Converting from Scientific Notation

Show your work for each problem.

1.
What is $5 \times 10^3$ as an ordinary number?
1. a) $500$
2. b) $50$
3. c) $0.005$
4. d) $5000$
Answer: $5000$
$10^3 = 1000$ , so $5 \times 10^3 = 5 \times 1000 = 5000$ : the point moves $3$ places right and three zeros fill the places.
2.
Write $7 \times 10^2$ as an ordinary number.
Answer: 700
$7 \times 10^2 = 7 \times 100 = 700$ . The point moves $2$ places right.
3.
Which number is equal to $4 \times 10^{-2}$ ?
1. a) $0.004$
2. b) $0.04$
3. c) $400$
4. d) $0.4$
Answer: $0.04$
Move the point $2$ places left: $4 \to 0.4 \to 0.04$ . So $4 \times 10^{-2} = 0.04$ (four hundredths).
4.
Write $3 \times 10^{-1}$ as a decimal number.
Answer: 0.3
Move the point $1$ place left: $3 \times 10^{-1} = 0.3$ . No zero is needed between the point and the $3$ .
5.
Write $6.1 \times 10^4$ as an ordinary number.
Answer: 61000
1. How many places does the decimal point move to the right? 4
2. How many digits does $6.1$ have after the decimal point? 1
3. How many zeros do you add after $61$ ? 3
4. Write the ordinary number. 61000
6.
Write $9 \times 10^{-3}$ as a decimal number.
Answer: $0.009$
1. How many places does the decimal point move to the left? 3
2. How many zeros stand between the decimal point and the $9$ ? 2
3. Write the decimal number. 0.009
7.
A calculator shows 2.5E-7. Which number is that?
1. a) $0.0000025$
2. b) $0.00000025$
3. c) $-25{,}000{,}000$
4. d) $25{,}000{,}000$
Answer: $0.00000025$
$2.5 \times 10^{-7}$ : move the point $7$ places left, so $7 - 1 = 6$ zeros come before $25$ : $0.00000025$ . It is positive and tiny, not a large or negative number.
8.
Write $2.75 \times 10^5$ as an ordinary number.
Answer: 275000
Move the point $5$ places right: $275$ uses $2$ of them, $3$ zeros fill the rest: $275{,}000$ .
9.
Write $5.6 \times 10^{-5}$ as a decimal number.
Answer: 0.000056
$5.6 \to 0.56 \to 0.056 \to 0.0056 \to 0.00056 \to 0.000056$ . So $5.6 \times 10^{-5} = 0.000056$ .
10.
The average distance from Mars to the Sun is $2.28 \times 10^8$ km. Write it as an ordinary number of kilometers.
Answer: 228000000
1. How many places does the decimal point move to the right? 8
2. How many digits does $2.28$ have after the decimal point? 2
3. How many zeros follow the digits $228$ ? 6
4. What is the distance in km as an ordinary number? 228000000
11.
Which number is equal to $5.06 \times 10^{-4}$ ?
1. a) $0.0000506$
2. b) $0.00506$
3. c) $5060$
4. d) $0.000506$
Answer: $0.000506$
Move the point $4$ places left: $5.06 \to 0.506 \to 0.0506 \to 0.00506 \to 0.000506$ . The zero inside $506$ stays where it is.
12.
A human hair is about $8 \times 10^{-5}$ m thick. Write the thickness as a decimal number of meters.
Answer: $0.00008$
1. How many places does the decimal point move to the left? 5
2. How many zeros stand between the decimal point and the $8$ ? 4
3. What is the thickness in meters? 0.00008
13.
A spreadsheet shows 4.05E7 as the total of a column. Write the total as an ordinary number.
Answer: 40500000
1. What is the exponent of 10? 7
2. How many zeros follow the digits $405$ ? 5
3. Write the total as an ordinary number. 40500000
14.
Write $9.99 \times 10^{-7}$ as a decimal number.
Answer: 0.000000999
Move the point $7$ places left. Six zeros come after the point, then the digits $999$ : $9.99 \times 10^{-7} = 0.000000999$ .
15.
Which conversion is correct?
1. a) $6.7 \times 10^{-3} = 0.0067$
2. b) $8 \times 10^{0} = 80$
3. c) $2.3 \times 10^{-2} = 0.23$
4. d) $4.1 \times 10^{3} = 41{,}000$
Answer: $6.7 \times 10^{-3} = 0.0067$
$6.7 \times 10^{-3} = 0.0067$ is correct. The others: $2.3 \times 10^{-2} = 0.023$ , $4.1 \times 10^3 = 4100$ , and $8 \times 10^0 = 8$ because $10^0 = 1$ .
16.
A dust mite is about $2.5 \times 10^{-4}$ m long. Write the length in meters as a decimal number, then in millimeters ( $1$ m $= 1000$ mm).
Answer: $0.25$
1. How many places does the decimal point move to the left? 4
2. What is the length in meters as a decimal number? 0.00025
3. What is the length in millimeters? 0.25
17.
One light-year is about $9.461 \times 10^{12}$ km. Write this distance as an ordinary number of kilometers.
Answer: 9461000000000
1. How many places does the decimal point move to the right? 12
2. How many digits does $9.461$ have after the decimal point? 3
3. How many zeros follow the digits $9461$ ? 9
4. Write the distance in km as an ordinary number. 9461000000000
18.
A hydrogen atom has a diameter of about $1.06 \times 10^{-10}$ m. When you write this diameter as a decimal number of meters, how many digits come after the decimal point?
Answer: 12
1. How many places does the decimal point move to the left? 10
2. How many zeros stand between the decimal point and the digit $1$ ? 9
3. How many digits come after the decimal point in total? 12