In this problem, you will practice adding, subtracting, multiplying, and dividing numbers written in scientific notation. For each part, use the following numbers:

Question

3.2×1041.95×1068.03×10−82.56×10−24.82×1012

Choose two of the scientific numbers from the list and add them. Show your work.
Choose two of the scientific numbers from the list and subtract them. Show your work.
Choose two of the scientific numbers from the list and multiply them. Show your work.
Choose two of the scientific numbers and divide them. Show your work.
Choose one of the scientific numbers from the list and add it to 32.8. Show your work.
Choose one of the scientific numbers from the list and subtract it from 1,845.19 (or subtract 1,845.19 from the number you choose). Show your work.
Choose one of the scientific numbers from the list and multiply it by 256.7. Show your work.
Choose one of the scientific numbers from the list and divide it by 0.3 (or divide 0.3 by the number you choose). Show your work.
The following table shows distances in miles between some cities in the United States. The distances have been written in scientific notation.

Atlanta Baltimore Boston Chicago Los Angeles Seattle Tampa
Atlanta 0
9.2735×102

1.50511×103

9.444×102

3.10801×103

3.50307×103

6.7037×102

Baltimore
9.2735×102

0
5.7785×102

9.7323×102

3.72245×103

3.7441×103

1.37072×103

Boston
1.50511×103

5.7785×102

0
1.36663×103

4.16643×103

3.99708×103

1.90293×103

Chicago
9.444×102

9.7323×102

1.36663×103

0
2.79980×103

2.78657×103

1.61477×103

Los Angeles
3.10801×103

3.72245×103

4.16643×103

2.79980×103

0
1.5442×103

3.45696×103

Seattle
3.50307×103

3.7441×103

3.99708×103

2.78657×103

1.5442×103

0
4.06249×103

Tampa
6.7037×102

1.37072×103

1.90293×103

1.61477×103

3.45696×103

4.06249×103

0
Plan a trip where you will start at one city, travel to a second city, travel to a third city, and then return to the original city. Assuming you could travel the direct straight-line distance between each pair of cities, how far will you travel all together? Write your total distance traveled in both decimal notation and scientific notation. Show all your work so it is clear what cities you are going to and in what order.
Sita lives in Los Angeles, and her grandmother lives in Seattle. Sita plans to visit her grandmother three times in the next year. Assuming she travels the direct straight-line distance between the cities on each trip, what is the total distance she will travel to make these three trips? Write and evaluate an expression that uses multiplication to calculate the total distance traveled. Give your answer in both decimal notation and scientific notation.
Using the table of distances, make up your own word problem that would require someone to use division to solve. Write the problem, and provide the solution.
Read the article "Planet." You will use information from this article to answer the following questions.
Planet
Make a list of the distances from each planet to the sun, written in scientific notation. If there is a shortest and longest distance, use the average distance from the sun.
Choose two of the distances of planets from the sun and add them. Show your work.
Choose two of the distances of planets from the sun and subtract them. Show your work.
Choose two of the distances of planets from the sun and multiply them. Show your work.
Choose two of the distances of planets from the sun and divide them. Show your work.
Compare and contrast adding/subtracting numbers written in scientific notation with multiplying/dividing numbers written in scientific notation. How are the steps in each case similar? How are they different? Do you think adding, subtracting, multiplying, or dividing numbers written in scientific notation is easiest? Why?

Answer 1

Sure! Let’s work through each of the tasks using the scientific numbers provided:

Adding Two Scientific Numbers

Choose: \(3.2 \times 10^4\) and \(1.95 \times 10^6\)

Convert to the same exponent:

• \(3.2 \times 10^4 = 0.032 \times 10^6\) (moving decimal two places to the left)
• Now sum: \(0.032 \times 10^6 + 1.95 \times 10^6 = (0.032 + 1.95) \times 10^6 = 1.982 \times 10^6\)

Subtracting Two Scientific Numbers

Choose: \(2.56 \times 10^{-2}\) and \(8.03 \times 10^{-8}\)

Convert to the same exponent:

• \(2.56 \times 10^{-2} = 256 \times 10^{-4}\) (moving decimal two places to the right)
• Now sum: \(256 \times 10^{-4} - 0.0000803 \times 10^{-2} = 2560 \times 10^{-4} - 0.0803 \times 10^{-4}\)
• Now subtract: \(2560 - 0.0803 = 2559.9197 \times 10^{-4} \)
• So: \(2.5599197 \times 10^{-2} = 2 \times 10^{-2}\) (approximately)

Multiplying Two Scientific Numbers

Choose: \(4.82 \times 10^{12}\) and \(1.95 \times 10^{6}\)

Multiply the coefficients and add the exponents:

• \(4.82 \times 1.95 = 9.399 \)
• \(10^{12} \times 10^6 = 10^{18}\)
• Result: \(9.399 \times 10^{18}\)

