Standard form turns up throughout GCSE Maths — from the first page of Paper 1 to the final calculation on Paper 3. Whether you're working with the distance between planets or the size of a bacterium, standard form is the tool examiners reach for. AQA (8300), Edexcel (1MA1) and OCR (J560) all test it, and it can appear on both the non-calculator paper and the calculator papers. Getting the method right is essential — and the good news is that once it clicks, it's one of the more reliable topics to score full marks on.
What You Need to Know First
Before diving into the method, make sure you're confident with:
- Powers of 10: 10¹ = 10, 10² = 100, 10³ = 1 000, 10⁻¹ = 0.1, 10⁻² = 0.01, 10⁻³ = 0.001
- Multiplying and dividing by powers of 10: every time you multiply by 10, the digits move one place to the left; dividing moves them right
- Index laws: a^m × a^n = a^(m+n) and a^m ÷ a^n = a^(m−n)
- Negative indices: 10⁻³ means 1/1000 — a negative power gives a small number, not a negative number
- How to enter standard form on a Casio calculator (EXP or ×10ˣ button) for Papers 2 and 3
Converting to and from Standard Form
A number is in standard form (also called standard index form) when it is written as:
A × 10ⁿ
where 1 ≤ A < 10 and n is an integer (positive, negative or zero).
The rule about A is the one students most often forget: A must be at least 1 and strictly less than 10. Not 0.3. Not 14. It must sit in that range.
Converting a large number to standard form
Follow three steps:
- Write the digits with the decimal point placed after the first non-zero digit.
- Count how many places the decimal point has moved — this is n.
- Write the result as A × 10ⁿ.
Worked Example 1: Write 47 000 000 in standard form.
Step 1: Place the decimal after the first digit → 4.7
Step 2: The decimal moved 7 places to the left (4.7000000 → 47 000 000)
Step 3: 47 000 000 = 4.7 × 10⁷
Worked Example 2: Write 305 000 in standard form.
Step 1: First non-zero digit is 3 → 3.05
Step 2: Decimal moved 5 places to the left
Step 3: 305 000 = 3.05 × 10⁵
Converting a small number to standard form
The method is identical, but because the decimal moves to the right, the power is negative.
Worked Example 3: Write 0.000 035 in standard form.
Step 1: First non-zero digit is 3 → 3.5
Step 2: The decimal moved 5 places to the right (0.000035 → 3.5)
Step 3: 0.000 035 = 3.5 × 10⁻⁵
Worked Example 4: Write 0.0072 in standard form.
Step 1: First non-zero digit is 7 → 7.2
Step 2: Decimal moved 3 places to the right
Step 3: 0.0072 = 7.2 × 10⁻³
Converting back from standard form
- Positive power → move the decimal point to the right.
- Negative power → move the decimal point to the left.
6.02 × 10⁴ → move right 4 places → 60 200
9.1 × 10⁻³ → move left 3 places → 0.0091
Calculating with Standard Form
Calculations are where marks are most often dropped. There are three types you need to handle — multiplying, dividing, and adding or subtracting.
Multiplying in standard form
- Multiply the A values together.
- Add the powers of 10.
- Adjust if your A result isn't between 1 and 10.
Worked Example 5: Calculate (3 × 10⁴) × (2 × 10⁵)
Step 1: 3 × 2 = 6
Step 2: 10⁴ × 10⁵ = 10⁹ (add the powers)
Step 3: 6 is between 1 and 10 — no adjustment needed.
Answer: 6 × 10⁹
Worked Example 6: Calculate (5 × 10³) × (4 × 10⁶)
Step 1: 5 × 4 = 20
Step 2: 10³ × 10⁶ = 10⁹
Step 3: 20 is not between 1 and 10. Write 20 = 2 × 10¹, so:
2 × 10¹ × 10⁹ = 2 × 10¹⁰
Answer: 2 × 10¹⁰
Dividing in standard form
- Divide the A values.
- Subtract the powers of 10.
