Write 3000 in standard form.
Standard form is \(a \times 10^n\) where 1 ≤ a < 10.
3000 = \(3 \times 10^3\).
Standard Form
Standard form is used to write very large or very small numbers in a compact way. This is particularly useful in science and higher GCSE Maths calculations.
Overview
Standard form is a short way of writing very large or very small numbers.
It is especially useful in science, calculator work and exam questions involving powers of 10.
\( 4.2 \times 10^6 = 4{,}200{,}000 \)
In standard form, the first number must be at least 1 but less than 10.
It is then multiplied by a power of 10.
What you should understand after this topic
- Understand what standard form looks like
- Convert into and out of standard form
- Understand how positive and negative powers of 10 work
- Compare numbers in standard form
- Calculate with standard form in exams
Key Definitions
Standard Form
A number written as \( a \times 10^n \), where \(1 \le a < 10\).
Coefficient
The first number in standard form, between 1 and 10.
Power of 10
The index showing how many places the decimal point moves.
Positive Power
Used for large numbers greater than or equal to 10.
Negative Power
Used for small numbers less than 1.
Convert
Change a number into or out of standard form.
Key Rules
Correct form
\( a \times 10^n \) where \(1 \le a < 10\).
Large number
Move the decimal left, so the power of 10 is positive.
Small number
Move the decimal right, so the power of 10 is negative.
Check the coefficient
The first number must never be 10 or more.
Quick Comparison
| Ordinary Number | Standard Form |
|---|---|
| 3000 | \( 3 \times 10^3 \) |
| 5600000 | \( 5.6 \times 10^6 \) |
| 0.4 | \( 4 \times 10^{-1} \) |
| 0.0072 | \( 7.2 \times 10^{-3} \) |
How to Solve
Step 1: Understand standard form
Standard form writes very large or very small numbers in a compact way.
\( a \times 10^n \)
\(a\) must be at least 1 and less than 10.
\(n\) must be an integer.
Exam tip: Always check the first number is between 1 and 10.
Step 2: Write large numbers in standard form
Move the decimal point left until the first number is between 1 and 10.
\( 820000 = 8.2 \times 10^5 \)
Large numbers have positive powers of 10.
Step 3: Write small numbers in standard form
Move the decimal point right until the first number is between 1 and 10.
\( 0.00063 = 6.3 \times 10^{-4} \)
Small decimals have negative powers of 10.
Step 4: Convert back to ordinary numbers
Positive power → move decimal right.
Negative power → move decimal left.
Positive power
\( 4.7 \times 10^3 = 4700 \)
Negative power
\( 9.1 \times 10^{-2} = 0.091 \)
Step 5: Check correct standard form
\(12.4\) is not allowed because it is not less than 10.
Correct
\( 3.4 \times 10^7 \)
Not correct
\( 12.4 \times 10^6 \)
Step 6: Compare numbers
Compare the powers of 10 first. If the powers match, compare the first numbers.
\( 3.1 \times 10^5 \quad \text{and} \quad 7.2 \times 10^4 \)
\(10^5\) is bigger than \(10^4\), so \(3.1 \times 10^5\) is larger.
Step 7: Multiply and divide in standard form
\( (2 \times 10^3)(4 \times 10^5) = 8 \times 10^8 \)
\( \frac{6 \times 10^7}{2 \times 10^3} = 3 \times 10^4 \)
See powers and roots for index rules.
Multiply
Multiply the first numbers and add the powers.
Divide
Divide the first numbers and subtract the powers.
Step 8: Exam method summary
- Check the number is written as \(a \times 10^n\).
- Make sure \(1 \leq a < 10\).
- Use positive powers for large numbers.
- Use negative powers for small decimals.
- After multiplying or dividing, convert back into correct standard form if needed.
Example Questions
Edexcel
Exam-style questions inspired by Edexcel GCSE Mathematics.
Edexcel
Write 4500 in standard form.
Edexcel
Write 0.00072 in standard form.
Edexcel
Write \( 3.6 \times 10^4 \) as an ordinary number.
Edexcel
Write \( 7.2 \times 10^{-3} \) as an ordinary number.
Edexcel
Calculate \( (2 \times 10^3) \times (4 \times 10^2) \). Give your answer in standard form.
AQA
Exam-style questions based on the AQA GCSE Mathematics specification, focusing on operations and fluency with standard form.
AQA
Write \( 9.4 \times 10^5 \) as an ordinary number.
AQA
Write \( 0.000056 \) in standard form.
AQA
Calculate \( \frac{6 \times 10^8}{3 \times 10^2} \). Give your answer in standard form.
AQA
Calculate \( 5.2 \times 10^3 + 3.1 \times 10^3 \). Give your answer in standard form.
AQA
A student writes 3.2 × 10⁴ as 32 × 10³.
AQA
Is the student correct?
Tick one box. Yes ☐ No ☐
Give a reason for your answer.
OCR
Exam-style questions aligned with OCR GCSE Mathematics, emphasising reasoning, accuracy, and real-world applications of standard form.
OCR
Write \( 5.03 \times 10^4 \) as an ordinary number.
