How to Multiply Numbers in Scientific Notation

Multiplying numbers in scientific notation is easier than multiplying the long-hand versions — that's the whole point of the notation. There is one rule: multiply the coefficients, add the exponents. This guide walks through the rule, the worked examples you'll actually meet, and the two places everyone slips up.

The rule

Any two numbers in scientific notation multiply like this:

(a × 10^m) × (b × 10ⁿ) = (a × b) × 10^m+n

It works because multiplication doesn't care about order or grouping — you can pull the two coefficients together and the two powers of ten together. The coefficients multiply like ordinary numbers, and the powers of ten combine by the exponent product rule: 10^m × 10ⁿ = 10^m+n. Ten to the fourth times ten to the third is seven factors of ten, so the exponents add.

Step by step

1. Multiply the coefficients. Treat them as plain decimals.
2. Add the exponents. Keep the signs — a negative exponent stays negative in the sum.
3. Normalize if needed. Proper scientific notation keeps the coefficient between 1 and 10. If step 1 gave you 10 or more, move the decimal one place left and add 1 to the exponent.

Worked examples

The clean case: (3 × 10⁴) × (2 × 10³). Coefficients: 3 × 2 = 6. Exponents: 4 + 3 = 7. Answer: 6 × 10⁷.

The normalize case: (4 × 10⁵) × (5 × 10⁻²). Coefficients: 4 × 5 = 20. Exponents: 5 + (−2) = 3. That gives 20 × 10³ — but 20 isn't a legal coefficient, so shift: 2 × 10⁴.

Negative exponents: (6 × 10⁻³) × (1.5 × 10⁻²). Coefficients: 6 × 1.5 = 9. Exponents: −3 + (−2) = −5. Answer: 9 × 10⁻⁵.

A real one: light travels 3 × 10⁸ m/s; a year is about 3.15 × 10⁷ seconds. Distance = (3 × 3.15) × 10⁸⁺⁷ = 9.45 × 10¹⁵ m — one light-year, no long multiplication required.

The mistakes to avoid

• Multiplying the exponents. 10⁴ × 10³ is 10⁷, never 10¹². Adding is the rule; multiplying exponents belongs to powers of powers, (10⁴)³.
• Forgetting to normalize. 20 × 10³ equals 2 × 10⁴, but only the second is scientific notation — and unnormalized answers get marked wrong.
• Dropping a sign. 5 + (−2) is 3, not 7. Write the exponent sum out with its signs before simplifying.
• Keeping too many digits. In science class, round the coefficient to the smallest number of significant figures either factor had.

To check any answer — or to see the normalization step done for you — put both numbers into the scientific notation calculator's math tab; it returns the normalized result with the exponent rule it applied. Division works by the mirror-image rule — subtract the exponents — while addition and subtraction need a different first move: matching the exponents.