Divide. Write Your Answer In Scientific Notation.\[$\frac{5 \times 10^2}{1 \times 10^9}\$\]

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Understanding 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. In this article, we will explore how to divide numbers in scientific notation.

The Problem: Divide in Scientific Notation

The problem we are given is to divide the number 5×1021×109\frac{5 \times 10^2}{1 \times 10^9}. To solve this problem, we need to understand the rules of dividing numbers in scientific notation.

Rules for Dividing Numbers in Scientific Notation

When dividing numbers in scientific notation, we follow these rules:

  1. Divide the numbers in front of the powers of 10.
  2. Subtract the exponents of the powers of 10.

Applying the Rules

Let's apply the rules to the given problem.

Step 1: Divide the Numbers in Front of the Powers of 10

The number in front of the power of 10 in the numerator is 5, and the number in front of the power of 10 in the denominator is 1. To divide these numbers, we simply divide 5 by 1, which gives us 5.

Step 2: Subtract the Exponents of the Powers of 10

The exponent of the power of 10 in the numerator is 10210^2, and the exponent of the power of 10 in the denominator is 10910^9. To subtract these exponents, we subtract 9 from 2, which gives us -7.

The Final Answer

Now that we have applied the rules, we can write the final answer in scientific notation.

5×1021×109=5×10−7\frac{5 \times 10^2}{1 \times 10^9} = 5 \times 10^{-7}

Conclusion

Dividing numbers in scientific notation involves following a set of rules. By understanding these rules and applying them to the given problem, we can arrive at the final answer. In this article, we have explored how to divide numbers in scientific notation and have applied the rules to a specific problem.

Example Problems

Here are a few example problems to help you practice dividing numbers in scientific notation.

Example 1

3×1042×106\frac{3 \times 10^4}{2 \times 10^6}

Step 1: Divide the Numbers in Front of the Powers of 10

The number in front of the power of 10 in the numerator is 3, and the number in front of the power of 10 in the denominator is 2. To divide these numbers, we simply divide 3 by 2, which gives us 1.5.

Step 2: Subtract the Exponents of the Powers of 10

The exponent of the power of 10 in the numerator is 10410^4, and the exponent of the power of 10 in the denominator is 10610^6. To subtract these exponents, we subtract 6 from 4, which gives us -2.

The Final Answer

Now that we have applied the rules, we can write the final answer in scientific notation.

3×1042×106=1.5×10−2\frac{3 \times 10^4}{2 \times 10^6} = 1.5 \times 10^{-2}

Example 2

2×1034×105\frac{2 \times 10^3}{4 \times 10^5}

Step 1: Divide the Numbers in Front of the Powers of 10

The number in front of the power of 10 in the numerator is 2, and the number in front of the power of 10 in the denominator is 4. To divide these numbers, we simply divide 2 by 4, which gives us 0.5.

Step 2: Subtract the Exponents of the Powers of 10

The exponent of the power of 10 in the numerator is 10310^3, and the exponent of the power of 10 in the denominator is 10510^5. To subtract these exponents, we subtract 5 from 3, which gives us -2.

The Final Answer

Now that we have applied the rules, we can write the final answer in scientific notation.

2×1034×105=0.5×10−2\frac{2 \times 10^3}{4 \times 10^5} = 0.5 \times 10^{-2}

Practice Problems

Here are a few practice problems to help you reinforce your understanding of dividing numbers in scientific notation.

Problem 1

6×1023×108\frac{6 \times 10^2}{3 \times 10^8}

Problem 2

9×1041×107\frac{9 \times 10^4}{1 \times 10^7}

Problem 3

1×1062×109\frac{1 \times 10^6}{2 \times 10^9}

Answer Key

Here is the answer key for the practice problems.

Problem 1

6×1023×108=2×10−6\frac{6 \times 10^2}{3 \times 10^8} = 2 \times 10^{-6}

Problem 2

9×1041×107=9×10−3\frac{9 \times 10^4}{1 \times 10^7} = 9 \times 10^{-3}

Problem 3

Frequently Asked Questions

In this article, we will answer some of the most frequently asked questions about dividing 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 divide numbers in scientific notation?

A: To divide numbers in scientific notation, you need to follow these steps:

  1. Divide the numbers in front of the powers of 10.
  2. Subtract the exponents of the powers of 10.

Q: What if the exponents are the same?

A: If the exponents are the same, you can simply divide the numbers in front of the powers of 10.

Q: What if the exponents are different?

A: If the exponents are different, you need to subtract the smaller exponent from the larger exponent.

Q: Can I use a calculator to divide numbers in scientific notation?

A: Yes, you can use a calculator to divide numbers in scientific notation. However, make sure to enter the numbers in the correct format, with the number between 1 and 10 and the power of 10.

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

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

Q: What is the difference between dividing numbers in scientific notation and standard notation?

A: The main difference between dividing numbers in scientific notation and standard notation is that in scientific notation, you need to follow the rules of dividing numbers with powers of 10.

Q: Can I use scientific notation to divide fractions?

A: Yes, you can use scientific notation to divide fractions. However, make sure to follow the rules of dividing fractions, which include inverting the second fraction and multiplying.

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 numbers with powers of 10
  • Not inverting the second fraction when dividing fractions
  • Not using the correct format for entering numbers in a calculator

Q: How can I practice dividing numbers in scientific notation?

A: You can practice dividing numbers in scientific notation by using online resources, such as worksheets and practice problems. You can also try dividing numbers in scientific notation on your own, using a calculator or by hand.

Q: What are some real-world applications of dividing numbers in scientific notation?

A: Some real-world applications of dividing numbers in scientific notation include:

  • Calculating the area and volume of objects in science and engineering
  • Determining the concentration of solutions in chemistry
  • Calculating the speed and distance of objects in physics

Conclusion

Dividing numbers in scientific notation can be a challenging task, but with practice and patience, you can become proficient in this skill. By following the rules of dividing numbers with powers of 10 and using online resources, you can improve your understanding of scientific notation and apply it to real-world problems.

Additional Resources

Here are some additional resources to help you learn more about dividing numbers in scientific notation:

  • Online tutorials and videos
  • Practice problems and worksheets
  • Scientific notation calculators and apps
  • Online communities and forums for discussing scientific notation

Answer Key

Here is the answer key for the practice problems.

Problem 1

6×1023×108=2×10−6\frac{6 \times 10^2}{3 \times 10^8} = 2 \times 10^{-6}

Problem 2

9×1041×107=9×10−3\frac{9 \times 10^4}{1 \times 10^7} = 9 \times 10^{-3}

Problem 3

1×1062×109=0.5×10−3\frac{1 \times 10^6}{2 \times 10^9} = 0.5 \times 10^{-3}