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Group Work: Exponents and Scientific Notation
We can use multiplication in place of repeated addition, and exponents in place of repeated
multiplication. Rewriting expressions in different forms can be a powerful tool for simplifying
expressions. Complete the following exercises on lined or scratch paper and hand in 1 per group
member!
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1. Is 35 the same as 3 · 5? Explain.
2. Exponents allow you to rewrite some multiplication problems in a simpler form. Some exponent
expressions can also be simplified. Complete the table below. Expand each expression into
factored form and then rewrite it with new exponents as shown in the example.
a. Work with your group to compare the bases and exponents of the original form to the base
and exponent of the simplified exponent form. Write a statement to describe the
relationships you see.
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3. Multiplying a number by 10 changes the number in a special way. Simplify each expression
below without using a calculator. As you work, pay attention to how the number changes when
you multiply it by powers of 10.
a. 9.23 · 10
b. 9.23 · 102
c. 9.23 · 103
d. 9.23 · 104
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4. Talk with your group about any patterns you see in your answers in problem 3. Based on those
patterns, what do you think 9.23 · 107 would be? Why?
a. Use the patterns you have found to find the product for each expression without
using your calculator.
i. 78.659 × l02
ii. 346.38 × l05
b. With your team, write a statement describing in general how you can quickly
multiply by powers of 10. You may want to include information about where the
decimal point moves after multiplying the number by a power of 10 or why you have
to add zeros. Work with your team to write a clear explanation for why this pattern
works.
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5. When astronomers describe distances in space, often the numbers are so large that they are
difficult to write. For example, the diameter of the sun is approximately one million, three hundred
ninety thousand kilometers, or 1,390,000 km. To make these large numbers easier to write,
astronomers and other scientists use scientific notation. In scientific notation, the diameter of the
sun is:
a. 1.39 × 106 km
b. Rewrite 106 as a single number without an exponent. What happens when you
multiply 1.39 by this number?
c. In scientific notation, the mass of the sun is approximately 1.99 × 1030 kg. What
does this number mean? Discuss with your team how to rewrite this number
without scientific notation, then write it.
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6. Scientific notation requires that one factor is a power of 10, and the other factor is a number
greater than or equal to 1 but less than 10. For example, 2.56 × 105 is correctly written in scientific
notation, but 25.6 × 104 is not. Scientific notation also uses the symbol “x” for multiplication
instead of “·” or parentheses. None of the numbers below is correctly written in scientific
notation. Explain why each one does not meet the criteria for scientific notation, and then write it
using correct scientific notation.
a. 25.6 × 104
b. 5.46 · 100
c. 0.93 × 108
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7. Write each number below in scientific notation.
a. 370,000,000
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b. 48,710,000,000
8. Scientific notation makes large numbers easier to write. It also provides you with quick
information about the sizes of the numbers. For example, Pluto and Haumea are both dwarf
planets. Pluto has a mass of 1.305 × 1022 kilograms and Haumea has a mass of 4.006 ×
l021 kilograms.
a. Without rewriting the mass of each planet, can you tell which of these dwarf planets is
larger? Explain your reasoning.
9. How do I turn a regular number into Scientific Notation Form??
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Step 1:
Step 2:
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Identify the number
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Place a decimal point in a location •
which would make a number between
330,000
3.30000
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Step 3:
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Step 4:
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1 and 9
Count the number of place values that•
the decimal point was moved.
Plug the numbers into the following •
formula:
5
3.30 x 105
Number(from step 2) X 10 place value
If the decimal point moves to the left…..the exponent will be positive
If the decimal point moves to the right…..the exponent will be negative
Ex1: 64,600
Step 1:
Step 2:
Step 3:
Step 4:
64, 600
6.46000
+4
6.46 x 10 4
Ex2: 32
Step 1:
Step 2:
Step 3:
Step 4:
32
3.2
+1
3.2 x 101
Ex3: .00122
Step 1:
Step 2:
Step 3:
Step 4:
.00122
1.22
-3
3.2 x 10-3
Ex4: .00123457145
Step 1:
Step 2:
Step 3:
Step 4:
.00123457145
1.23457145
3
1.45 x 10-3
Convert to Scientific Notation Form
a. 10 =
b. 1,000,000,000,000,000,000 =
c. 0.001 =
d. 0.000000001 =
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10. Multiplying Scientific Notation To multiply two numbers expressed in scientific notation,
simply multiply the numbers out front and add the exponents.
Generically speaking, this process is expressed as:
(n x 10a) • (m x 10b) = (n · m) x 10a+b
Ex1:
(5.1 x 104) • (2.5 x 103) = 12.75 x 107 Oops!!
This new answer is no longer in proper scientific notation.
Proper scientific notation is 1.275 x 108
Verify the given answers without a calculator. Show work! Then solve the given problems using a
calculator and verify that you understand how to get the correct answers. Solve the 3rd question by
hand and with a calculator.
Question
2(6 x 1023)
(3 x 103) • (6 x 1023)
400(6 x 1023)
Answer
1.2 x 1024
1.8 x 1027
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11. Dividing Scientific Notation To divide two numbers expressed in scientific notation, simply
divide the numbers out front and subtract the exponents. Generically speaking, this process is
expressed as:
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Ex2:
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Ex3:
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= 1.83 x 10-2
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Watch out for those negative exponents!!!
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Verify the given answers without a calculator. Show work! Then solve the given problems using a
calculator and verify that you understand how to get the correct answers. Solve the last question by
hand and with a calculator.
Question
12 x 1023
6 x 1023
6 x 1019
6 x 1023
18 x 1045
6 x 1023
3 x 1023
6 x 1023
Answer
2
1 x 10-4
3 x 1022