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Name _______________________________________
HOLT
Date _________________ Class _________________
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TEACHING RESOURCES
4.
OPERATIONS WITH SCIENTIFIC NOTATION
In chemistry you will work with a variety of numbers that range from very large to
very small. Avogadro’s constant is an example of a very large number that you
will use. Avogadro’s constant is 602 213 674 000 000 000 000 000. In scientific
notation this number is written as 6.022 136 74 3 1023. As you can see, scientific
notation makes writing this number easier. The mass of an electron is an example
of a very small number that is used in chemistry. The mass of an electron is
0.000 000 000 000 000 000 000 000 000 000 910 9 kg. It is much easier to write
9.109 3 10 231 kg.
EXAMPLE 1
Writing Numbers in Scientific Notation
Write 157 000 g in scientific notation.
Move the decimal to the left or right until there is only one digit to the left of
the decimal.
1 5 7 0 0 0. : 1.570 00
Determine the power of 10 by counting the number of places you moved the
decimal. If you moved the decimal to the left, the exponent is positive. If you
moved the decimal to the right, the exponent is negative.
1 5 7 0 0 0. : 1.570 00 3 105
5 4 3 2 1
Determine the units, and place them after the number. Units do not change
when a number is converted to scientific notation.
1.570 00 3 105 g
PRACTICE
1. Work through the following problem.
What is 0.000 837 mm in scientific notation?
a. Move the decimal to the left or right until there is only one digit to the left
of the decimal.
0 . 0 0 0 8 3 7 : ____________________
b. Determine the power of 10 by counting the number of places you moved
the decimal. If you moved the decimal to the left, the exponent is positive.
If you moved the decimal to the right, the exponent is negative.
0 . 0 0 0 8 3 7 : _________________ 3 10 ____
c. Determine the units, and place them after the number.
2. Rewrite the following values in scientific notation.
a. 530 000 L 5 ________________________
b. 0.000 53 L 5 ________________________
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4. OPERATIONS WITH SCIENTIFIC NOTATION continued
c. 0.000 000 92 g 5 ____________________________
d. 6 000 000 000 mL 5 _________________________
e. 12 600 000 000 mg 5 ________________________
3. Rewrite the following expressions in decimal form.
a. 1.87 3 1025 km 5 ___________________________
b. 5.99 3 107 mL 5 ____________________________
c. 9.001 3 103 mg 5 ___________________________
d. 6.98 3 105 g 5 _____________________________
e. 4.44 3 1025 kg 5 ___________________________
SIGNIFICANT FIGURES AND SCIENTIFIC NOTATION
When a number is written in scientific notation, all of the significant figures are
represented in the first factor so that there is no confusion about which zeros are
significant and which are not. Look at the following examples for expressing data
in scientific notation with the correct number of significant figures.
6800 5 6.8 3 103
(If both zeros are not significant, there are two significant
figures.)
6800 5 6.80 3 103
(If one zero is significant, there are three significant
figures.)
28 000 5 2.8000 3 104 (If all the zeros are significant, there are five significant
figures.)
When performing calculations in scientific notation, you must often round numbers to obtain the correct number of significant figures. Review the following rules
for rounding to the correct number of significant figures.
If the digit immediately to the
right of the last significant figure
you want to retain is
Example
(rounded to three significant
figures)
• greater than 5, then increase the
76 570 L 5 7.66 3 104 L
last digit by 1.
• less than 5, then do not change the
2574 g 5 2.57 3 103 g
last digit.
• 5, followed by nonzero digit(s), then
9 425 100 s 5 9.43 3 107 s
increase the last digit by 1.
• 5, not followed by nonzero digit(s),
and preceded by an odd digit, then
increase the last digit by 1.
• 5, not followed by nonzero digit(s),
and the preceding significant digit
is even, then do not change the last
digit.
2
TEACHING RESOURCES: Math Skills
56 350 m 5 5.64 3 104 m
(because 3 is odd)
246 500 cm 5 2.46 3 105 cm
(because 6 is even)
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4. OPERATIONS WITH SCIENTIFIC NOTATION continued
PRACTICE
4. Write each of the following in scientific notation with the indicated number
of significant figures. Round where appropriate.
a. 525 mL (one significant figure) ________________________________
b. 7527 kg (three significant figures) ______________________________
c. 0.045 12 m (two significant figures) _____________________________
d. 0.000 676 6 cm (three significant figures) ________________________
e. 268.5 mg (four significant figures) ______________________________
CALCULATIONS USING SCIENTIFIC NOTATION
When adding or subtracting numbers in scientific notation, all of the powers of 10
must be the same magnitude. When multiplying powers of 10, the exponents are
added together.
100 3 1000 5 100 000
102 3 103 5 105
10213 5 105
When dividing powers of 10, the exponents are subtracted.
100
1
5
1000
10
5 0.1
102
5 10223 5 1021
103
Following are some examples that demonstrate how to work through calculations
with numbers in scientific notation.
