Performing calculations using scientific notation

Multiplying two numbers in scientific notation

Multiply the coefficients together, and add the exponents of the powers of 10:

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

Simple example

(3 × 10⁴) × (2 × 10³) = (3 × 2) × 10^(4+3) = 6 × 10⁷

Example requiring adjustment

(5 × 10³) × (4 × 10²) = 20 × 10⁵

Here, 20 is not between 1 and 10. We must therefore rewrite 20 as 2 × 10¹ and then group the terms:

20 × 10⁵ = 2 × 10¹ × 10⁵ = 2 × 10⁶

Dividing two numbers in scientific notation

We divide the coefficients and subtract the exponents:

(a × 10^n) / (b × 10^m) = (a/b) × 10^(n-m)

Example: (8 × 10⁶) / (2 × 10³) = 4 × 10³

What is it actually used for?

Scientific notation is used in astronomy (distances between stars), physics (size of atoms, speed of light: approximately 3 × 10⁸ m/s) and chemistry, as it allows us to easily compare the order of magnitude of several measurements without having to count the zeros.

Common pitfalls

• After multiplication or division, always check that the final coefficient satisfies 1 ≤ a < 10; if not, you must adjust it as in the example above.
• Do not forget to add (or subtract) the exponents of the powers of 10 as well as dealing with the coefficients: these are two separate operations to be carried out in parallel.