Divide The Following And Express Your Answer In Scientific Notation:${ \frac{1.6 \times 10^3}{8.0 \times 10^7} }$(A) ${ 2 \times 10^{-5}\$} (B) ${ 2 \times 10^{-4}\$} (C) ${ 0.2 \times 10^{10}\$} (D) [$0.2

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Scientific notation is a way of expressing numbers in the form of a product of a number between 1 and 10 and a power of 10. It is commonly used in mathematics and science to simplify calculations and express very large or very small numbers. In this article, we will learn how to divide numbers in scientific notation.

What is Scientific Notation?

Scientific notation is a way of expressing numbers in the form of a product of a number between 1 and 10 and a power of 10. It is written in the form:

a Γ— 10^n

where a is a number between 1 and 10, and n is an integer.

Dividing Numbers in Scientific Notation

To divide numbers in scientific notation, we need to follow the rules of dividing exponents. When dividing two numbers in scientific notation, we divide the coefficients (the numbers in front of the powers of 10) and subtract the exponents.

Example: Dividing Numbers in Scientific Notation

Let's consider the following example:

1.6Γ—1038.0Γ—107\frac{1.6 \times 10^3}{8.0 \times 10^7}

To divide this expression, we need to divide the coefficients (1.6 and 8.0) and subtract the exponents (3 and 7).

1.68.0Γ—103βˆ’7\frac{1.6}{8.0} \times 10^{3-7}

1.68.0Γ—10βˆ’4\frac{1.6}{8.0} \times 10^{-4}

Now, let's simplify the expression:

1.68.0=0.2\frac{1.6}{8.0} = 0.2

So, the final answer is:

0.2Γ—10βˆ’40.2 \times 10^{-4}

Simplifying the Answer

To simplify the answer, we can express it in the form of a number between 1 and 10 and a power of 10.

0.2Γ—10βˆ’4=2Γ—10βˆ’50.2 \times 10^{-4} = 2 \times 10^{-5}

Therefore, the final answer is:

2Γ—10βˆ’5\boxed{2 \times 10^{-5}}

Conclusion

Dividing numbers in scientific notation is a simple process that involves dividing the coefficients and subtracting the exponents. By following the rules of dividing exponents, we can simplify complex expressions and arrive at the final answer.

Common Mistakes to Avoid

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

  • Not following the rules of dividing exponents: When dividing two numbers in scientific notation, we need to divide the coefficients and subtract the exponents.
  • Not simplifying the answer: To simplify the answer, we need to express it in the form of a number between 1 and 10 and a power of 10.
  • Not checking the units: When dividing numbers in scientific notation, we need to check the units to ensure that they are correct.

Practice Problems

Here are some practice problems to help you understand how to divide numbers in scientific notation:

  1. 4.0Γ—1022.0Γ—105\frac{4.0 \times 10^2}{2.0 \times 10^5}
  2. 3.0Γ—1046.0Γ—103\frac{3.0 \times 10^4}{6.0 \times 10^3}
  3. 2.0Γ—1064.0Γ—102\frac{2.0 \times 10^6}{4.0 \times 10^2}

Answer Key

Here are the answers to the practice problems:

  1. 2Γ—10βˆ’4\boxed{2 \times 10^{-4}}
  2. 0.5Γ—101\boxed{0.5 \times 10^1}
  3. 5Γ—104\boxed{5 \times 10^4}

Conclusion

In this article, we will answer some of the most frequently asked questions on dividing numbers in scientific notation.

Q: What is scientific notation?

A: Scientific notation is a way of expressing numbers in the form of a product of a number between 1 and 10 and a power of 10. It is written in the form:

a Γ— 10^n

where a is a number between 1 and 10, and n is an integer.

Q: How do I divide numbers in scientific notation?

A: To divide numbers in scientific notation, you need to follow the rules of dividing exponents. When dividing two numbers in scientific notation, you divide the coefficients (the numbers in front of the powers of 10) and subtract the exponents.

