How to calculate with scientific notation

For a positive number, scientific notation has the form a × 10ⁿ, with 1 ≤ a < 10 and n an integer. For a negative number, the absolute value of the coefficient lies in that range. For example, 480000 = 4.8 × 10⁵, while 0.00048 = 4.8 × 10⁻⁴. Zero is a separate case and does not have a nonzero coefficient in this range.

Count decimal-place shifts rather than zeros. Moving the decimal point five places left turns 480000 into 4.8, so multiply 4.8 by 10⁵ to restore the original value. Moving it four places right turns 0.00048 into 4.8, so multiply by 10⁻⁴ to undo that enlargement.

A negative exponent does not mean a negative number. The value 10⁻³ is 1/1000, so 2 × 10⁻³ = 0.002 is positive. By contrast, −2 × 10³ = −2000 is negative because the coefficient has a minus sign.

Calculate (3 × 10⁵)(4 × 10⁻³). Multiply 3 by 4 to get 12. Add the exponents: 5 + (−3) = 2. This gives 12 × 10², which equals 1200.

The coefficient 12 is too large for scientific notation. Rewrite 12 as 1.2 × 10¹, then combine the powers: 1.2 × 10¹ × 10² = 1.2 × 10³. Moving the coefficient's decimal point left by one place requires increasing the exponent by one.

For a quick size check, 3 × 10⁵ is 300000 and 4 × 10⁻³ is 0.004. Multiplying by 0.004 makes the first quantity much smaller, so 1200 is plausible. An answer of 1.2 × 10⁸ would fail that check.

Calculate (6 × 10⁻⁴)/(3 × 10²). Divide 6 by 3 to get 2, then subtract the denominator's exponent: −4 − 2 = −6. The answer is 2 × 10⁻⁶.

Keep brackets when the denominator has a negative exponent. For (8 × 10³)/(2 × 10⁻²), the coefficient is 4 and the exponent is 3 − (−2) = 5. The answer is 4 × 10⁵. Dividing by 0.02 makes the original positive number larger, which fits this result.

You may need to adjust a coefficient smaller than one. For (2 × 10³)/(5 × 10⁵), the first result is 0.4 × 10⁻². Rewrite 0.4 as 4 × 10⁻¹ to get 4 × 10⁻³. Moving the coefficient's decimal point right by one place requires decreasing the exponent by one.

Addition uses a different rule from multiplication. To calculate 3.2 × 10⁵ + 4.5 × 10⁴, first rewrite the second term as 0.45 × 10⁵. Now add the coefficients: (3.2 + 0.45) × 10⁵ = 3.65 × 10⁵. Do not add the exponents.

For 6.1 × 10⁻³ − 8 × 10⁻⁴, rewrite the second term as 0.8 × 10⁻³. Subtraction gives 5.3 × 10⁻³. In decimal form, this is 0.0061 − 0.0008 = 0.0053.

The terms must also describe compatible quantities. Convert to the same unit before adding measurements expressed in different units. Matching the powers of ten cannot make a length and a time into quantities you can add. When multiplying or dividing measurements, carry the units through that operation as well.

A calculator display such as 2E−6 means 2 × 10⁻⁶. A dedicated EXP or EE key enters the power-of-ten part of a number; it is not another multiplication by ten to add afterward. Check your calculator's key labels, since entry methods differ.

When typing with ordinary multiplication and exponent keys, keep each complete quantity in brackets. For example, enter (6 × 10^(−4))/(3 × 10^2). The brackets ensure the entire denominator is used. Use a negative-number entry key where your calculator requires it, rather than assuming subtraction and a negative sign use the same key.

Keep extra digits during intermediate calculations and round the final result to the precision requested by the question. Scientific notation does not itself tell you how many significant figures to use. If a final value is 9.96 × 10³ and the instruction is two significant figures, rounding gives 1.0 × 10⁴. The trailing zero records those two significant figures.

Copy this template: original calculation; units checked; operation on coefficients; operation on exponents, or shared power for addition; unadjusted result; coefficient adjusted into range; final rounding; estimate or decimal check.

Try (2.5 × 10⁻³)(8 × 10⁶) before reading on. The coefficient product is 20, and the exponent sum is −3 + 6 = 3. So the result is 20 × 10³ = 2 × 10⁴. If the question asks for two significant figures, write 2.0 × 10⁴.

For addition practice, 7.2 × 10⁴ + 9 × 10³ becomes (7.2 + 0.9) × 10⁴ = 8.1 × 10⁴. If your result is smaller than either positive term, check whether you accidentally applied a multiplication rule or changed the value while matching exponents.