Scientific notation

Video lesson4 worked examples

Theory

You must be able to perform calculations using scientific notation.

It is heavily linked to science and technology to represent very large or very small numbers (e.g., ms⁻¹).

The Golden Rule: a number in scientific notation is written as $a \times 10^n$ where $a$ is between 1 and 10 (that is, $1 \le a < 10$ ). Always check the front number is in that range — especially after a calculation, where it can drift out.

⚠️ Common Examiner Traps

Front number out of range: $42 \times 10^3$ is not standard form. Rewrite it as $4.2 \times 10^4$ .
Sign of the index: small numbers below 1 take a negative power, e.g. $0.0003 = 3 \times 10^{-4}$ .
Not re-standardising after multiplying: if multiplying the front numbers gives, say, $20 \times 10^5$ , adjust it to $2 \times 10^6$ .
Calculator display: a screen showing $4.2^{06}$ or “4.2E6” means $4.2 \times 10^6$ — write it out in full.

Worked examples

Example 1

Percentage of a Number in Scientific Notation

Calculate 6.1% of $3.27 \times 10^{-22}$ grams.

Step 1: Convert the percentage to a fraction: $\tfrac{6.1}{100}$ .

Step 2: Multiply by the standard form number: $\frac{6.1}{100} \times 3.27 \times 10^{-22}$ .

Answer: $1.99 \times 10^{-23}$ grams.

Example 2

Dividing in Scientific Notation

Calculate the following, giving your answer in scientific notation:

\frac{8 \times 10^5}{2 \times 10^2}

Step 1: Divide the front numbers: $8 \div 2 = 4$ .

Step 2: Subtract the indices for the powers of 10: $10^5 \div 10^2 = 10^3$ .

Answer: $4 \times 10^3$ .

Example 3

Writing Numbers in Scientific Notation

Write (a) $4{,}530{,}000$ and (b) $0.00072$ in scientific notation.

Step 1 (a): Place the decimal point after the first non-zero digit to get a front number between 1 and 10: $4.53$ . Count how many places the point moved — 6 places left — so the index is $+6$ .

4{,}530{,}000 = 4.53 \times 10^6

Step 2 (b): Again make the front number $7.2$ . This time the point moves 4 places right to get there, so the index is negative:

0.00072 = 7.2 \times 10^{-4}

Example 4

🔗 Bringing it together

Calculate $(5 \times 10^4) \times (6 \times 10^3)$ , giving your answer in scientific notation.

Step 1: Multiply the front numbers, and add the indices for the powers of 10: $5 \times 6 = 30$ and $10^4 \times 10^3 = 10^7$ , giving $30 \times 10^7$ .

Step 2: This is not yet in scientific notation, because $30$ is not between 1 and 10. Re-standardise by writing $30 = 3 \times 10^1$ :

30 \times 10^7 = 3 \times 10^1 \times 10^7 = 3 \times 10^8

Answer: $3 \times 10^8$ .

Ready to practise?

11 Scientific Notation questions with answers and video solutions