left of the decimal point. In the
case of the number 0.04264 × 10−4 , we’ll first convert the number 0.04264 into
scientific notation, ignoring 10−4 for a moment. To do that, we must move the
decimal point in the number 0.04264 two places to the right, then compensate
by multiplying by multiplying by a power of ten so that the result is still
identical to the original number. That is:
0.04264 × 10−4 = 4.264 × 10−2 × 10−4
To convert to a single power of 10, repeat the base and add the exponents.
= 4.264 × 10−2+(−4)
= 4.264 × 10−6
Thus, 0.04264 × 10−4 = 4.264 × 10−6 .
27. Converting a number into scientifc notation requires that we convert the
given number into the form a × 10k , where k is an integer and 1 ≤ |a| < 10.
The requirement 1 ≤ |a| < 10 says that the magnitude of the number a must
be greater than or equal to 1, but strictly less than 10. This means that
there must be a single nonzero digit to the left of the decimal point. In the
case of the number 130000 × 103 , we’ll first convert the number 130000 into
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scientific notation, ignoring 103 for a moment. To do that, we must move the
decimal point in the number 130000 five places to the left, then compensate by
multiplying by multiplying by a power of ten so that the result is still identical
to the original number. That is:
130000 × 103 = 1.3 × 105 × 103
To convert to a single power of 10, repeat the base and add the exponents.
= 1.3 × 105+3
= 1.3 × 108
Thus, 130000 × 103 = 1.3 × 108 .
29. Converting a number into scientifc notation requires that we convert the
given number into the form a × 10k , where k is an integer and 1 ≤ |a| < 10.
The requirement 1 ≤ |a| < 10 says that the magnitude of the number a must
be greater than or equal to 1, but strictly less than 10. This means that there
must be a single nonzero digit to the left of the decimal point. In the case of the
number 30.04×105, we’ll first convert the number 30.04 into scientific notation,
ignoring 105 for a moment. To do that, we must move the decimal point in
the number 30.04 one place to the left, then compensate by multiplying by
multiplying by a power of ten so that the result is still identical to the original
number. That is:
30.04 × 105 = 3.004 × 101 × 105
To convert to a single power of 10, repeat the base and add the exponents.
= 3.004 × 101+5
= 3.004 × 106
Thus, 30.04 × 105 = 3.004 × 106 .
31. Converting a number into scientifc notation requires that we convert the
given number into the form a × 10k , where k is an integer and 1 ≤ |a| < 10.
The requirement 1 ≤ |a| < 10 says that the magnitude of the number a must
be greater than or equal to 1, but strictly less than 10. This means that there
must be a single nonzero digit to the left of the decimal point. In the case of the
number 0.011×101, we’ll first convert the number 0.011 into scientific notation,
ignoring 101 for a moment. To do that, we must move the decimal point in
the number 0.011 two places to the right, then compensate by multiplying by
multiplying by a power of ten so that the result is still identical to the original
number. That is:
0.011 × 101 = 1.1 × 10−2 × 101
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To convert to a single power of 10, repeat the base and add the exponents.
= 1.1 × 10−2+1
= 1.1 × 10−1
Thus, 0.011 × 101 = 1.1 × 10−1 .
33. The notation 1.134E -1 is the calculator’s way of expressing scientific notation. That is, the notation 1.134E -1 is equivalent to the symbolism 1.134×10−1.
Because the power of ten is negative, we move the decimal point 1 place to the
left. Thus:
1.134E -1 = 1.134 × 10−1
= 0.1134
35. The notation 1.556E -2 is the calculator’s way of expressing scientific notation. That is, the notation 1.556E -2 is equivalent to the symbolism 1.556×10−2.
Because the power of ten is negative, we move the decimal point 2 places to
the left. Thus:
1.556E -2 = 1.556 × 10−2
= 0.01556
37. The notation 1.748E -4 is the calculator’s way of expressing scientific notation. That is, the notation 1.748E -4 is equivalent to the symbolism 1.748×10−4.
Because the power of ten is negative, we move the decimal point 4 places to
the left. Thus:
1.748E -4 = 1.748 × 10−4
= 0.0001748
39. We’ll use the approximations 2.5 ≈ 3 and 1.6 ≈ 2, which enable us to
write:
(2.5 × 10−1 )(1.6 × 10−7 ) ≈ (3 × 10−1 )(2 × 10−7 )
≈ 6 × 10−1+(−7)
≈ 6 × 10−8
Next, enter (2.5 × 10−1 )(1.6 × 10−7 ) as 2.5E-1*1.6E-7 on your calculator,
yielding the result shown in the following window.
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Hence, (2.5 × 10−1 )(1.6 × 10−7 ) = 4 × 10−8 .
41. We’ll use the approximations 1.4 ≈ 1 and 1.8 ≈ 2, which enable us to
write:
(1.4 × 107 )(1.8 × 10−4 ) ≈ (1 × 107 )(2 × 10−4 )
≈ 2 × 107+(−4)
≈ 2 × 103
Next, enter (1.4×107)(1.8×10−4 ) as 1.4E7*1.8E-4 on your calculator, yielding
the result shown in the following window.
Hence, (1.4 × 107 )(1.8 × 10−4 ) = 2.52 × 103 .
43. We’ll use the approximations 3.2 ≈ 3 and 2.5 ≈ 3, which enables us to
write:
3 × 10−5
3.2 × 10−5
≈
2.5 × 10−7
3 × 10−7
3 10−5
≈ · −7
3 10
≈ 1 · 10−5−(−7)
≈ 1 × 102
Push the MODE button, then highlight SCI mode and press ENTER. Move
your cursor to the same row containing the FLOAT command, then highlight
the number 2 and press ENTER. Press 2ND MODE to quit the MODE menu.
Next, enter (3.2 × 10−5 )/(2.5 × 10−7 ) as 3.2E-5/2.5E-7 on your calculator,
yielding the result shown in the following window.
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CHAPTER 7. RATIONAL FUNCTIONS
Hence, (3.2 × 10−5 )/(2.5 × 10−7 ) ≈ 1.28 × 102 . Don’t forget to return your
calculator to its original mode by selecting NORMAL and FLOAT in the
MODE menu.
45. We’ll use the approximations 5.9 ≈ 6 and 2.3 ≈ 2, which enables us to
write:
5.9 × 103
6 × 103
≈
2.3 × 105
2 × 105
6 103
≈ · 5
2 10
≈ 3 · 103−5
≈ 3 × 10−2
Push the MODE button, then highlight SCI mode and press ENTER. Move
your cursor to the same row containing the FLOAT command, then highlight
the number 2 and press ENTER. Press 2ND MODE to quit the MODE menu.
Next, enter (5.9 × 103)/(2.3 × 105) as 5.9E3/2.3E5 on your calculator, yielding
the result shown in the following window.
Hence, (5.9 × 103 )/(2.3 × 105 ) ≈ 2.57 × 10−2 . Don’t forget to return your
calculator to its original mode by selecting NORMAL and FLOAT in the
MODE menu.
47. We need to form the ratio of biomass to the mass of the Earth
6.8
1013
6.8 × 1013 kg
=
·
5.9736 × 1024 kg
5.9736 1024
≈ 1.14 × 10−11
Taking our number out of scientific notation we get 0.0000000000114. Changing
this to a percent by moving the decimal two places to the right we get that
the ratio of biomass to the mass of the Earth is about 0.00000000114%. Not
a very large portion of the mass of our planet. Kind of makes you feel a bit
insignificant.
Second Edition: 2012-2013