Calculate in standard form and estimate to a sensible accuracy
Convert numbers to and from standard form, calculate with powers of ten and round only when the context calls for it.
Use this pathway for Cambridge IGCSE International Mathematics 0607, 2028–2030. Core and Extended objectives are labelled in each lesson; Extended adds requirements beyond Core. Paper 1 and Paper 2 are non-calculator. Papers 3–6 require a graphic display calculator (GDC). These original lessons use self-checked practice; they are not a Cambridge grade, full timed paper or teacher review.
Before you begin
- Check whether your school has entered you for Core or Extended. Core and Extended share most Number ideas, but some notation and methods are Extended only.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
Which coefficient is allowed in A × 10ⁿ?
What you will learn
- Write A × 10ⁿ with 1 ≤ A < 10 and integer n
- Multiply and divide standard-form numbers
- Round to stated decimal places or significant figures and judge reasonableness
The first factor fixes the standard-form scale
In standard form A × 10ⁿ, the coefficient satisfies 1 ≤ A < 10 and n is an integer. Moving the decimal point left makes n positive; moving it right makes n negative.
For multiplication, multiply coefficients and add powers. For division, divide coefficients and subtract powers. Renormalise the coefficient if it falls outside the allowed interval. Core standard-form calculations are assessed on calculator papers only.
Estimate first and delay rounding
Round a value to the place or number of significant figures requested. The first non-zero digit is the first significant figure; leading zeros only locate the decimal point.
An estimate helps catch an incorrect power of ten or misplaced decimal point. Keep the calculator value during intermediate steps and round the final answer to a context-appropriate accuracy.
Pause and explain
What is 4.8 × 10⁻² as an ordinary decimal?
Worked example
Write 0.0000725 in standard form and calculate (3 × 10⁵)(4 × 10⁻³).
Show the worked solution
- Move the decimal five places right to obtain 7.25, so the exponent is −5.
- Multiply the coefficients and add exponents: 3 × 4 = 12 and 5 + (−3) = 2.
- Renormalise 12 × 10² as 1.2 × 10³.
Answer 0.0000725 = 7.25 × 10⁻⁵; (3 × 10⁵)(4 × 10⁻³) = 1.2 × 10³.
Try a topic test
Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.
Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
Calculate (6 × 10⁷) ÷ (3 × 10²), then write the answer in standard form.
Give me a hint
Divide the coefficients and subtract the denominator's exponent.
Compare my reasoning
- 6 ÷ 3 = 2.
- 7 − 2 = 5.
- The coefficient is already in the interval 1 ≤ A < 10.
2 × 10⁵.
Look for these in your work
- Coefficients divided
- Exponent subtraction correct
- Answer is standardised
Round 0.004786 to 2 significant figures and 3 decimal places. Estimate 19.7 × 0.48 to check the size of the product.
Give me a hint
Count from the first non-zero digit for significant figures; estimate with 20 × 0.5.
Compare my reasoning
- To 2 significant figures, 0.004786 becomes 0.0048.
- To 3 decimal places, it becomes 0.005.
- The product should be close to 10; the exact value is 9.456.
2 significant figures: 0.0048; 3 decimal places: 0.005; estimated product ≈ 10.
Look for these in your work
- Rounding instructions distinguished
- Leading zeros not counted as significant
- Estimate gives a useful magnitude
A lab sample has mass 3.2 × 10⁻⁴ g. A batch contains 2.5 × 10⁶ samples. Find the total mass in grams and standard form.
Give me a hint
Multiply the masses and add the powers of ten, then normalise.
Compare my reasoning
- 3.2 × 2.5 = 8.0.
- 10⁻⁴ × 10⁶ = 10².
- The result is already normalised.
8.0 × 10² g, or 800 g.
Look for these in your work
- Units retained
- Powers combined
- Magnitude checked
Common mistakes
- Writing a coefficient below 1 or at least 10
- Adding exponents when dividing
- Rounding before the last step
- Using decimal places when significant figures are requested
What can you explain now?
How is 6 300 000 written in standard form?
Compare with the explanation
6.3 × 10⁶
The coefficient is between 1 and 10 and the decimal moved six places.
After trying it yourself, choose your next review. This is your self-assessment.
Your review choice appears on Today. Sign in to sync it across devices.
Make it stick
What bounds must A satisfy in standard form, and how do the exponents change in multiplication and division?
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.