EXAMPLE 2
Adding and Subtracting Numbers in Scientific Notation
2.51 3 104 m 1 1.61 3 103 m 5 _____________
All of the powers of 10 must be the same magnitude. The power of 10 in
2.51 3 104 is 4. The power of 10 in 1.61 3 103 is 3. To make the power of 10 equal
to 4 in both numbers, move the decimal one place to the left in 1.61 3 103.
1.61 3 103 : 0.161 3 104
Notice that these two numbers have the same value in decimal form.
Add or subtract the first factors. In this case, we are adding.
2.51 1 0.161 5 2.671
Attach the second factor to the number.
2.671 3 104
Determine the number of significant figures, and round if necessary. Be sure
all the numbers have the same power of 10 when you find the leftmost uncertain
digit. The two numbers used to determine the number of significant figures in the
answer in this problem are 2.51 3 104 and 0.161 3 104 (not 1.61 3 103).
2.67 3 104
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4. OPERATIONS WITH SCIENTIFIC NOTATION continued
If the first factor is greater than 10 or less than 1, move the decimal and
change the power of 10. This number does not need to be changed.
Determine the units, and place them after the number. Notice that the units are
the same for both numbers. In order for two numbers to be added or subtracted,
they must have the same units. The units do not change during addition or
subtraction.
2.67 3 104 m
PRACTICE
5. Work through the following problem.
9.53 3 105 kg 2 9.11 3 105 kg 5 _______________
a. All of the powers of 10 must be the same magnitude.
b. Add or subtract the first factors.
c. Attach the second factor to the number.
d. Determine the number of significant figures, and round if necessary.
e. If the first factor is greater than 10 or less than 1, move the decimal and
change the power of 10.
f. Determine the units, and place them after the number.
4
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4. OPERATIONS WITH SCIENTIFIC NOTATION continued
6. Solve the following problems. Write the answers with the correct number of
significant figures, and round if necessary.
a. (2.3 3 103 g) 1 (3.5 3 103 g) 5 ________________________________
b. (2.3 3 103 cm) 1 (3.5 3 102 cm) 5 _____________________________
c. (7.88 3 104 mol) 2 (1.55 3 104 mol) 5 __________________________
d. (5.67 3 105 kg) 2 (8.91 3 102 kg) 5 ____________________________
e. (4.72 3 1025 mg) 1 (4.66 3 1025 mg) 5 ________________________
f. (6.79 3 103 L) 1 (6.79 3 102 L) 5 ______________________________
g. (6.3 3 107 km) 1 (7.8 3 107 km) 5 _____________________________
h. (6.23 3 1022 m) 2 (6.01 3 1022 m) 5 __________________________
EXAMPLE 3
Multiplying and Dividing in Scientific Notation
(2.3 3 103 m)(5.7 3 104 m) 5 _______________
When multiplying, add the powers of 10. When dividing, subtract the powers
of 10. In this case, we are multiplying.
103 3 104 5 103 4 5 107
1
Multiply or divide the first factors. In this case, we are multiplying.
2.3 3 5.7 5 13.11
Attach the second factor to the number.
13.11 3 107
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4. OPERATIONS WITH SCIENTIFIC NOTATION continued
Determine the number of significant figures, and round if necessary.
13. 3 107
If the first factor is greater than 10 or less than 1, move the decimal and
change the power of 10.
13. 3 107 : 1.3 3 108
Determine the units, and place them after the number. Because m 3 m 5 m2,
the unit is m2. The answer is 1.3 3 108 m2.
PRACTICE
7. Work through the following problem.
5.22 3 106 g
5 _____________
2.9 3 102 L
a. When multiplying, add the powers of 10. When dividing, subtract the
powers of 10.
b. Multiply or divide the first factors.
c. Attach the second factor to the number.
d. Determine the number of significant figures, and round if necessary.
e. If the first factor is greater than 10 or less than 1, move the decimal and
change the power of 10.
f. Determine the units, and place them after the number.
6
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4. OPERATIONS WITH SCIENTIFIC NOTATION continued
8. Solve the following problems. Write the answers with the correct number of
significant figures, and round if necessary.
a. (4.7 3 104 km) 3 (1.1 3 102 km) 5 ____________________________
b. (1.17 3 102 m) 3 (8.49 3 103 m) 5 ____________________________
c. (4.5 3 103 M) 3 (1.5 3 1022 M) 5 ____________________________
d. (2.68 3 105 cm) 3 (4.11 3 1022 cm) 5 _________________________
e. (6.1 3 1024 mol/L) 3 (2.4 3 1023 L) 5 ________________________
f.
5.2 3 105 kg
5 ______________________________
1.3 3 102 kg
g.
3.21 3 104 mg
5 ____________________________
5.33 3 104 mg
h.
6.79 3 1025 kg
5 ____________________________
1.7 3 102 cm3
i.
4.34 3 106 L
5 __________________________
2.01 3 1023 min
j.
