Date ___________ Hr ___ Name______________________________________________
WS 1.1 Scientific Notation (Powers of Ten)
Scientific notation is a method to express very large or very small numbers in a compact way. To
write a number in scientific notation use a:
Number ( ≥ 1 but < 10 ) X 10Power ( power = pos, neg, or 0)
Number
Scientific Notation
1 0 0.0
5 5 ,0 0 0
SAMPLE PROBLEM – Positive Exponents
A single railroad engine pulled 250 freight cars on the Erie Railroad from May 1914 until 1929.
The mass pulled by the engine was 15,545,000 kg. Express this value for mass in scientific
notation.
SOLUTION - Move the decimal point to the left until the number is ≥ 1 but < 10 (only one digit
remains to the left of the decimal point). Count the number of places the decimal point was moved. To
change 15,545,000 to 1.554 5000, the decimal point is moved seven places to the left. When the
decimal point is moved to the left, the exponent is positive. The zeros at the end of the number can be
dropped from the answer.
mass =1.5545 x 107 kg
Rewrite in Scientific Notation
1. 1,000
2. 147,000
3. 12,300
4. 1,800,000,000
Rewrite as a Regular Number
5. 1 .0 x 105
6. 6.32 x 108
7. 7.74 x 103
8. 2.22 x 106
PRACTICE
9. Scientists have estimated that the area
of Earth covered by water is 70.98%,
or 362,031,100 km2.
Rewrite this area in scientific notation.
notation.
10. The brightest comet on record,
the Great Comet of 1843, had a tail that
trailed for 205,000,000 mi.
Rewrite the tail length in scientific
11. Mount Everest, an eastern Himalayan
peak on the Tibet-Nepal border, was discovered
to be the world’s highest mountain in March
1856. The Survey Department of the Government
of India computed its height to be 2.9 x 104 ft.
Rewrite this height as a number.
12. The speed of light is the fastest
motion possible. For light travelling in
the a vacuum speed is 3.0 x 108 m/s.
Rewrite this speed as a number.
SAMPLE PROBLEM – Negative Exponents
The diameter of the hydrogen atom has been measured to be about 0.00000001 m. Express this
diameter in scientific notation.
SOLUTION - Move the decimal point to the right until the number is ≥ 1 but < 10 (first nonzero digit
is to the left of the decimal point). Count the number of places the decimal point was moved. To
change 0.000 00001 to 1.0, the decimal point is moved eight places to the right. When the decimal
point is moved to the right, the exponent is negative.
d = 1.0 x 10–8 m
Rewrite in Scientific Notation
13. 0.001
14. 0.000035
15. 0.0579
16. 0.000000000992
Rewrite as a Regular Number
17. 1 .0 x 10-2
18. 6.32 x 10-6
19. 7.74 x 10-3
20. 2.22 x 10-8
PRACTICE
21. In 1981 the IBM Zurich research laboratory
invented the scanning tunneling microscope (STM).
It has a magnifying ability of 100 million with
resolution capability down to 0.0000000002 m,
about the diameter of a sulfur atom.
Rewrite the diameter in scientific notation.
22. One of the smallest of all free-living
organisms, Mycoplasma laidlawii, was
first discovered in sewage in 1936.
During its early existence, its diameter
can be as small as only 0.0000001 m.
Rewrite this diameter in scientific notation.
23. The mass of the smallest bacterium is about
2 x 10-15 grams.
Rewrite this mass as a number.
24. The density of helium is
1.78 x 10-4 g/cm3.
Rewrite helium’s density without using
scientific notation.
25. Although you cannot see it, the lights in a
room flicker on and off several times a second.
The flickering results from the electricity changing
direction rapidly, causing the lights to brighten
and dim repeatedly. This cycle occurs once
every 0.0166 s.
Rewrite this time interval in scientific notation.
26. The speed with which the shutter of
a camera opens and shuts can be changed
on certain models of cameras. On most
35 mm cameras, the fastest shutter speed
is 0.001 s.
Rewrite this time in scientific notation.
Using Scientific Notation
Do the following problems using your calculator. The keys used to enter a power of ten into your
calculator will vary depending on the company and model of your calculator. The following are
typical for most calculators models:
Example: 1,700,000 x 350,000,000 = 1.7 x 106 x 3.5x 108
TI (Texas Instruments)
Note: If your EE button is yellow or blue, and above another
key, you must push the yellow 2nd function button first to make the EE active
You type in : 1.7 EE 6 x 3.5 EE 8 =
Your display : 1.7E6 x 3.5E8
Casio, Sanyo, etc......
You type in :
Your display :
1.7 EXP 6 x 3.5 EXP 8 =
1.7E6 x 3.5E8
OR
1.7
06
x 3.5 08
PROBLEM
The Concorde, a supersonic passenger jet that flies from the East Coast of the United States to
London and Paris, must fly at the speed of a regular jet while flying over land. This lower speed is
used to avoid problems associated with the sonic boom of supersonic jets. Once over the Atlantic
Ocean, the Concorde can increase its speed. If the Concorde flies at 8.85 x 102 km/h over land and
2.300 x 103 km/h over sea, what is the difference in these speeds?
SOLUTION
Step 1: Write the Formula for
the in difference in speeds
Step 2: Plug the known values
into the equation.
Step 3: Do the math (Chug).
Difference in speeds = sea speed – land speed
Difference in speeds = (2.300 x 103 km/h) – (8.85 x 102 km/h)
Use calculator OR
Rewrite either value so that the exponent 2.300 x 103 km/h = 23.00 x 102 km/h
term of both numbers is the same.
Can only subtracts same exponents.
Difference in speeds = (23.00 – 8.85) x 102 km/h
Move the decimal point to the left
until the number is ≥ 1 but < 10.
Difference in speeds = 14.15 x 102 km/h
Difference in speeds = 1.415 x 103 km/h
PRACTICE
27. 2.300 x 103 - 8.85 x 102 =
28. 4.5 x 104 x 7.7 x 107 =
29. 2.54 x 102 / 1.93 x 101 =
30. 7.35 x 10-8 + 4.703 x 10-6=
Put all numbers and answers in scientific notation.
31. The heaviest commonly used United States coin is the half dollar, which has a mass of 11.340 g.
The lightest United States coin is the dime, which has a mass of 2.268 g. Report the sum of these
masses in scientific notation.
32. A signal is transmitted between four microwave signal towers. The distances between the
towers is 2.50 km, 2.500x 101 km, and 5.0 x 10–1 km. What is the total distance traveled by the
signal?
33. The adult house fly lives for only about 1 month, or 8 x 10-2 y. The oldest recorded age of a
tortoise was 1.88 x 102 y. What is the difference, in scientific notation, between these two ages?
34. The ink needed to dot an i in this book has a mass of around 0.0000000011 kg. What is the
mass of ink needed to dot all of the i’s in this problem?
35. The Republic of China presented one of the world’s largest flags to the city of Kaohsiung in
1989. The flag of the Republic of China measured 1.26 x 104 cm by 8.40 x 103 cm. What is the
area of this flag in square centimeters?