Scientific notation is a compact way to write very large or very small numbers using powers of ten. It has the form a × 10^n, where the coefficient a is at least 1 and less than 10. This format is important in science because measurements like atomic sizes, light speeds, and planetary distances often contain many zeros.
Operations become easier when the powers of ten are handled separately from the coefficients.
Key Facts
- Scientific notation format: a × 10^n, where 1 ≤ a < 10 and n is an integer.
- Multiplication rule: (a × 10^m)(b × 10^n) = (ab) × 10^(m + n).
- Division rule: (a × 10^m) ÷ (b × 10^n) = (a ÷ b) × 10^(m - n).
- Addition and subtraction require matching exponents first: 3.2 × 10^5 + 4.1 × 10^5 = 7.3 × 10^5.
- If the coefficient is 10 or greater, move the decimal left and increase the exponent: 45 × 10^6 = 4.5 × 10^7.
- If the coefficient is less than 1, move the decimal right and decrease the exponent: 0.62 × 10^-3 = 6.2 × 10^-4.
Vocabulary
- Scientific notation
- A way to write a number as a coefficient multiplied by a power of ten.
- Coefficient
- The decimal number in scientific notation that must be at least 1 and less than 10.
- Exponent
- The integer power on 10 that shows how many places the decimal point has shifted.
- Standard form
- The usual way of writing a number without powers of ten, such as 45000 or 0.0062.
- Significant figures
- The meaningful digits in a measured value that show its precision.
Common Mistakes to Avoid
- Adding exponents when adding numbers is wrong because exponent rules for powers of ten apply to multiplication, not addition. First rewrite the numbers with matching exponents, then add the coefficients.
- Leaving a coefficient outside the range 1 ≤ a < 10 is wrong because the answer is not in proper scientific notation. Adjust the decimal point and exponent until the coefficient is in the correct range.
- Forgetting to subtract exponents during division is wrong because dividing powers of ten uses 10^m ÷ 10^n = 10^(m - n). Keep the coefficient division and exponent subtraction as separate steps.
- Ignoring significant figures is wrong because final answers should reflect the precision of the given measurements. Round multiplication and division answers to the same number of significant figures as the least precise measurement.
Practice Questions
- 1 Multiply and write the answer in proper scientific notation: (3.0 × 10^4)(2.5 × 10^6).
- 2 Add and write the answer in proper scientific notation: 6.4 × 10^5 + 2.8 × 10^4.
- 3 Explain why 58.2 × 10^-3 is not properly written in scientific notation, then describe how to correct it.