Calculate \left(6.4 \times 10^8\right) \div\left(2 \times 10^{-4}\right ].Give Your Answer In Standard Index Form.

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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. For example, the number 400 can be expressed in scientific notation as 4 x 10^2. This makes it easier to perform calculations with large numbers.

The Problem

We are given the expression (6.4×108)÷(2×10−4)\left(6.4 \times 10^8\right) \div\left(2 \times 10^{-4}\right). Our goal is to calculate the result of this expression and express it in standard index form.

Step 1: Divide the Coefficients

To divide two numbers in scientific notation, we first divide the coefficients (the numbers in front of the powers of 10). In this case, we have 6.4 ÷ 2 = 3.2.

Step 2: Subtract the Exponents

Next, we subtract the exponents of the powers of 10. In this case, we have 10^8 ÷ 10^-4. To divide powers of 10, we subtract the exponents, so we get 10^(8-(-4)) = 10^(8+4) = 10^12.

Step 3: Combine the Results

Now that we have divided the coefficients and subtracted the exponents, we can combine the results to get the final answer. We have 3.2 x 10^12.

Expressing the Answer in Standard Index Form

To express the answer in standard index form, we need to move the decimal point of the coefficient to the left until we have a number between 1 and 10. In this case, we have 3.2, which is already between 1 and 10. Therefore, our final answer is 3.2 x 10^12.

Conclusion

In this article, we have learned how to calculate large numbers in scientific notation. We have seen how to divide two numbers in scientific notation by dividing the coefficients and subtracting the exponents. We have also seen how to express the answer in standard index form by moving the decimal point of the coefficient to the left until we have a number between 1 and 10.

Examples and Practice Problems

Here are a few examples and practice problems to help you practice calculating large numbers in scientific notation:

  • (5×106)÷(3×102)\left(5 \times 10^6\right) \div\left(3 \times 10^2\right)
  • (2×109)÷(4×10−3)\left(2 \times 10^9\right) \div\left(4 \times 10^{-3}\right)
  • (7×104)÷(2×101)\left(7 \times 10^4\right) \div\left(2 \times 10^1\right)

Try to solve these problems on your own before looking at the answers below.

Answers

  • (5×106)÷(3×102)\left(5 \times 10^6\right) \div\left(3 \times 10^2\right) = 1.67 x 10^4
  • (2×109)÷(4×10−3)\left(2 \times 10^9\right) \div\left(4 \times 10^{-3}\right) = 5 x 10^11
  • (7×104)÷(2×101)\left(7 \times 10^4\right) \div\left(2 \times 10^1\right) = 3.5 x 10^3

Tips and Tricks

Here are a few tips and tricks to help you calculate large numbers in scientific notation:

  • Make sure to divide the coefficients and subtract the exponents correctly.
  • Use the correct order of operations (PEMDAS) when performing calculations.
  • Express the answer in standard index form by moving the decimal point of the coefficient to the left until you have a number between 1 and 10.

By following these tips and tricks, you should be able to calculate large numbers in scientific notation with ease.

Conclusion

Frequently Asked Questions

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 convert a number to scientific notation?

A: To convert a number to scientific notation, you need to move the decimal point of the number to the left until you have a number between 1 and 10. Then, multiply the number by a power of 10 that is equal to the number of places you moved the decimal point.

Q: How do I divide two numbers in scientific notation?

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

Q: What is the order of operations when performing calculations in scientific notation?

A: The order of operations when performing calculations in scientific notation is the same as the order of operations in regular arithmetic: PEMDAS (Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction).

Q: How do I express the answer in standard index form?

A: To express the answer in standard index form, you need to move the decimal point of the coefficient to the left until you have a number between 1 and 10.

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

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

  • Not dividing the coefficients correctly
  • Not subtracting the exponents correctly
  • Not following the order of operations (PEMDAS)
  • Not expressing the answer in standard index form

Q: How can I practice calculating large numbers in scientific notation?

A: You can practice calculating large numbers in scientific notation by working through examples and practice problems. You can also try using online resources or calculators to help you practice.

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

A: Scientific notation has many real-world applications, including:

  • Calculating large numbers in physics and engineering
  • Expressing very small or very large numbers in chemistry and biology
  • Performing calculations in finance and economics
  • Expressing large numbers in computer science and programming

Q: How can I use scientific notation to simplify complex calculations?

A: You can use scientific notation to simplify complex calculations by expressing large numbers in a more manageable form. This can make it easier to perform calculations and reduce the risk of errors.

Q: What are some tips for mastering scientific notation?

A: Some tips for mastering scientific notation include:

  • Practicing regularly to build your skills and confidence
  • Using online resources and calculators to help you practice
  • Breaking down complex calculations into smaller, more manageable steps
  • Double-checking your work to ensure accuracy

By following these tips and practicing regularly, you should be able to master scientific notation and perform complex calculations with ease.