Q: What is the rule for dividing exponents?

A: The rule for dividing exponents is:

a^m Γ· a^n = a^(m-n)

where a is a number and m and n are integers.

Q: How do I simplify the answer after dividing numbers in scientific notation?

A: To simplify the answer, you need to express it in the form of a number between 1 and 10 and a power of 10.

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

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

  • Not following the rules of dividing exponents
  • Not simplifying the answer
  • Not checking the units

Q: How do I check the units when dividing numbers in scientific notation?

A: To check the units, you need to ensure that the units of the two numbers being divided are the same. If the units are different, you need to convert one of the numbers to have the same unit as the other number.

Q: What are some examples of dividing numbers in scientific notation?

A: Here are some examples of dividing numbers in scientific notation:

  • 1.6Γ—1038.0Γ—107\frac{1.6 \times 10^3}{8.0 \times 10^7}
  • 4.0Γ—1022.0Γ—105\frac{4.0 \times 10^2}{2.0 \times 10^5}
  • 3.0Γ—1046.0Γ—103\frac{3.0 \times 10^4}{6.0 \times 10^3}

Q: How do I solve these examples?

A: To solve these examples, you need to follow the rules of dividing exponents. Here are the steps to solve each example:

  • 1.6Γ—1038.0Γ—107\frac{1.6 \times 10^3}{8.0 \times 10^7}: Divide the coefficients (1.6 and 8.0) and subtract the exponents (3 and 7). The final answer is 2Γ—10βˆ’42 \times 10^{-4}.
  • 4.0Γ—1022.0Γ—105\frac{4.0 \times 10^2}{2.0 \times 10^5}: Divide the coefficients (4.0 and 2.0) and subtract the exponents (2 and 5). The final answer is 2Γ—10βˆ’32 \times 10^{-3}.
  • 3.0Γ—1046.0Γ—103\frac{3.0 \times 10^4}{6.0 \times 10^3}: Divide the coefficients (3.0 and 6.0) and subtract the exponents (4 and 3). The final answer is 0.5Γ—1010.5 \times 10^1.

Q: What are some practice problems to help me understand how to divide numbers in scientific notation?

A: Here are some practice problems to help you understand how to divide numbers in scientific notation:

  • 2.0Γ—1064.0Γ—102\frac{2.0 \times 10^6}{4.0 \times 10^2}
  • 5.0Γ—1032.0Γ—105\frac{5.0 \times 10^3}{2.0 \times 10^5}
  • 3.0Γ—1046.0Γ—103\frac{3.0 \times 10^4}{6.0 \times 10^3}

Q: How do I solve these practice problems?

A: To solve these practice problems, you need to follow the rules of dividing exponents. Here are the steps to solve each practice problem:

  • 2.0Γ—1064.0Γ—102\frac{2.0 \times 10^6}{4.0 \times 10^2}: Divide the coefficients (2.0 and 4.0) and subtract the exponents (6 and 2). The final answer is 5Γ—1045 \times 10^4.
  • 5.0Γ—1032.0Γ—105\frac{5.0 \times 10^3}{2.0 \times 10^5}: Divide the coefficients (5.0 and 2.0) and subtract the exponents (3 and 5). The final answer is 2.5Γ—10βˆ’22.5 \times 10^{-2}.
  • 3.0Γ—1046.0Γ—103\frac{3.0 \times 10^4}{6.0 \times 10^3}: Divide the coefficients (3.0 and 6.0) and subtract the exponents (4 and 3). The final answer is 0.5Γ—1010.5 \times 10^1.

Conclusion

Dividing numbers in scientific notation is a simple process that involves dividing the coefficients and subtracting the exponents. By following the rules of dividing exponents, you can simplify complex expressions and arrive at the final answer. With practice, you can become proficient in dividing numbers in scientific notation and solve complex problems with ease.