Scientific Notation Worksheets

Both directions, both sizes

Write 45,300 as 4.53 × 10⁴. Or go the other way: 3.6 × 10⁵ is 360,000. Mixed alternates.

Sixteen questions by default, and the set covers small numbers too — negative exponents for anything below 1.

The form

One digit before the decimal point, then the rest, then a power of ten. The leading number must be at least 1 and less than 10.

That constraint is what makes the notation useful: it is a standard form, so two numbers written this way can be compared at a glance by their exponents. 45.3 × 10³ equals the right amount and is not in scientific notation, and marking it wrong is not pedantry — the whole point is that there is exactly one way to write each number.

Counting the moves

The exponent is how many places the point moved, and the sign says which way.

Moving left, to make a big number smaller, gives a positive exponent. Moving right, to make a small number bigger, gives a negative one. 0.00072 becomes 7.2 × 10⁻⁴, because the point travelled four places right.

The reliable check is size: a positive exponent means a number bigger than 10, a negative one means smaller than 1. A pupil who has written 0.00072 as 7.2 × 10⁴ can catch it in a second by asking whether their answer is large or small.

Why anyone bothers

Because the alternative is counting zeros, and people are bad at counting zeros.

The speed of light is 3 × 10⁸ metres per second. A hydrogen atom is about 1 × 10⁻¹⁰ metres across. Written out, those are strings of digits nobody can read accurately or compare — and in science, orders of magnitude are usually what matter, which is precisely what the exponent shows.

The link to index laws

Multiplying two numbers in scientific notation means multiplying the fronts and adding the exponents. Dividing means subtracting them.

That is not a new rule, it is the index laws applied. Pupils who see the connection stop treating this as a separate topic, and their arithmetic with very large and very small numbers gets suddenly easier.

Level and use

Typically thirteen to sixteen in maths, and it turns up in science well before that in practice. Sixteen questions is a lesson; run one-direction sheets first and mix them once both are secure.

Related: exponents and powers.