Standard form (scientific notation) explained

Count how many places the decimal point moves to get a number between 1 and 10:

• $7{,}200{,}000$: point moves 6 places left, so $7.2 \times 10^6$.
• $0.00052$: point moves 4 places right, so $5.2 \times 10^{-4}$.
• $3.9$ is already between 1 and 10: $3.9 \times 10^0$.

Check any conversion with our standard form calculator.

• $6.04 \times 10^5 = 604{,}000$ (move the point 5 places right).
• $2.1 \times 10^{-3} = 0.0021$ (move the point 3 places left).

A negative power doesn't make the number negative; it makes it small.

Multiply the numbers and add the powers; divide the numbers and subtract the powers. These are the laws of indices.

• $(3 \times 10^4) \times (2 \times 10^5) = 6 \times 10^9$
• $(5 \times 10^3) \times (4 \times 10^6) = 20 \times 10^9 = 2 \times 10^{10}$ (adjust so the first number is below 10)
• $(8 \times 10^7) \div (2 \times 10^3) = 4 \times 10^4$
• $(3 \times 10^2) \div (6 \times 10^5) = 0.5 \times 10^{-3} = 5 \times 10^{-4}$

You can't just add the powers. Convert to ordinary numbers (or make the powers match), then convert back:

$3 \times 10^4 + 5 \times 10^3 = 30{,}000 + 5{,}000 = 35{,}000 = 3.5 \times 10^4$

Use the $\times 10^x$ or EXP key rather than typing "× 10 ^". To enter $3.2 \times 10^{-5}$, press 3.2, then $\times 10^x$, then $-5$. Many calculators show answers in standard form automatically when they're very big or very small.

• Leaving the first number above 10: $20 \times 10^9$ isn't standard form.
• Getting the sign of the power wrong for small numbers.
• Adding powers when adding numbers.
• Writing $10^{-3}$ as a negative number.

1. Write $83{,}000$ in standard form.
2. Write $0.000907$ in standard form.
3. Work out $(6 \times 10^5) \times (3 \times 10^{-2})$.
4. Work out $(9 \times 10^8) \div (3 \times 10^{-2})$.

Answers: $8.3 \times 10^4$, $9.07 \times 10^{-4}$, $1.8 \times 10^4$, $3 \times 10^{10}$. Rounding often comes with these: see significant figures.

You'll meet standard form in science lessons too. The mass of an electron is about $9.11 \times 10^{-31}$ kg and the average distance from the Earth to the Sun is about $1.5 \times 10^8$ km. Writing them out in full would mean 30 zeros after the decimal point, or eight digits before it, which is easy to get wrong.