What Is The Product Of $8.2 \times 10^9$ And $4.5 \times 10^{-5}$ In Scientific Notation?A. $ 36.9 × 10 − 45 36.9 \times 10^{-45} 36.9 × 1 0 − 45 [/tex]B. $12.7 \times 10^4$C. $3.69 \times 10^5$D. $3.69 \times

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Understanding Scientific Notation and Multiplication

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. In this article, we will explore the product of two numbers given in scientific notation and determine the result in scientific notation.

The Product of Two Numbers in Scientific Notation

To find the product of two numbers in scientific notation, we need to multiply the coefficients (the numbers in front of the powers of 10) and add the exponents of the powers of 10. The formula for this is:

a×10m×b×10n=(a×b)×10m+na \times 10^m \times b \times 10^n = (a \times b) \times 10^{m+n}

where aa and bb are the coefficients, and mm and nn are the exponents.

Applying the Formula to the Given Numbers

Let's apply the formula to the given numbers:

8.2×109×4.5×10−58.2 \times 10^9 \times 4.5 \times 10^{-5}

First, we multiply the coefficients:

8.2×4.5=36.98.2 \times 4.5 = 36.9

Next, we add the exponents:

9+(−5)=49 + (-5) = 4

So, the product of the two numbers is:

36.9×10436.9 \times 10^4

Comparing the Result to the Answer Choices

Now, let's compare our result to the answer choices:

A. $36.9 \times 10^{-45}$

B. $12.7 \times 10^4$

C. $3.69 \times 10^5$

D. $3.69 \times 10^4$

Our result, $36.9 \times 10^4$, matches answer choice D.

Conclusion

In this article, we explored the product of two numbers given in scientific notation and determined the result in scientific notation. We applied the formula for multiplying numbers in scientific notation and compared our result to the answer choices. The correct answer is D. $3.69 \times 10^4$.

Understanding the Importance of Scientific Notation

Scientific notation is a powerful tool for expressing very large or very small numbers in a more manageable form. It is commonly used in science, engineering, and other fields where large or small numbers are encountered. By understanding how to multiply numbers in scientific notation, we can perform calculations more efficiently and accurately.

Real-World Applications of Scientific Notation

Scientific notation has many real-world applications. For example, it is used to express the size of atoms and molecules, the distance to stars and galaxies, and the amount of energy released in nuclear reactions. It is also used in finance to express large or small amounts of money, and in medicine to express the concentration of drugs or other substances.

Common Mistakes When Multiplying Numbers in Scientific Notation

When multiplying numbers in scientific notation, it is easy to make mistakes. One common mistake is to forget to multiply the coefficients or to add the exponents incorrectly. Another mistake is to write the result in the wrong form, such as writing a number in scientific notation when it should be written in standard form.

Tips for Multiplying Numbers in Scientific Notation

To avoid making mistakes when multiplying numbers in scientific notation, follow these tips:

  • Make sure to multiply the coefficients correctly.
  • Add the exponents correctly.
  • Write the result in the correct form, either in scientific notation or in standard form.
  • Check your work by plugging the result back into the original equation.

Conclusion

In conclusion, multiplying numbers in scientific notation is a straightforward process that requires attention to detail and a basic understanding of the formula. By following the tips outlined in this article, you can avoid common mistakes and perform calculations more efficiently and accurately.
Frequently Asked Questions About Multiplying Numbers in Scientific Notation

Q: What is scientific notation?

A: 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.

Q: How do I multiply numbers in scientific notation?

A: To multiply numbers in scientific notation, you need to multiply the coefficients (the numbers in front of the powers of 10) and add the exponents of the powers of 10. The formula for this is:

a×10m×b×10n=(a×b)×10m+na \times 10^m \times b \times 10^n = (a \times b) \times 10^{m+n}

Q: What if the exponents are negative?

A: If the exponents are negative, you need to add them as you would with positive exponents. For example:

2×10−3×3×10−2=(2×3)×10−3−2=6×10−52 \times 10^{-3} \times 3 \times 10^{-2} = (2 \times 3) \times 10^{-3-2} = 6 \times 10^{-5}

Q: What if the coefficients are decimals?

A: If the coefficients are decimals, you need to multiply them as you would with whole numbers. For example:

4.2×103×2.5×102=(4.2×2.5)×103+2=10.5×1054.2 \times 10^3 \times 2.5 \times 10^2 = (4.2 \times 2.5) \times 10^{3+2} = 10.5 \times 10^5

Q: Can I multiply numbers in scientific notation by hand?

A: Yes, you can multiply numbers in scientific notation by hand. However, it may be more efficient to use a calculator or computer to perform the calculation.

Q: How do I convert a number from scientific notation to standard form?

A: To convert a number from scientific notation to standard form, you need to multiply the coefficient by the power of 10. For example:

3.4×104=3.4×10,000=34,0003.4 \times 10^4 = 3.4 \times 10,000 = 34,000

Q: How do I convert a number from standard form to scientific notation?

A: To convert a number from standard form to scientific notation, you need to express the number as a number between 1 and 10 multiplied by a power of 10. For example:

34,000=3.4×10434,000 = 3.4 \times 10^4

Q: What are some common mistakes to avoid when multiplying numbers in scientific notation?

A: Some common mistakes to avoid when multiplying numbers in scientific notation include:

  • Forgetting to multiply the coefficients
  • Adding the exponents incorrectly
  • Writing the result in the wrong form
  • Not checking the work by plugging the result back into the original equation

Q: How do I check my work when multiplying numbers in scientific notation?

A: To check your work when multiplying numbers in scientific notation, you need to plug the result back into the original equation. For example:

2×103×3×102=(2×3)×103+2=6×1052 \times 10^3 \times 3 \times 10^2 = (2 \times 3) \times 10^{3+2} = 6 \times 10^5

Check the result by plugging it back into the original equation:

2×103×3×102=6×1052 \times 10^3 \times 3 \times 10^2 = 6 \times 10^5

This should be true if the calculation was performed correctly.

Conclusion

In conclusion, multiplying numbers in scientific notation is a straightforward process that requires attention to detail and a basic understanding of the formula. By following the tips outlined in this article, you can avoid common mistakes and perform calculations more efficiently and accurately.