What is the rule for adding scientific notation?

The exponents must match before you can add. Rewrite one number so both share the same power of ten, add the coefficients, and keep that power: 3.1 × 10^4 + 4.2 × 10^3 becomes 3.1 × 10^4 + 0.42 × 10^4 = 3.52 × 10^4. Normalize at the end if the coefficient left the 1–10 range.

Why do the exponents have to be the same?

For the same reason you can't add 3 meters to 5 centimeters without converting: the power of ten is the unit each coefficient counts in. 10^4 and 10^3 are different-sized units, so their coefficients don't combine until both numbers are expressed in the same one.

Which number should I rewrite?

Either works, but it's safest to rewrite the smaller-exponent number up to the larger exponent — its coefficient shifts below 1 (0.42), which is fine mid-calculation. The answer then usually needs no normalization, or at most a small one.

Why does subtracting sometimes change the exponent a lot?

When two numbers are close, their difference is much smaller than either — 5.0 × 10^3 − 4.8 × 10^3 = 0.2 × 10^3, which normalizes to 2 × 10^2. The subtraction cancelled the leading digits, so the answer legitimately lives at a lower power of ten. In measured data this cancellation also amplifies relative error, which is why experimentalists avoid subtracting nearly equal quantities.

Scientific Notation

How to Add and Subtract in Scientific Notation

LK Forge · 4 min read · Updated July 2026

Addition and subtraction are the odd ones out in scientific notation. Multiplication and division work on the parts separately — coefficients here, exponents there. Adding doesn't: you can only add coefficients that count the same power of ten. So the whole method is: match the exponents, combine the coefficients, normalize.

Why matching comes first

The power of ten is a unit. 3.1 × 10⁴ means "3.1 lots of ten thousand" and 4.2 × 10³ means "4.2 lots of a thousand" — adding 3.1 + 4.2 there would be adding ten-thousands to thousands, like adding meters to centimeters without converting. Line the numbers up in the same unit and the coefficients add like ordinary decimals. (It's the same reason long addition lines up place values in columns.)

Step by step

Match the exponents. Rewrite the smaller-exponent number at the larger exponent: each step up in exponent moves its decimal point one place left. 4.2 × 10³ = 0.42 × 10⁴.
Add or subtract the coefficients. The shared power of ten comes along unchanged.
Normalize. If the result's coefficient is 10 or more, or under 1, shift the decimal and adjust the exponent to restore proper form.

Worked examples

Addition: 3.1 × 10⁴ + 4.2 × 10³ = 3.1 × 10⁴ + 0.42 × 10⁴ = 3.52 × 10⁴.

Addition with carry: 9.6 × 10⁵ + 8 × 10⁴ = 9.6 × 10⁵ + 0.8 × 10⁵ = 10.4 × 10⁵ → normalize → 1.04 × 10⁶.

Subtraction with cancellation: 5.0 × 10³ − 4.8 × 10³ = 0.2 × 10³ → normalize the other way → 2 × 10².

Negative exponents: 6 × 10⁻³ + 5 × 10⁻⁴ = 6 × 10⁻³ + 0.5 × 10⁻³ = 6.5 × 10⁻³. The matching step works identically below zero.

The mistakes to avoid

Adding the exponents. That's multiplication's rule. In addition the shared exponent passes through untouched — only the coefficients combine.
Adding coefficients across different exponents. 3.1 + 4.2 = 7.3 is meaningless when one counts 10⁴ and the other 10³. Match first, always.
Shifting the decimal the wrong way. Raising the exponent by one moves the decimal one place left. Check with a small number: 42 = 4.2 × 10¹ = 0.42 × 10².
Skipping the final normalization. 10.4 × 10⁵ and 0.2 × 10³ are correct values in illegal clothing — normalize before writing the answer down.

The math tab of the scientific notation calculator does the matching, the combining and the normalization in one step — useful for checking homework line by line. For the other two operations, the rules are simpler: multiplication adds the exponents and division subtracts them.

Multiply, divide, add or subtract any two numbers in scientific notation — normalized answer with the exponent rule shown.

Open the Scientific Notation Calculator