What is standard form?
Standard form is a way of writing very large or very small numbers concisely, using powers of 10 (see the Place Value page for how these powers relate to place value columns). Every number in standard form is written as:
A × 10ⁿ
where A must be at least 1 but less than 10 (1 ≤ A < 10), and n is an integer (positive for large numbers, negative for small numbers).
Writing large numbers in standard form
Worked Example 1
Write 5,600,000 in standard form.
1
Place the decimal point after the first non-zero digit: 5.6
2
Count how many places the decimal point moved to get from 5.6 to 5,600,000: 6 places
3
Since the original number is large, the power of 10 is positive: 5.6 × 10⁶
Answer5.6 × 10⁶
Writing small numbers in standard form
Worked Example 2
Write 0.00043 in standard form.
1
Place the decimal point after the first non-zero digit: 4.3
2
Count how many places the decimal point moved to get from 0.00043 to 4.3: 4 places
3
Since the original number is small (less than 1), the power of 10 is negative: 4.3 × 10⁻⁴
Answer4.3 × 10⁻⁴
The exponent tells you how far a number is from 1, and in which direction.
Converting standard form to an ordinary number
Worked Example 3
Write 3.2 × 10⁵ as an ordinary number.
1
A positive power of 10 means the number is large — move the decimal point 5 places to the right
2
3.2 → 32 → 320 → 3200 → 32000 → 320000
Answer320,000
Calculating with standard form
To multiply or divide numbers in standard form, deal with the two parts separately: multiply/divide the front numbers, and add/subtract the powers of 10. Afterwards, check the front number is still between 1 and 10 — if not, adjust it.
Worked Example 4
Work out (a) (3 × 10⁴) × (2 × 10³) and (b) (8 × 10⁶) ÷ (4 × 10²)
1
(a) Multiply the front numbers: 3 × 2 = 6. Add the powers: 4 + 3 = 7. Result: 6 × 10⁷
2
(b) Divide the front numbers: 8 ÷ 4 = 2. Subtract the powers: 6 − 2 = 4. Result: 2 × 10⁴
Answer(a) 6 × 10⁷ (b) 2 × 10⁴
Watch out: if multiplying front numbers gives 10 or more (e.g. 6 × 5 = 30), the result isn't in standard form yet — rewrite 30 × 10⁷ as 3 × 10⁸, adjusting the power to compensate.
Practice questions
Work through each question before checking the answers.
Foundation (Grade 3–5)
Q1Write 47,000 in standard form.Foundation
Q2Write 0.0056 in standard form.Foundation
Q3Write 2.3 × 10³ as an ordinary number.Foundation
Q4Write 6 × 10⁻² as an ordinary number.Foundation
Q5Write 380,000 in standard form.Foundation
Higher (Grade 5–7)
Q6Work out (4 × 10³) × (2 × 10⁵), giving your answer in standard form.Higher
Q7Work out (9 × 10⁷) ÷ (3 × 10²), giving your answer in standard form.Higher
Q8Work out (2 × 10⁴) + (3 × 10³), giving your answer in standard form.Higher
Q9Work out (6 × 10⁴) × (5 × 10³), giving your answer in standard form.Higher
Q10Write 0.0000091 in standard form.Higher
Higher — Hard (Grade 8–9)
Q11Work out (2.5 × 10⁶) × (4 × 10⁻²), giving your answer in standard form.Grade 8–9
Q12The mass of a proton is approximately 1.67 × 10⁻²⁷ kg. Find the mass of 1000 protons, giving your answer in standard form.Grade 8–9
Q13Given that a = 3 × 10⁵ and b = 6 × 10⁸, find b ÷ a in standard form.Grade 8–9
Q14Simplify (2 × 10³)², giving your answer in standard form.Grade 8–9
Q15A country has a population of 6.4 × 10⁷ and an area of 2 × 10⁵ km². Find the population density in people per km², giving your answer in standard form.Grade 8–9
Answers
Foundation (Q1–Q5)
Q14.7 × 10⁴
Q25.6 × 10⁻³
Q32300
Q40.06
Q53.8 × 10⁵
Higher (Q6–Q10)
Q68 × 10⁸
Q73 × 10⁵
Q82.3 × 10⁴(20,000 + 3,000 = 23,000)
Q93 × 10⁸(6 × 5 = 30, so 30 × 10⁷ = 3 × 10⁸)
Q109.1 × 10⁻⁶
Higher — Hard (Q11–Q15)
Q111 × 10⁵(2.5 × 4 = 10, so 10 × 10⁴ = 1 × 10⁵)
Q121.67 × 10⁻²⁴ kg(1.67 × 10⁻²⁷ × 10³)
Q132 × 10³
Q144 × 10⁶(2² = 4, (10³)² = 10⁶)
Q153.2 × 10² (320 people per km²)
Common mistakes
Common Mistake 1
Writing the front number outside the range 1 ≤ A < 10
56 × 10⁵ is not standard form, because 56 is not less than 10. It must be rewritten as 5.6 × 10⁶ — always check the front number is between 1 and 10.
Common Mistake 2
Using the wrong sign for the power
Large numbers (greater than 1) use a positive power; small numbers (less than 1) use a negative power. Mixing these up is one of the most common errors in this topic.
Common Mistake 3
Forgetting to re-normalise after a calculation
If multiplying the front numbers gives 10 or more, the answer isn't finished — 30 × 10⁷ must become 3 × 10⁸, since 30 is outside the allowed range for a front number.
Common Mistake 4
Adding powers when the numbers should be added, not multiplied
The "add the powers" rule only applies when multiplying. For addition or subtraction of standard form numbers, convert to ordinary numbers first, add or subtract, then convert back.
Exam tips
💡 Exam Tip 1
Count decimal point moves carefully
The power of 10 always equals the number of places the decimal point moves. Counting on your fingers or writing out each shift explicitly avoids off-by-one errors.
💡 Exam Tip 2
Check your answer's size makes sense
Before finalising an answer, estimate roughly how big the original number was. If your standard form answer doesn't match that rough size, you've likely made a sign or place-value error.
💡 Exam Tip 3
Convert to ordinary numbers for addition and subtraction
Standard form's power rules only work directly for multiplication and division. For addition or subtraction, it's usually safest to convert both numbers to ordinary form first.
💡 Exam Tip 4
Know your calculator's standard form button
Most calculators have an "×10ⁿ" or "EXP" button for entering standard form directly — practising with it before the exam avoids fumbling with the syntax under time pressure.
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