Standard Form
The Skill
Standard form is a way of writing very large or very small numbers concisely.
The Format
$$A \times 10^n$$
where:
- $1 \leq A < 10$ (A is between 1 and 10, including 1 but not 10)
- $n$ is an integer (positive for large numbers, negative for small numbers)
Converting to Standard Form
Large numbers: Move the decimal point left until you have a number between 1 and 10. Count the moves — that's your power.
$$45000 = 4.5 \times 10^4$$
Small numbers: Move the decimal point right. The power is negative.
$$0.00032 = 3.2 \times 10^{-4}$$
Converting from Standard Form
Move the decimal point by the power of 10.
- Positive power: move right (number gets bigger)
- Negative power: move left (number gets smaller)
$$6.7 \times 10^3 = 6700$$ $$2.1 \times 10^{-2} = 0.021$$
Calculations in Standard Form
Multiplication: Multiply the numbers, add the powers. $$(3 \times 10^4) \times (2 \times 10^3) = 6 \times 10^7$$
Division: Divide the numbers, subtract the powers. $$(8 \times 10^6) \div (4 \times 10^2) = 2 \times 10^4$$
Addition/Subtraction: Convert to the same power of 10 first, or convert to ordinary numbers.
The Traps
Common misconceptions and how to avoid them.
Having A greater than or equal to 10 "The Ten Trap"
The Mistake in Action
Write 35000 in standard form.
Wrong: $35 \times 10^3$
Why It Happens
Students correctly identify the power but don't move the decimal point far enough. They stop at a "round" number like 35 instead of continuing to get a number between 1 and 10.
The Fix
In standard form, $A$ must be at least 1 but less than 10.
$35 \times 10^3$ is not valid because $35 \geq 10$.
Move the decimal one more place: $$35000 = 3.5 \times 10^4$$
Check: Is $3.5$ between 1 and 10? Yes ✓
Spot the Mistake
Write 35000 in standard form
$35 \times 10^3$
Click on the line that contains the error.
Not adjusting after calculation "The Unadjusted Answer"
The Mistake in Action
Calculate $(4 \times 10^5) \times (5 \times 10^3)$. Give your answer in standard form.
Wrong: $4 \times 5 = 20$ $10^5 \times 10^3 = 10^8$ Answer: $20 \times 10^8$
Why It Happens
Students correctly multiply the numbers and add the powers, but forget that the final answer must also be in standard form — with A between 1 and 10.
The Fix
After calculating, check if A is still between 1 and 10. If not, adjust.
$20 \times 10^8$ is not in standard form because $20 \geq 10$.
Adjust: $20 = 2 \times 10^1$
So: $20 \times 10^8 = 2 \times 10^1 \times 10^8 = 2 \times 10^9$
Answer: $2 \times 10^9$
Spot the Mistake
$(4 \times 10^5) \times (5 \times 10^3)$
$= 20 \times 10^8$
Click on the line that contains the error.
Using wrong sign for small numbers "The Sign Slip"
The Mistake in Action
Write 0.00045 in standard form.
Wrong: $4.5 \times 10^4$
Why It Happens
Students move the decimal point correctly but use a positive power instead of negative. They confuse the direction of movement.
The Fix
For numbers less than 1, the power is negative.
Think of it this way:
- Big numbers (≥10) → positive power (×10 makes things bigger)
- Small numbers (<1) → negative power (×10^{-n} makes things smaller)
$$0.00045 = 4.5 \times 10^{-4}$$
Check: $10^{-4} = 0.0001$, and $4.5 \times 0.0001 = 0.00045$ ✓
Spot the Mistake
Write 0.00045 in standard form
$4.5 \times 10^4$
Click on the line that contains the error.
The Deep Dive
Apply your knowledge with these exam-style problems.
Level 1: Fully Worked
Complete solutions with commentary on each step.
Question
Write 7,400,000 in standard form.
Solution
Step 1: Place the decimal point to get a number between 1 and 10. $$7,400,000 \rightarrow 7.4$$
Step 2: Count how many places you moved the decimal point. $7,400,000.$ to $7.4$ = 6 places to the left
Step 3: Write in standard form. Moving left = positive power. $$7,400,000 = 7.4 \times 10^6$$
Check: $10^6 = 1,000,000$, and $7.4 \times 1,000,000 = 7,400,000$ ✓
Answer: $7.4 \times 10^6$
Question
Write 0.000082 in standard form.
Solution
Step 1: Place the decimal point to get a number between 1 and 10. $$0.000082 \rightarrow 8.2$$
Step 2: Count how many places you moved the decimal point. $0.000082$ to $8.2$ = 5 places to the right
Step 3: Write in standard form. Moving right = negative power. $$0.000082 = 8.2 \times 10^{-5}$$
Check: $10^{-5} = 0.00001$, and $8.2 \times 0.00001 = 0.000082$ ✓
Answer: $8.2 \times 10^{-5}$
Question
Calculate $(3 \times 10^4) \times (6 \times 10^5)$. Give your answer in standard form.
Solution
Step 1: Multiply the numbers. $$3 \times 6 = 18$$
Step 2: Add the powers. $$10^4 \times 10^5 = 10^{4+5} = 10^9$$
Step 3: Combine. $$18 \times 10^9$$
Step 4: Adjust (18 is not between 1 and 10). $$18 = 1.8 \times 10^1$$ $$18 \times 10^9 = 1.8 \times 10^1 \times 10^9 = 1.8 \times 10^{10}$$
Answer: $1.8 \times 10^{10}$
Level 2: Scaffolded
Fill in the key steps.
Question
Write $3.6 \times 10^{-3}$ as an ordinary number.
Level 3: Solo
Try it yourself!
Question
Calculate $(8.4 \times 10^7) \div (2.1 \times 10^3)$. Give your answer in standard form.
Show Solution
Step 1: Divide the numbers. $$8.4 \div 2.1 = 4$$
Step 2: Subtract the powers. $$10^7 \div 10^3 = 10^{7-3} = 10^4$$
Step 3: Combine. $$4 \times 10^4$$
Check: Is 4 between 1 and 10? Yes ✓
Answer: $4 \times 10^4$
Question
The distance from Earth to the Sun is approximately $1.5 \times 10^8$ km. Light travels at $3 \times 10^5$ km per second. How long does it take light to travel from the Sun to Earth? Give your answer in standard form.
Show Solution
Time = Distance ÷ Speed
$$\text{Time} = \frac{1.5 \times 10^8}{3 \times 10^5}$$
Step 1: Divide the numbers. $$1.5 \div 3 = 0.5$$
Step 2: Subtract the powers. $$10^8 \div 10^5 = 10^{8-5} = 10^3$$
Step 3: Combine. $$0.5 \times 10^3$$
Step 4: Adjust (0.5 is not between 1 and 10). $$0.5 \times 10^3 = 5 \times 10^{-1} \times 10^3 = 5 \times 10^2$$
Answer: $5 \times 10^2$ seconds (or 500 seconds ≈ 8.3 minutes)
Examiner's View
Mark allocation: Conversions are 1-2 marks. Calculations are 2-3 marks.
Common errors examiners see:
- A value not between 1 and 10 (e.g., $32 \times 10^5$)
- Wrong sign on the power
- Not adjusting the power when A changes after calculation
- Adding powers when multiplying (instead of adding)
What gains marks:
- Writing the final answer in correct standard form
- Showing clear working for calculations
- Checking that $1 \leq A < 10$
AQA Notes
AQA often asks you to interpret standard form in context (populations, distances, atoms).