- Adjust if needed.
Worked Example 7: Calculate (9.6 × 10⁸) ÷ (3 × 10³)
Step 1: 9.6 ÷ 3 = 3.2
Step 2: 10⁸ ÷ 10³ = 10⁵ (subtract the powers)
Step 3: 3.2 is between 1 and 10 — no adjustment needed.
Answer: 3.2 × 10⁵
Worked Example 8: Calculate (2.4 × 10²) ÷ (6 × 10⁵)
Step 1: 2.4 ÷ 6 = 0.4
Step 2: 10² ÷ 10⁵ = 10⁻³
Step 3: 0.4 is not between 1 and 10. Write 0.4 = 4 × 10⁻¹, so:
4 × 10⁻¹ × 10⁻³ = 4 × 10⁻⁴
Answer: 4 × 10⁻⁴
Adding and subtracting in standard form
This is the trickiest type — and it's the one that trips up even students who handle multiplication and division well. The rule is simple but easily forgotten: the powers must match before you add or subtract.
Worked Example 9: Calculate (4 × 10⁵) + (3 × 10⁴)
The powers are different (5 and 4), so convert both to the same power. Use 10⁴:
4 × 10⁵ = 40 × 10⁴
Now add: (40 × 10⁴) + (3 × 10⁴) = 43 × 10⁴
Adjust: 43 × 10⁴ = 4.3 × 10⁵
Answer: 4.3 × 10⁵
Worked Example 10: Calculate (7.2 × 10⁶) − (5 × 10⁵)
Convert to the same power. Use 10⁵:
7.2 × 10⁶ = 72 × 10⁵
Subtract: (72 × 10⁵) − (5 × 10⁵) = 67 × 10⁵
Adjust: 67 × 10⁵ = 6.7 × 10⁶
Answer: 6.7 × 10⁶
On Paper 1 (non-calculator) you'll work through these steps by hand. On Papers 2 and 3, use your calculator — but enter standard form using the EXP or ×10ˣ button. Type 4 EXP 5 to enter 4 × 10⁵. Never type 4 × 10 ^ 5 separately, as this can give the wrong result.
Common Mistakes Students Make
- A value outside the required range: Writing 14 × 10³ instead of 1.4 × 10⁴. Always check that A is between 1 and 10 after multiplying or dividing — this final adjustment is a step students skip when they're in a hurry.
- Treating a negative power as a negative number: 3.5 × 10⁻² = 0.035, not −350. A negative power means the number is small (a decimal), not that the value is negative.
- Miscounting decimal place movements: Write out the full number and count carefully. Students often miscount when the original number has zeros as placeholders.
- Adding A values without matching powers first: In addition and subtraction, you cannot combine 4 × 10⁵ and 3 × 10⁴ until the powers are the same. Students who skip this step get the wrong answer almost every time.
- Copying the calculator display directly: A calculator might show 3.2E8 or 3.2×10⁸ on screen. Neither of those is a correct written answer — write 3.2 × 10⁸ in full.
Exam Tips for Standard Form
- Show the intermediate steps on Paper 1: Write out each step of your calculation. AQA and Edexcel award method marks for correct working, even if you slip up on the final arithmetic. A correct method with a small error can still earn you two out of three marks.
- For conversion questions, one mark is usually for A and one for n: Even if your power is wrong, you may still earn the mark for a correct A value. Don't leave blank answers — partial credit is real.
- After every answer, check A is in range: Make it a habit. Take five seconds at the end of each question to confirm 1 ≤ A < 10 before writing your final answer.
- Use context to sense-check: Distances in space, light years, and masses of stars involve very large positive powers. Sizes of atoms, molecules, and bacteria involve negative powers. If you get a negative power for a planetary distance, something's gone wrong.
- On the calculator paper, still show the key steps: Write (3 × 10⁴) × (2 × 10⁵) = and then your answer. This earns the method mark even if you accidentally miskey the calculation.
Practise Now
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