7.37 3 1023 g
5 ________________________
2.43 3 1025 g/mol
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4. OPERATIONS WITH SCIENTIFIC NOTATION continued
SCIENTIFIC NOTATION ON A CALCULATOR
Many scientific calculators can perform calculations and give answers in scientific
notation. When entering scientific notation on a calculator, enter only the first factor
and the exponent of the power of 10. The exponent entry key is usually labeled
or
. This automatically includes the 10. When performing calculations,
most scientific calculators will display the result in scientific notation when there
are too many digits for the display. However, you must be careful because the calculator will not determine the number of significant figures. It will also cut off any
digits that will not fit in the display.
EXAMPLE 4
Solving Problems in Scientific Notation with a Calculator
Calculate (2.50 3 104 )(2.51 3 105) on your calculator.
Press
.
or
, then press
. (This is the exponent of 104.)
Press
.
Press
.
Press
or
, then press
. (This is the exponent of 105.)
Press the
6.275 09 .
Press
, and the calculator will display
9
This is equivalent to 6.275 3 10 .
Round the answer to three significant figures. The answer is 6.28 3 109.
PRACTICE
9. The mass of an electron is 9.109 3 10231 kg. The mass of a proton is
1.673 3 10227 kg. The mass of a neutron is 1.675 3 10227 kg.
a. Which has more mass, an electron or a proton?
b. How much more mass does a neutron have than a proton?
c. A hydrogen atom is made of one electron and one proton. What is the
mass of one hydrogen atom?
d. Tritium is a type of hydrogen that has one electron, one proton, and two
neutrons. What is the mass of one tritium atom?
e. Helium has two protons, two neutrons, and two electrons. What is the mass
of one helium atom?
f. Which has more mass, a helium atom or a tritium atom?
g. What is the combined mass of one helium atom and one tritium atom?
8
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4. OPERATIONS WITH SCIENTIFIC NOTATION continued
10. Some solid ionic compounds dissolve easily in water, while others dissolve
only very slightly. When a compound only slightly dissolves, the amount that
will dissolve in water is referred to as the solubility-product constant. To find
the solubility product of a compound, multiply the concentration of each ion.
The equation for the solubility product for silver chloride, AgCl, is as follows.
Ksp 5 [Ag1][Cl2]
If [Ag+] = 1.33 3 1025 and [Cl2] = 1.33 3 10–5, what is the Ksp for silver
chloride?
Answer
11. Assume Avogadro’s constant is 6.02 3 1023. When you have that number of
molecules of glucose, C6H12O6 , you have a quantity called a mole. If 1 mole
of glucose has a mass of 180. g, find the mass in grams of 4.23 3 1022 molecules of glucose by working through the following problem.
mass (g)
180. g
5
22
4.23 3 10 molecules
6.02 3 1023 molecules
mass (g) 5
(4.23 3 1022 molecules)(180. g)
6.02 3 1023 molecules
Answer
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ANSWER KEY
4.
TEACHING RESOURCES
Operations with Scientific Notation
1.
2.
a. 8.37
f. 7.47 3 103 L
b. 8.37 3 1024
g. 1.4 3 108 km
c. 8.37 3 1024 mm
h. 2.20 3 1023 m
a. 5.3 3 105 L
b. 5.3 3 1024 L
27
c. 9.2 3 10
3.
7.
b. 1.8
c. 1.8 3 104
g
d. 6 3 109 mL
d. 1.8 3 104 (no change)
e. 1.26 3 1010 mg
e. 1.8 3 104 (no change)
a. 0.000 018 7 km
f. 1.8 3 104 g/L
b. 59 900 000 mL
4.
8.
b. 9.93 3 105 m2
d. 698 000 g
c. 6.8 3 101 M2
e. 0.000 044 4 kg
d. 1.10 3 104 cm2
e. 1.5 3 1026 mol
a. 5 3 102 mL
f. 4.0 3 103
g. 6.02 3 1021
c. 4.5 3 1022 m
h. 4.0 3 1027 kg/cm3
d. 6.77 3 1024 cm
i. 2.16 3 109 L/min
e. 2.685 3 102 mg
j. 3.03 3 102 mol
a. They are already the same.
b. 0.42
9.
c. 0.42 3 105
e. 6.698 3 10227 kg
f. 4.2 3 104 kg
f. a helium atom
a. 5.8 3 103 g
g. 1.172 3 10226 kg
b. 2.7 3 10 cm
3
c. 6.33 3 104 mol
10.
1.77 3 10210
11.
1.26 3 101 g
d. 5.66 3 105 kg
e. 9.38 3 10
b. 2.000 3 10230 kg
d. 5.024 3 10227 kg
e. 4.2 3 104
25
a. proton
c. 1.674 3 10227 kg
d. 0.42 3 105 (no change)
6.
a. 5.2 3 106 km2
c. 9001 mg
b. 7.53 3 103 kg
5.
a. 104
mg
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