Name:
SOLUTIONS
Astronomy 110: Homework 1
Scientific Notation Basics
106
• How do you write the number 1,000,000 using scientific notation? _________
109
• How do you write the number 1,000,000,000 using scientific notation? _________
1012
• How do you write the number 1,000,000,000,000 using scientific notation? _________
1,000
• How many times larger is 1 billion than 1 million? And 1 trillion than 1 billion? _________
1,000
• How many times larger is the number 1024 than the number 1021? _________
• Roughly position the numbers 10-6, 109, 10-12, and 1024 on the following number line:
–
10-12
0
109
10-6
1024
+
Doing Calculations Using Scientific Notation
Now that we know how to represent large (and small) numbers using scientific notation, it’s time to learn a
little about making calculations with powers of 10. There are two “rules” to know:
• Multiplying two powers of ten: 10A • 10B = 10A+B
• Dividing two powers of ten: 10A / 10B = 10A–B
“A” and “B” here are simply symbols that can represent any numbers. For example, 105 • 103 = 105+3 = 108.
And 108 / 105 = 103. Do you see why these “rules” work? Here are a few problems that should help you see
that they must be true. Remember, 10 = 101.
101 x 102 = 103 → 1+2=3
What does 10 x 100 = 1,000 look like when written in scientific notation? __________________________
102 x 102 = 104 → 2+2=4
What does 100 x 100 = 10,000 look like when written in scientific notation? __________________________
103 x 106 = 109 → 3+6=9
What does 1,000 x 1,000,000 = 1,000,000,000 look like? __________________________
102 / 101 = 101 → 2–1=1
How about 100 / 10 = 10? __________________________
104 / 102 = 102 → 4–2=2
And 10,000 / 100 = 100? __________________________
109 / 103 = 106 → 9–3=6
And finally 1,000,000,000 / 1,000 = 1,000,000? __________________________
Hopefully!
Does this make sense? __________________________
(If not, look at it again.)
Real calculations don’t just involve numbers that are powers of 10. In other words, you won’t usually see a
problem as simple as 107 • 106, but instead something like (3.1x107 • 1.4x106) / 5.7x109. Note that the “x” and
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“•” in the previous expression both mean “multiplication.” What do you do in situations like this? The most
convenient way to do it is usually to separate out the powers of 10 from everything else, like this:
(3.1x107 • 1.4x106) / 5.7x109 = [(3.1 • 1.4) / 5.7] • [(107 • 106) / 109]
We can then do the parts in square brackets separately, to get
[(3.1 • 1.4) / 5.7] • [(107 • 106) / 109] = 0.76 x 104
This answer is correct, but it’s nice to write the first number here as a number that’s greater than 1 but less
than 10. If you think about it (or if you multiply the first number by 10 and divide the second number by 10),
you’ll see that the above is equivalent to 7.6x103 = 7,600. That’s not so terrible, right? OK, now your turn.
What’s (4.2x106 / 7.5x109) • 3.3x1010?
(4.2x106 / 7.5x109) • 3.3x1010 = [(4.2 / 7.5) • 3.3] • [(106 / 109) • 1010] ≈ 1.8 x 107
How about 4.2x106 / (7.5x109 • 3.3x1010)?
4.2x106 / (7.5x109 • 3.3x1010) = [4.2 / (7.5 • 3.3)] • [106 / (109 • 1010)] ≈ 1.7 x 10-14
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