Multiplying Scientific Notation and Normalizing

🔑 Key Concepts

  • Multiply the coefficients first.
  • Use the product rule for powers of 10: \(10^a \cdot 10^b = 10^{a+b}\).
  • If the coefficient is greater than or equal to 10, rewrite the answer in proper scientific notation.
  • Normalizing the answer may increase the exponent by 1.

💡 Main Idea

When multiplying numbers in scientific notation, multiply the decimal coefficients and add the exponents on the powers of 10. If the coefficient is not between 1 and 10, normalize the answer by moving the decimal point and adjusting the exponent.

✏️ Worked Examples

🧠 Math Vocabulary

  • Scientific notation: A way to write very large or very small numbers using a coefficient and a power of 10.
  • Coefficient: The decimal factor in scientific notation, such as \(3.4\) in \(3.4 \times 10^3\).
  • Power of 10: A number written as \(10^n\), where the exponent controls the size of the number.
  • Product rule: When multiplying powers with the same base, add the exponents.
  • Normalize: Rewrite the coefficient so it is at least 1 but less than 10.
  • Negative exponent: An exponent that represents a very small value or repeated division by 10.

📚 What You Should Already Know

Students should already know how to multiply decimals, use powers of 10, apply basic exponent rules, and recognize whether a number is written in proper scientific notation.

🚀 What Comes Next

Next, students can divide numbers in scientific notation, solve mixed multiplication and division problems, and apply scientific notation to real-world quantities such as distances, speed, and measurement.

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