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1. In this problem, you will practice adding, subtracting, multiplying, and dividing numbers written in scientific notation. For each part, use the following numbers. a. Choose two of the scientific numbers from the list and add them. 3.2 X 10^4 1.95 X 10^6 8.03 X 10^-8 2.56 X 10^-2 4.82 X 10^12 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 from the list 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.
g. Choose one of the scientific numbers from the list and multiply it by 256.7.
Show your work.
h. 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.

please DO NOT add the ], #, and *

9 answers

Let’s work through the operations step by step using numbers from the list:

Numbers from the list:

  1. 3.2 × 10^4
  2. 1.95 × 10^6
  3. 8.03 × 10^-8
  4. 2.56 × 10^-2
  5. 4.82 × 10^12

a. Addition

Choose 3.2 × 10^4 and 1.95 × 10^6:

To add these numbers, we need them to have the same exponent:

  1. Convert 3.2 × 10^4 to 10^6:
    3.2 × 10^4 = 0.032 × 10^6

  2. Now add:
    0.032 × 10^6 + 1.95 × 10^6 = (0.032 + 1.95) × 10^6
    = 1.982 × 10^6

b. Subtraction

Choose 1.95 × 10^6 and 3.2 × 10^4:

Again, we will bring the two numbers to the same exponent:

  1. Convert 3.2 × 10^4 to 10^6:
    3.2 × 10^4 = 0.032 × 10^6

  2. Now subtract:
    1.95 × 10^6 - 0.032 × 10^6 = (1.95 - 0.032) × 10^6
    = 1.918 × 10^6

c. Multiplication

Choose 2.56 × 10^-2 and 4.82 × 10^12:

To multiply, multiply the coefficients and add the exponents:

  1. Coefficients:
    2.56 × 4.82 = 12.3072

  2. Exponents:
    (-2 + 12) = 10

  3. Combine:
    12.3072 × 10^10.

d. Division

Choose 4.82 × 10^12 and 1.95 × 10^6:

To divide, divide the coefficients and subtract the exponents:

  1. Coefficients:
    4.82 ÷ 1.95 ≈ 2.464

  2. Exponents:
    (12 - 6) = 6

  3. Combine:
    2.464 × 10^6.

e. Addition with a Decimal

Choose 2.56 × 10^-2 and add it to 32.8:

Convert 2.56 × 10^-2 to a decimal:

2.56 × 10^-2 = 0.0256

Now add:

32.8 + 0.0256 = 32.8256.

f. Subtraction from 1,845.19

Choose 1.95 × 10^6 and subtract from 1,845.19:

  1. Convert 1.95 × 10^6 to a decimal:
    1.95 × 10^6 = 1,950,000.

  2. Now subtract:
    1,845.19 - 1,950,000 = -1,948,154.81.

g. Multiplication by 256.7

Choose 3.2 × 10^4:

  1. Coefficients:
    3.2 × 256.7 ≈ 820.64.

  2. Keep exponent as 10^4.

  3. Combine:
    820.64 × 10^4
    = 8.2064 × 10^6 (shifting the decimal to maintain scientific notation).

h. Division by 0.3

Choose 8.03 × 10^-8:

  1. Division:
    (8.03 ÷ 0.3) × 10^-8 ≈ 26.767 × 10^-8.

  2. Adjust to scientific notation:
    2.6767 × 10^-7.

In summary, we have performed addition, subtraction, multiplication, and division, converting where necessary for scientific notation.

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?

Adding/subtracting and multiplying/dividing numbers in scientific notation involve different processes, but they also share similarities. Here’s a comparative analysis of both operations:

Similarities:

  1. Scientific Notation Format: Both operations involve numbers expressed in the format \( a \times 10^n \), where \( a \) is a coefficient and \( n \) is an exponent. Regardless of the operation being performed, this format is foundational.

  2. Coefficient Manipulation: For both addition/subtraction and multiplication/division, you manipulate the coefficient \( a \). In both cases, you perform arithmetic operations on these coefficients (addition or subtraction for addition/subtraction; multiplication or division for multiplication/division).

  3. Exponential Rules: Both operations require some consideration of the exponents. While they use different rules for handling exponents (addition/subtraction depending on matching exponents versus multiplication/division involving adding or subtracting exponents), they still require you to understand how to handle exponents effectively.

Differences:

  1. Exponent Matching:

    • Addition/Subtraction: Requires the exponents to be the same before performing the operation. If the exponents are not the same, you'll first convert one (or both) numbers so that their exponents match.
    • Multiplication/Division: Does not require matching exponents. You can multiply or divide the numbers directly once you handle the coefficients.

  2. Operations with Exponents:

    • Addition/Subtraction: The exponents must be equal, and only the coefficients are added or subtracted.
    • Multiplication: You multiply the coefficients and then add the exponents.
    • Division: You divide the coefficients and then subtract the exponent of the divisor from the exponent of the dividend.

  3. Final Format:

    • The outcome from adding or subtracting might also require normalization to ensure the coefficient remains in the proper scientific notation range (1 ≤ \( a \) < 10), while this is typically not a concern for multiplication/division unless the results fall outside that range.

Which is Easiest?

In general, many might find multiplication and division of numbers in scientific notation easier than addition and subtraction. This is because:

  • Fewer Steps Required: For multiplication and division, you can often directly perform the operations on the coefficients and exponents with minimal conversion required.
  • Less Risk of Error: When adding and subtracting, the requirement to match exponents can introduce an extra layer of complexity. If exponents are not managed carefully, it may lead to mistakes.

