How to Multiply Numbers in Scientific Notation
Scientific notation writes a number as a coefficient (or mantissa) multiplied by a power of ten: a × 10ⁿ, where 1 ≤ |a| < 10 and n is an integer. To multiply two numbers written this way, you don't need to expand them back to standard form — you multiply the coefficients together and add the exponents: (a × 10^m) × (b × 10^n) = (a × b) × 10^(m + n). If the resulting coefficient (a × b) is not already between 1 and 10, you then renormalize it by shifting the decimal point and adjusting the exponent to match.
How the calculation works
Enter the coefficient and exponent for each number. This calculator multiplies the two coefficients (a × b) and adds the two exponents (m + n) to get the raw, unnormalized product. It then checks whether the resulting coefficient falls in the standard range of 1 to 10: if the coefficient is 10 or greater, it divides by 10 and adds 1 to the exponent (repeating as needed); if the coefficient is less than 1 (and nonzero), it multiplies by 10 and subtracts 1 from the exponent. The result is the normalized answer in proper scientific notation, alongside the same value expanded into ordinary decimal form.
Worked example
Multiply (2.5 × 10³) × (4 × 10⁻²). First multiply the coefficients: 2.5 × 4 = 10. Then add the exponents: 3 + (−2) = 1. That gives an unnormalized product of 10 × 10¹. Since 10 is outside the 1–10 range, divide the coefficient by 10 and add 1 to the exponent: 1.0 × 10². As a decimal, that is 100.
Common mistakes
- Multiplying the exponents instead of adding them: the exponent rule for multiplication is 10^m × 10^n = 10^(m+n) — exponents add, they never multiply.
- Forgetting to renormalize: a coefficient of 10 or more (or less than 1) is not valid scientific notation — always shift the decimal and adjust the exponent until 1 ≤ |a| < 10.
- Sign errors with negative exponents: adding a negative exponent means subtracting, e.g. 3 + (−2) = 1, not 3 − 2 = 5 or 3 + 2 = 5.
Real-world applications
- Physics and chemistry calculations routinely multiply very large or very small measured quantities, such as combining Avogadro's number with a molar mass, or a distance in meters with a frequency in hertz.
- Engineering and computing use scientific notation to multiply quantities that span many orders of magnitude, such as processor clock speeds or data storage capacities, without writing out long strings of zeros.
- Astronomy relies on scientific notation multiplication for distances, masses, and luminosities of stars and galaxies, where standard decimal notation would be unwieldy.