Dividing Two Scientific Numbers

Choose: \(1.95 \times 10^{6}\) and \(3.2 \times 10^{4}\)

Divide the coefficients and subtract the exponents:

• \(1.95 \div 3.2 \approx 0.609375\)
• \(10^{6} \div 10^4 = 10^{6-4} = 10^{2}\)
• Result: \(0.609375 \times 10^{2} = 60.9375\)

Adding a Scientific Number to a Decimal

Choose: \(8.03 \times 10^{-8}\) and \(32.8\)

Convert \(32.8\) into scientific notation: \(3.28 \times 10^1\)

Convert to the same exponent:

• \(3.28 \times 10^1 = 32.8\) is much larger than \(8.03 \times 10^{-8}\), and thus:
• The sum is approximately \(3.28 \times 10^1\) (since the second number is negligible).

Subtracting a Scientific Number from a Decimal

Choose: \(1.95 \times 10^{6}\) from \(1845.19\)

Convert to scientific notation:

• \(1845.19 \approx 1.845 \times 10^3\)

Convert to the same exponent:

• \(1.95 \times 10^{6}\) is larger, so:
• \(1.95 \times 10^{6} - 1.845 \times 10^{3} = 1.95 \times 10^{6} - 0.001845 \times 10^{6}\)
• Result: \(1.949998155 \times 10^{6}\)

Multiply a Scientific Number by a Decimal

Choose: \(4.82 \times 10^{12}\) multiplied by \(256.7\)

Multiply:

• \(4.82 \times 256.7 = 1238.694\)
• So, \(1238.694 \times 10^{12} \approx 1.238694 \times 10^{15}\)

Divide a Scientific Number by a Decimal

Choose: \(2.56 \times 10^{-2}\) divided by \(0.3\)

Calculate:

• \(2.56 \div 0.3 \approx 8.53333\)

Thus:

• \(8.53333 \times 10^{-2} = 0.085333\)

Traveling Distances Round Trip

Let's start with the trip from Atlanta to Baltimore, then Baltimore to Boston, and finally back to Atlanta:

1. Atlanta to Baltimore: \(9.2735 \times 10^2\)
2. Baltimore to Boston: \(5.7785 \times 10^2\)
3. Boston to Atlanta: \(1.50511 \times 10^3\)

To find the total distance traveled: \[ \text{Total} = 9.2735 \times 10^2 + 5.7785 \times 10^2 + 1.50511 \times 10^3 \] Convert all to the same exponent: \[ = 0.92735 \times 10^3 + 0.57785 \times 10^3 + 1.50511 \times 10^3 \] Adding: \[ = (0.92735 + 0.57785 + 1.50511) \times 10^3 = (3.01031) \times 10^3 = 3010.31 \]

Sita's Travel to Her Grandmother

The distance from Los Angeles to Seattle is \(3.50307 \times 10^3\). For three trips, total distance is: \[ \text{Total distance} = 3 \times (3.50307 \times 10^3) = 10.50921 \times 10^3 \approx 1.050921 \times 10^4 \]

Division Word Problem

Problem: If the distance from Atlanta to Chicago is \(9.444 \times 10^2\), how many trips of \(0.3 \times 10^3\) can you take?

Solution: \[ \frac{9.444 \times 10^2}{0.3 \times 10^3} = \frac{9.444}{0.3} = 31.48 \text{ trips} \]

Distance From Each Planet to the Sun

Here’s a hypothetical list:

1. Mercury: \(5.79 \times 10^7\)
2. Venus: \(1.08 \times 10^8\)
3. Earth: \(1.5 \times 10^8\)
4. Mars: \(2.28 \times 10^8\)

Adding Two Distances

Choose: \(1.08 \times 10^8\) and \(1.5 \times 10^8\) Convert: \[ 1.08 + 1.5 = 2.58 \times 10^8. \]

Subtracting Two Distances

Choose: \(2.28 \times 10^8\) minus \(5.79 \times 10^7\) Convert: \[ 2.28 \times 10^8 - 0.579 \times 10^8 = 1.701 \times 10^8. \]

Multiplying Distances

Choose: \(1.08 \times 10^8\) and \(1.5 \times 10^8\) Result: \[ 1.08 \times 1.5 = 1.62 \times 10^{16} \]

Dividing Distances

Choose: \(1.5 \times 10^8\) divided by \(5.79 \times 10^7\) \[ \frac{1.5}{5.79} \approx 2.59 \text{ times} \] The result is \(2.59\).

Comparison of Addition/Subtraction vs. Multiplication/Division

• Similarities: Both operations involve manipulating the base numbers and exponents in scientific form.
• Differences: Addition and subtraction require matching exponents first, while multiplication and division involve adjusting the exponents based on the operation.
• Easiest: Many find multiplication and division easier due to straightforward coefficient manipulation, while addition and subtraction can be more cumbersome with exponent alignment.

This covers all tasks! Let me know if you need clarification on any part.