Simplify: $6.7 \times 10^{-4}$A. \[$-0.000067\$\]B. \[$0.000067\$\]C. \[$0.00067\$\]D. \[$-6.7000\$\]

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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 mathematics, physics, and engineering to simplify complex calculations and expressions.

Understanding the Given Expression

The given expression is 6.7×10−46.7 \times 10^{-4}A. In this expression, 6.7 is the coefficient, and 10−410^{-4} is the exponent. The negative exponent indicates that the number is very small, and the coefficient is multiplied by 10 to the power of -4.

Simplifying the Expression

To simplify the expression, we need to multiply the coefficient by 10 to the power of -4. This can be done by moving the decimal point in the coefficient four places to the left.

6.7 × 10^(-4) = 0.000067

Analyzing the Options

Now that we have simplified the expression, let's analyze the options:

A. {-0.000067$}$ - This option is incorrect because the original expression does not have a negative sign.

B. ${0.000067\$} - This option is correct because it matches the simplified expression.

C. ${0.00067\$} - This option is incorrect because it has a decimal point in the wrong place.

D. {-6.7000$}$ - This option is incorrect because it has a negative sign and the decimal point is in the wrong place.

Conclusion

In conclusion, the correct answer is B. ${0.000067\$}. This option matches the simplified expression and is the correct representation of 6.7×10−46.7 \times 10^{-4}A.

Why is Scientific Notation Important?

Scientific notation is an essential tool in mathematics, physics, and engineering. It allows us to express very large or very small numbers in a more manageable form, making it easier to perform calculations and understand complex concepts.

Real-World Applications of Scientific Notation

Scientific notation has numerous real-world applications, including:

  • Physics: Scientific notation is used to express the speed of light, the Planck constant, and other fundamental physical constants.
  • Engineering: Scientific notation is used to express the dimensions of complex systems, such as electrical circuits and mechanical systems.
  • Computer Science: Scientific notation is used to express the size of data structures, such as arrays and linked lists.

Tips for Working with Scientific Notation

Here are some tips for working with scientific notation:

  • Use the correct exponent: Make sure to use the correct exponent when multiplying or dividing numbers in scientific notation.
  • Move the decimal point correctly: When multiplying or dividing numbers in scientific notation, make sure to move the decimal point correctly.
  • Use the correct notation: Use the correct notation when expressing numbers in scientific notation, including the use of positive and negative exponents.

Common Mistakes to Avoid

Here are some common mistakes to avoid when working with scientific notation:

  • Incorrect exponent: Make sure to use the correct exponent when multiplying or dividing numbers in scientific notation.
  • Incorrect decimal point: Make sure to move the decimal point correctly when multiplying or dividing numbers in scientific notation.
  • Incorrect notation: Make sure to use the correct notation when expressing numbers in scientific notation, including the use of positive and negative exponents.

Conclusion

Q&A: Simplifying 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 simplify an expression in scientific notation? A: To simplify an expression in scientific notation, you need to multiply the coefficient by 10 to the power of the exponent. For example, if you have 6.7×10−46.7 \times 10^{-4}, you would multiply 6.7 by 10 to the power of -4.

Q: What is the correct way to express a number in scientific notation? A: The correct way to express a number in scientific notation is to use a number between 1 and 10, multiplied by a power of 10. For example, 6.7×10−46.7 \times 10^{-4} is a correct expression in scientific notation.

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 coefficient by 10 to the power of the exponent. For example, if you have 6.7×10−46.7 \times 10^{-4}, you would multiply 6.7 by 10 to the power of -4, which would give you 0.000067.

Q: What is the difference between positive and negative exponents in scientific notation? A: Positive exponents in scientific notation indicate that the number is very large, while negative exponents indicate that the number is very small. For example, 6.7×1046.7 \times 10^{4} is a very large number, while 6.7×10−46.7 \times 10^{-4} is a very small number.

Q: How do I add or subtract numbers in scientific notation? A: To add or subtract numbers in scientific notation, you need to have the same exponent. For example, if you have 6.7×10−46.7 \times 10^{-4} and 3.4×10−43.4 \times 10^{-4}, you can add them by adding the coefficients and keeping the same exponent.

Q: How do I multiply or divide numbers in scientific notation? A: To multiply or divide numbers in scientific notation, you need to multiply or divide the coefficients and add or subtract the exponents. For example, if you have 6.7×10−46.7 \times 10^{-4} and 3.4×1043.4 \times 10^{4}, you would multiply the coefficients and add the exponents.

Q: What are some common mistakes to avoid when working with scientific notation? A: Some common mistakes to avoid when working with scientific notation include:

  • Incorrect exponent
  • Incorrect decimal point
  • Incorrect notation
  • Not following the rules for adding or subtracting numbers in scientific notation
  • Not following the rules for multiplying or dividing numbers in scientific notation

Q: How can I practice working with scientific notation? A: You can practice working with scientific notation by:

  • Using online resources and calculators to practice converting numbers from scientific notation to standard notation
  • Using online resources and calculators to practice adding, subtracting, multiplying, and dividing numbers in scientific notation
  • Practicing converting numbers from standard notation to scientific notation
  • Practicing adding, subtracting, multiplying, and dividing numbers in scientific notation

Conclusion

In conclusion, scientific notation is an essential tool in mathematics, physics, and engineering. It allows us to express very large or very small numbers in a more manageable form, making it easier to perform calculations and understand complex concepts. By following the tips and avoiding common mistakes, we can master the art of working with scientific notation and apply it to real-world problems.