However, the perception of ease can also vary based on personal strengths with math. Some individuals may feel more comfortable with the straightforward arithmetic and normalized format of adding/subtracting, while others may prefer the consistency of multiplying/dividing without needing to adjust exponents. Overall, the choice of which operation feels easier can depend on the individual's familiarity and comfort with scientific notation.

are you serious bot, please make it into a paragraph and make it sound as an 8th grader

Adding and subtracting numbers written in scientific notation is quite different from multiplying and dividing them, but there are also some similarities. First, both processes work with numbers in the format \( a \times 10^n \), where \( a \) is the coefficient and \( n \) is the exponent. When adding or subtracting, you need to make sure the exponents are the same before you can add or subtract the coefficients. If they aren’t the same, you'll have to convert one of the numbers. In contrast, when multiplying or dividing, you don't need to match the exponents; you can just multiply or divide the coefficients directly and then add or subtract the exponents. Many people often find multiplication and division easier because they're straightforward and need fewer steps. You can just focus on multiplying or dividing the numbers without worrying about changing exponents. Overall, whether adding/subtracting or multiplying/dividing is easier really depends on how comfortable someone is with the different rules involved in scientific notation.

honestly I think multiplying is much easier

It's totally understandable to think that multiplying numbers in scientific notation is easier! When you multiply, you can simply multiply the coefficients and add the exponents without worrying about making them match first, which simplifies the process. There's less chance for mistakes since you can dive right into the operation without extra steps. That straightforward approach can make it feel more manageable, especially since many people find multiplication to be a simpler math operation in general. Everyone has their own preferences and strengths, so it makes sense that you would find multiplying in scientific notation to be the easiest!

3. Read the article "Planet." You will use information from this article to answer the
following questions.
a. 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.
b. Choose two of the distances of planets from the sun and add them. Show your work.
c. Choose two of the distances of planets from the sun and subtract them. Show your work.
d. Choose two of the distances of planets from the sun and multiply them. Show your work.
e. Choose two of the distances of planets from the sun and divide them. Show your work.

To help with your request, I'll provide some typical average distances from the Sun for the planets in our solar system, written in scientific notation. Here is that distance data:

  1. Mercury: \( 5.79 \times 10^7 \) km
  2. Venus: \( 1.08 \times 10^8 \) km
  3. Earth: \( 1.50 \times 10^8 \) km
  4. Mars: \( 2.28 \times 10^8 \) km
  5. Jupiter: \( 7.79 \times 10^8 \) km
  6. Saturn: \( 1.43 \times 10^9 \) km
  7. Uranus: \( 2.87 \times 10^9 \) km
  8. Neptune: \( 4.50 \times 10^9 \) km

a. Distances from Each Planet to the Sun

Here is the list of distances from each planet to the Sun, written in scientific notation:

  • Mercury: \( 5.79 \times 10^7 \) km
  • Venus: \( 1.08 \times 10^8 \) km
  • Earth: \( 1.50 \times 10^8 \) km
  • Mars: \( 2.28 \times 10^8 \) km
  • Jupiter: \( 7.79 \times 10^8 \) km
  • Saturn: \( 1.43 \times 10^9 \) km
  • Uranus: \( 2.87 \times 10^9 \) km
  • Neptune: \( 4.50 \times 10^9 \) km

To find the average distance for Mercury and Neptune, we find that the shortest distance is Mercury (5.79 × 10^7 km) and the longest is Neptune (4.50 × 10^9 km).

b. Adding Distances

Choose Earth (\( 1.50 \times 10^8 \) km) and Mars (\( 2.28 \times 10^8 \) km):

To add these, we need to make sure the exponents are the same:

  1. Both are already expressed in \( 10^8 \):

    • Earth: \( 1.50 \times 10^8 \)
    • Mars: \( 2.28 \times 10^8 \)

  2. Now, add the coefficients:

    • \( 1.50 + 2.28 = 3.78 \)

  3. Therefore, the sum is:

    • \( 3.78 \times 10^8 \) km

c. Subtracting Distances

Choose Jupiter (\( 7.79 \times 10^8 \) km) and Earth (\( 1.50 \times 10^8 \) km):

  1. Both are already in \( 10^8 \):

    • Jupiter: \( 7.79 \times 10^8 \)
    • Earth: \( 1.50 \times 10^8 \)

  2. Now subtract the coefficients:

    • \( 7.79 - 1.50 = 6.29 \)

  3. Therefore, the difference is:

    • \( 6.29 \times 10^8 \) km

d. Multiplying Distances

Choose Venus (\( 1.08 \times 10^8 \) km) and Saturn (\( 1.43 \times 10^9 \) km):

To multiply, multiply the coefficients and add the exponents:

  1. Multiply the coefficients:

    • \( 1.08 \times 1.43 = 1.5444 \)

  2. Add the exponents:

    • \( 8 + 9 = 17 \)

  3. Therefore, the product is:

    • \( 1.5444 \times 10^{17} \) km²

e. Dividing Distances

Choose Uranus (\( 2.87 \times 10^9 \) km) and Mars (\( 2.28 \times 10^8 \) km):

To divide, divide the coefficients and subtract the exponents:

  1. Divide the coefficients:

    • \( 2.87 \div 2.28 \approx 1.26 \)

  2. Subtract the exponents:

    • \( 9 - 8 = 1 \)

  3. Therefore, the quotient is:

    • \( 1.26 \times 10^1 \) or \( 1.26 \times 10 \) km

In conclusion, for each operation, we carefully handled the distances using scientific notation rules, ensuring correct addition, subtraction, multiplication, and division.