OCR
Write \( 0.00000081 \) in standard form.
OCR
Calculate \( (3 \times 10^{-4}) \times (2 \times 10^6) \). Give your answer in standard form.
OCR
Calculate \( \frac{8 \times 10^5}{4 \times 10^2} \). Give your answer in standard form.
OCR
The speed of light is 300,000,000 metres per second. Write this number in standard form.
Exam Checklist
Step 1
Make sure the first number is between 1 and 10.
Step 2
Count decimal moves carefully.
Step 3
Use a positive power for large numbers and a negative power for small numbers.
Step 4
After calculations, check whether the final answer still needs rewriting.
Most common exam mistakes
Coefficient error
Writing something like \( 14 \times 10^5 \) instead of proper standard form.
Sign error
Using a positive power when the number is less than 1.
Place value error
Moving the decimal the wrong number of places.
Calculation error
Not adding or subtracting the powers correctly.
Common Mistakes
These are common mistakes students make when working with standard form in GCSE Maths.
Using an invalid coefficient
Incorrect
A student writes a number with a coefficient of 10 or more (or less than 1).
Correct
In standard form, the coefficient must be between 1 and 10 (1 ≤ a < 10). For example, 3.2 × 10^5 is correct, but 32 × 10^4 is not.
Using the wrong sign for the power
Incorrect
A student writes a positive power instead of a negative one, or vice versa.
Correct
Large numbers use positive powers of 10, while small numbers (less than 1) use negative powers.
Moving the decimal in the wrong direction
Incorrect
A student shifts the decimal incorrectly when converting.
Correct
Moving the decimal to the left gives a positive power. Moving it to the right gives a negative power.
Not writing the final answer in standard form
Incorrect
A student performs a calculation but leaves the answer not fully simplified.
Correct
After calculations, always rewrite the answer so the coefficient is between 1 and 10.
Confusing multiplication and division rules
Incorrect
A student mixes up how to combine powers of 10.
Correct
When multiplying, add powers: \(10^a \times 10^b = 10^{a+b}\). When dividing, subtract powers: \(10^a \div 10^b = 10^{a-b}\).
Try It Yourself
Practise writing numbers in standard form and performing calculations.
Foundation
Foundation Practice
Convert between ordinary numbers and standard form.
Write 50000 in standard form.
Move decimal 4 places.
50000 = \(5 \times 10^4\).
Write 0.004 in standard form.
Count decimal moves to the right.
0.004 = \(4 \times 10^{-3}\).
Write 0.0006 in standard form.
Move decimal 4 places right.
0.0006 = \(6 \times 10^{-4}\).
Write \(2 \times 10^3\) as an ordinary number.
Multiply by 1000.
\(2 × 1000 = 2000\).
Write \(7 \times 10^2\) as an ordinary number.
Multiply by 100.
700.
Write \(3 \times 10^{-2}\) as a decimal.
Move decimal left.
0.03.
Write \(8 \times 10^{-3}\) as a decimal.
Move decimal left 3 places.
0.008.
Which is correctly written in standard form?
First number must be between 1 and 10.
Only \(4 \times 10^5\) is valid.
Write 900000 in standard form.
Move decimal 5 places.
\(9 \times 10^5\).
Higher
Higher Practice
Perform calculations using standard form.
Find: \((2 \times 10^3)(3 \times 10^2)\)
Multiply numbers and add powers.
2 × 3 = 6 and 10³ × 10² = 10⁵ → \(6 × 10^5\).
Find: \((4 \times 10^2)(5 \times 10^3)\).
Multiply and add indices.
4 × 5 = 20 → \(2 × 10^6\).
Find: \((6 \times 10^4) ÷ (2 \times 10^2)\)
Divide numbers and subtract powers.
6 ÷ 2 = 3 and 10⁴ ÷ 10² = 10².
Find: \((8 \times 10^5) ÷ (4 \times 10^2)\).
Divide and subtract indices.
8 ÷ 4 = 2 and 10⁵ ÷ 10² = 10³.
Write \(0.00045\) in standard form.
Move decimal to make 4.5.
\(4.5 × 10^{-4}\).
Write 7200000 in standard form.
Move decimal 6 places.
\(7.2 × 10^6\).
A student writes \(50 \times 10^3\). What is wrong?
Check standard form rule.
First number must be between 1 and 10.
Find: \((3 \times 10^2)(2 \times 10^{-1})\).
Add powers carefully.
3 × 2 = 6 and 10² × 10⁻¹ = 10¹.
Which is equal to \(1.2 \times 10^3\)?
Multiply by 1000.
1.2 × 1000 = 1200.
Estimate: \((4.9 \times 10^3)(2.1 \times 10^2)\).
Round to 1 significant figure.
5 × 10³ × 2 × 10² = 10 × 10⁵ = \(1 × 10^6\).
Games
Practise this topic with interactive games.
Games coming soon.
Standard Form Video Tutorial
Frequently Asked Questions
What is standard form?
A way of writing numbers as a number between 1 and 10 multiplied by a power of 10.
When do I use negative powers?
For very small numbers.
Why is standard form useful?
It simplifies calculations with very large or small numbers.