Stephen Evaluated \[$\left(6.34 \times 10^{-7}\right)\left(4.5 \times 10^3\right)\$\]. His Work Is Shown Below. Which Two Statements Describe The Errors Stephen Made?Step 1: \[$\left(6.34 \times 10^{-7}\right)\left(4.5 \times

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Introduction

Scientific notation is a way of expressing very large or very small numbers in a compact form. It is commonly used in mathematics, physics, and engineering to simplify calculations and make them more manageable. In this article, we will discuss the concept of scientific notation and how to multiply numbers in this notation. We will also evaluate the work of Stephen, who attempted to multiply two numbers in scientific notation, and identify the errors he made.

What is Scientific Notation?

Scientific notation is a way of expressing a number as a product of a number between 1 and 10 and a power of 10. For example, the number 456,000 can be expressed in scientific notation as 4.56 × 10^5. Similarly, the number 0.000456 can be expressed in scientific notation as 4.56 × 10^-4.

Multiplying Numbers in Scientific Notation

When multiplying numbers in scientific notation, we multiply the coefficients (the numbers between 1 and 10) and add the exponents of the powers of 10. For example, to multiply 4.56 × 10^5 and 2.34 × 10^3, we multiply the coefficients 4.56 and 2.34 to get 10.648, and add the exponents 5 and 3 to get 8. Then, we write the result as 10.648 × 10^8.

Stephen's Work

Stephen evaluated the product of two numbers in scientific notation: (6.34 × 10^-7) and (4.5 × 10^3). His work is shown below:

  1. Step 1: (6.34 × 10^-7) × (4.5 × 10^3)
  2. Step 2: (6.34 × 4.5) × (10^-7 × 10^3)
  3. Step 3: 28.47 × 10^(-7 + 3)
  4. Step 4: 28.47 × 10^(-4)

Errors in Stephen's Work

Based on the correct procedure for multiplying numbers in scientific notation, we can identify the errors in Stephen's work.

  • Error 1: In Step 2, Stephen multiplied the coefficients 6.34 and 4.5 to get 28.47, which is correct. However, he incorrectly added the exponents -7 and 3 to get -4. The correct result should be -4 + 7 = 3.
  • Error 2: In Step 4, Stephen wrote the result as 28.47 × 10^(-4), which is incorrect. The correct result should be 28.47 × 10^3.

Conclusion

In conclusion, Stephen made two errors in his work. He incorrectly added the exponents in Step 2 and wrote the result in the wrong form in Step 4. To avoid these errors, it is essential to follow the correct procedure for multiplying numbers in scientific notation, which involves multiplying the coefficients and adding the exponents.

Key Takeaways

  • Scientific notation is a way of expressing very large or very small numbers in a compact form.
  • When multiplying numbers in scientific notation, we multiply the coefficients and add the exponents of the powers of 10.
  • To avoid errors, it is essential to follow the correct procedure for multiplying numbers in scientific notation.

Practice Problems

  1. Multiply (3.21 × 10^4) and (2.15 × 10^2).
  2. Multiply (4.56 × 10^-3) and (6.78 × 10^4).
  3. Multiply (9.87 × 10^6) and (3.21 × 10^-2).

Answer Key

  1. (3.21 × 10^4) × (2.15 × 10^2) = 6.9185 × 10^6
  2. (4.56 × 10^-3) × (6.78 × 10^4) = 3.09348 × 10^2
  3. (9.87 × 10^6) × (3.21 × 10^-2) = 3.17267 × 10^5
    Frequently Asked Questions (FAQs) on 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 compact form. It is commonly used in mathematics, physics, and engineering to simplify calculations and make them more manageable.

Q: How do I multiply numbers in scientific notation?

A: When multiplying numbers in scientific notation, you multiply the coefficients (the numbers between 1 and 10) and add the exponents of the powers of 10. For example, to multiply 4.56 × 10^5 and 2.34 × 10^3, you multiply the coefficients 4.56 and 2.34 to get 10.648, and add the exponents 5 and 3 to get 8. Then, you write the result as 10.648 × 10^8.

Q: What are the rules for adding exponents in scientific notation?

A: When adding exponents in scientific notation, you add the exponents of the powers of 10. For example, to add 10^5 and 10^3, you add the exponents 5 and 3 to get 8. Then, you write the result as 10^8.

Q: Can I multiply numbers in scientific notation with different powers of 10?

A: Yes, you can multiply numbers in scientific notation with different powers of 10. For example, to multiply 4.56 × 10^5 and 2.34 × 10^3, you multiply the coefficients 4.56 and 2.34 to get 10.648, and add the exponents 5 and 3 to get 8. Then, you write the result as 10.648 × 10^8.

Q: How do I divide numbers in scientific notation?

A: When dividing numbers in scientific notation, you divide the coefficients (the numbers between 1 and 10) and subtract the exponents of the powers of 10. For example, to divide 4.56 × 10^5 by 2.34 × 10^3, you divide the coefficients 4.56 and 2.34 to get 1.96, and subtract the exponents 5 and 3 to get 2. Then, you write the result as 1.96 × 10^2.

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

A: Yes, you can convert a number from scientific notation to standard notation by multiplying the coefficient by the power of 10. For example, to convert 4.56 × 10^5 to standard notation, you multiply the coefficient 4.56 by the power of 10 10^5 to get 456,000.

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

A: Yes, you can convert a number from standard notation to scientific notation by expressing the number as a product of a number between 1 and 10 and a power of 10. For example, to convert 456,000 to scientific notation, you express the number as 4.56 × 10^5.

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:

  • Adding the exponents instead of subtracting them when dividing numbers.
  • Writing the result in the wrong form, such as writing 10^8 as 10^5.
  • Failing to multiply the coefficients correctly.
  • Failing to add the exponents correctly.

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

A: You can practice multiplying numbers in scientific notation by working through examples and exercises. You can also use online resources, such as calculators and worksheets, to help you practice. Additionally, you can try multiplying numbers in scientific notation with different powers of 10 to help you understand the concept better.