Jupiter's Moon Io Orbits The Planet At A Distance Of $421,700 , \text{km}$. What Is The Correct Way To Write This Distance In Scientific Notation?A. $4.217 \times 10^{-5} , \text{km}$ B. $42.17 \times 10^4 ,

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What is Scientific Notation?

Scientific notation is a way of expressing very large or very small numbers in a more manageable form. It consists of a number between 1 and 10, multiplied by a power of 10. This notation is commonly used in scientific and mathematical contexts to simplify calculations and make it easier to understand and compare large or small numbers.

Writing Distances in Scientific Notation

When writing distances in scientific notation, we need to express the number in a way that is easy to read and understand. The correct way to write a distance in scientific notation depends on the magnitude of the number.

Example: Writing the Distance of Io from Jupiter

The distance of Io from Jupiter is given as $421,700 , \text{km}$. To write this distance in scientific notation, we need to express it as a number between 1 and 10, multiplied by a power of 10.

Step 1: Move the Decimal Point

To write the distance in scientific notation, we need to move the decimal point to the left until we have a number between 1 and 10. In this case, we need to move the decimal point 5 places to the left.

Step 2: Determine the Power of 10

When we move the decimal point to the left, we need to determine the power of 10 that we need to multiply the number by. In this case, we need to multiply the number by $10^5$.

The Correct Answer

Therefore, the correct way to write the distance of Io from Jupiter in scientific notation is:

4.217×105 km4.217 \times 10^5 \, \text{km}

Why Not Option A?

Option A, $4.217 \times 10^{-5} , \text{km}$, is incorrect because it has a negative exponent. The exponent should be positive, since we moved the decimal point to the left.

Why Not Option B?

Option B, $42.17 \times 10^4 , \text{km}$, is also incorrect because it has the wrong exponent. The exponent should be 5, not 4.

Conclusion

In conclusion, the correct way to write the distance of Io from Jupiter in scientific notation is $4.217 \times 10^5 , \text{km}$. This notation makes it easier to understand and compare large numbers, and it is commonly used in scientific and mathematical contexts.

Common Mistakes to Avoid

When writing distances in scientific notation, it's easy to make mistakes. Here are some common mistakes to avoid:

  • Using a negative exponent when the number is large.
  • Using the wrong exponent.
  • Not moving the decimal point far enough to the left.

Tips for Writing Distances in Scientific Notation

Here are some tips for writing distances in scientific notation:

  • Move the decimal point to the left until you have a number between 1 and 10.
  • Determine the power of 10 that you need to multiply the number by.
  • Use a positive exponent.
  • Double-check your work to make sure you have the correct exponent.

Practice Problems

Here are some practice problems to help you practice writing distances in scientific notation:

  • Write the distance of the Earth from the Sun in scientific notation.
  • Write the distance of the Moon from the Earth in scientific notation.
  • Write the distance of a star from the Earth in scientific notation.

Conclusion

In conclusion, writing distances in scientific notation is an important skill to have in science and mathematics. By following the steps outlined in this article, you can write distances in scientific notation with ease. Remember to move the decimal point to the left until you have a number between 1 and 10, determine the power of 10 that you need to multiply the number by, and use a positive exponent. With practice, you'll become a pro at writing distances in scientific notation!