Overview
This lecture reviews scientific notation, including how to convert between scientific and decimal notation, perform arithmetic operations, and handle roots and powers with scientific notation.
Introduction to Scientific Notation
- Scientific notation is a way to write very large or very small numbers using powers of 10.
- Example: 1,000,000,000 = 1 × 10⁹ and 0.000064 = 6.4 × 10⁻⁵.
Converting Scientific Notation to Decimal Notation
- Multiply the decimal by 10 raised to the exponent (e.g., 2.3 × 10² = 230).
- For positive exponents, move the decimal to the right; for negative exponents, move it to the left.
- Example: 4.28 × 10⁴ = 42,800; 3.4 × 10⁻² = 0.034.
Converting Decimal Numbers to Scientific Notation
- Move the decimal so one nonzero digit is left of the decimal point.
- Each move left increases the exponent by one; each move right decreases it by one.
- Example: 32,500 = 3.25 × 10⁴; 0.024 = 2.4 × 10⁻².
Rules for Manipulating Exponents
- Moving the decimal left increases the exponent; moving right decreases it.
- Scientific notation requires a coefficient between 1 and 10 (e.g., 4.28 × 10⁶, not 42.8 × 10⁵).
Multiplying and Dividing in Scientific Notation
- To multiply: multiply coefficients, add exponents (e.g., 4 × 10³ × 2 × 10⁵ = 8 × 10⁸).
- To divide: divide coefficients, subtract exponents (e.g., 12 × 10⁶ ÷ 3 × 10⁻⁴ = 4 × 10¹⁰).
- Adjust the result to keep the coefficient between 1 and 10.
Adding and Subtracting in Scientific Notation
- Terms must have the same exponent before combining coefficients.
- Convert exponents if needed (move decimal, adjust exponent) before adding or subtracting.
- Example: 5 × 10³ + 4 × 10³ = 9 × 10³.
Roots and Powers in Scientific Notation
- To find roots, take the root of the coefficient and divide the exponent by the root’s index.
- Example: √(4 × 10⁶) = 2 × 10³; ³√(8 × 10⁹) = 2 × 10³.
- To raise to a power, raise the coefficient and multiply the exponent by the power.
Mental Shortcuts with Powers of Ten
- Recognize 10³ = thousand, 10⁶ = million, 10⁹ = billion to convert quickly.
Key Terms & Definitions
- Scientific Notation — A way to express numbers as a product of a decimal and a power of ten.
- Decimal Notation — Standard way of writing numbers without exponents.
- Exponent — Indicates how many times to multiply the base (10) by itself.
- Coefficient — The number (1–10) multiplied by the power of ten in scientific notation.
Action Items / Next Steps
- Practice converting between scientific and decimal notation.
- Complete assigned homework on scientific notation operations.
- Review rules for arithmetic with exponents before the next class.