Convert and calculate in scientific notation
Normalise a positive value as a × 10ⁿ · Calculate products, quotients and sums · Check powers of ten with an estimate
Exam scope and assessment notes
Core and Extended foundations. Historical study of Cambridge IGCSE Mathematics (US) 0444, 2025 Version 2. The final assessment was June 2025; the syllabus was available only to schools in Arizona, USA. This is not a current exam-entry option. Original self-check teaching and explained practice are not Cambridge papers, official marks or predicted grades. Linked teaching foundations do not establish full syllabus mastery. Calibrated cumulative assessment, reliable drawn-work review and qualified mathematics-teacher review remain unfinished.
Before you begin
- Decimal place value
- Integer exponent laws
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What should you establish before applying the method in this lesson?
What you will learn
- Normalise a positive value as a × 10ⁿ
- Calculate products, quotients and sums
- Check powers of ten with an estimate
Normalise the coefficient
For a positive non-zero number, scientific notation is a × 10ⁿ with 1 ≤ a < 10 and integer n. Moving the decimal left gives a positive exponent for large values; small positive values need negative exponents. Zero is not represented in this normalised positive form.
0.00072 = 7.2 × 10⁻⁴. A coefficient such as 72 must be renormalised: 72 × 10³ = 7.2 × 10⁴.
Calculate with the right operation
Multiply or divide coefficients and combine equal-base powers; normalise the final coefficient. For addition or subtraction, first use a common power of ten. Adding exponents does not add the represented numbers.
Sources for this lesson
Pause and explain
Is 12 × 10³ normalised scientific notation?
Worked example
Calculate (3 × 10⁵)(4 × 10⁻²).
Show the worked solution
- Coefficients give 12; powers give 10³.
- 12 × 10³ = 1.2 × 10⁴.
Answer 1.2 × 10⁴.
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.
Write 0.000045 and 6200000 in scientific notation.
Give me a hint
Count decimal moves.
Compare my reasoning
- 0.000045 = 4.5 × 10⁻⁵.
- 6200000 = 6.2 × 10⁶.
4.5 × 10⁻⁵ and 6.2 × 10⁶.
Look for these in your work
- Use a coefficient in range
- Check the exponent sign
Calculate (8 × 10⁶)/(2 × 10⁻³).
Give me a hint
Divide coefficients and subtract exponents.
Compare my reasoning
- 8/2 = 4.
- 6 − (−3) = 9.
4 × 10⁹.
Look for these in your work
- Subtract the signed exponent
- Normalise the coefficient
A device uses 2.4 × 10⁻³ J then 7 × 10⁻⁴ J. Find the total in scientific notation.
Give me a hint
Use 10⁻³ for both terms.
Compare my reasoning
- 7 × 10⁻⁴ = 0.7 × 10⁻³.
- (2.4 + 0.7) × 10⁻³ = 3.1 × 10⁻³.
3.1 × 10⁻³ J.
Look for these in your work
- Align powers before adding
- Keep the unit
Common mistakes
- Normalise after calculating.
- Use matching powers for addition.
What can you explain now?
Is 12 × 10³ normalised scientific notation?
Compare with the explanation
No; write 1.2 × 10⁴.
The coefficient must be at least 1 and less than 10.
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Make it stick
Is 12 × 10³ normalised scientific notation?
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.