Number and ratio · GCSE Maths
Standard form calculations
GCSE Maths standard-form calculations: multiply and divide by combining a-values and powers of 10, then add or subtract by matching powers, rewriting into standard form.
Multiply: multiply the a-values and add the indices. Divide: divide the a-values and subtract the indices. Add or subtract: match the powers of 10 first. Then rewrite so 1 ≤ a < 10.
The important bits
What you need to know
- 1
Standard form is a × 10ⁿ with 1 ≤ a < 10 and n an integer. 45 000 = 4.5 × 10⁴ and 0.0032 = 3.2 × 10⁻³. An answer such as 12 × 10³ is not yet in standard form.
- 2
To multiply, multiply the a-values and add the powers of 10: (3 × 10⁵) × (4 × 10⁻²) = 12 × 10³, then rewrite as 1.2 × 10⁴.
- 3
To divide, divide the a-values and subtract the powers: (8 × 10⁶) ÷ (2 × 10⁻³) = 4 × 10⁹. Subtracting indices includes the minus: 6 − (−3) = 9.
- 4
To add or subtract, the powers of 10 must match. Convert the smaller power up, or convert both to ordinary numbers if they are friendly, then return to standard form.
- 5
Example of addition: 3.1 × 10⁴ + 4.2 × 10³ = 3.1 × 10⁴ + 0.42 × 10⁴ = 3.52 × 10⁴. Do not add 3.1 + 4.2 and 4 + 3 as if they were like terms with different indices.
- 6
After any operation, check 1 ≤ a < 10. If a ≥ 10, divide a by 10 and add 1 to n. If a < 1, multiply a by 10 and subtract 1 from n. Repeat if needed.
- 7
On a calculator use the ×10ˣ or EXP button. Typing 3 × 10^5 as 3 × 10 × 5 is a different (wrong) number. Still write the non-calculator method; papers test it.
- 8
Sense-check the size: 6 × 10⁷ is sixty million, not six million. A negative index is a small number, not a negative number: 5 × 10⁻³ = 0.005.
Quotations worth analysing
Short evidence. Real method.
“Give your answer in standard form.”
12 × 10³ is equal in value but not standard form. Rewrite as 1.2 × 10⁴ or the final accuracy mark is lost.
“Add the indices when multiplying; subtract when dividing.”
This is a^m × a^n = a^{m+n} with base 10. Dividing uses a^m ÷ a^n = a^{m−n}, including 6 − (−2) = 8.
“Convert so the powers of 10 are the same.”
You cannot add 10⁴ to 10³ by adding the indices. Match the powers, add the a-values, then restore standard form.
Go deeper
Multiply and divide, then force standard form
(2.5 × 10⁴) × (6 × 10⁵) = 15 × 10⁹. That product is correct in value and wrong as a finished answer, because 15 is not between 1 and 10. Write 1.5 × 10¹⁰: you moved the point one place left, so the power of 10 goes up by 1. Division is the same discipline. (4.8 × 10⁻³) ÷ (1.6 × 10²) = 3 × 10⁻⁵, because 4.8 ÷ 1.6 = 3 and −3 − 2 = −5. If the a-division is not tidy, keep extra decimal places until the rewrite: (7.2 × 10⁷) ÷ (8 × 10⁻⁴) = 0.9 × 10¹¹ = 9.0 × 10¹⁰. The 0.9 is the warning light: a is less than 1, so shift it. Calculator work still needs this rewrite if the display offers 15 × 10⁹ or 0.9e11.
Go deeper
Addition is place value, not index laws
Index laws do not add numbers with different powers. 3 × 10⁴ + 2 × 10⁴ is 5 × 10⁴ because the powers already match. 3 × 10⁴ + 2 × 10³ is 3 × 10⁴ + 0.2 × 10⁴ = 3.2 × 10⁴, or as ordinary numbers 30 000 + 2000 = 32 000 = 3.2 × 10⁴. Students who write 5 × 10⁷ have added both the a-values and the indices, which is two topics colliding. If the powers differ by many places, converting to ordinary numbers can be safer on a non-calculator paper: 5.1 × 10² + 4 × 10⁻¹ = 510 + 0.4 = 510.4 = 5.104 × 10². Subtraction follows the same matching. Watch the sign of n: 6.2 × 10⁻³ − 4.1 × 10⁻⁴ needs 6.2 × 10⁻³ − 0.41 × 10⁻³ = 5.79 × 10⁻³.
Go deeper
The rewrite is a separate mark
A large fraction of dropped marks on this topic are not arithmetic errors; they are unfinished form. After multiplying, pause and ask: is a between 1 and 10, including 1 and excluding 10? 1.0 × 10⁵ is allowed; 10 × 10⁴ is not; 0.99 × 10⁵ is not. Each time you move the decimal point one place in a, you change n by 1 in the opposite story: smaller a, larger n. Science-style questions (mass of a dust particle, distance to a star) still want this rewrite, and they still want a sensible degree of accuracy. If the inputs have two significant figures, do not offer ten decimal places in a. The method remains: operate, rewrite, check the size against an ordinary-number estimate.
See the idea in action
Calculate (3 × 10⁵) × (4 × 10⁻²) and (3.1 × 10⁴) + (4.2 × 10³). Give both answers in standard form. Multiplication: Step 1: Multiply the a-values: 3 × 4 = 12. Step 2: Add the indices: 10⁵ × 10⁻² = 10³. Step 3: 12 × 10³ is not standard form, so write 1.2 × 10⁴. Addition: Step 4: Match powers: 4.2 × 10³ = 0.42 × 10⁴. Step 5: 3.1 × 10⁴ + 0.42 × 10⁴ = 3.52 × 10⁴, already in standard form. Check: 300 000 × 0.04 = 12 000 = 1.2 × 10⁴, and 31 000 + 4200 = 35 200 = 3.52 × 10⁴.
Exam technique
Turn knowledge into marks
If a is 10 or more, or less than 1, you have not finished. Adjust a and change the power of 10 by one for each place you move the point.
Common mistakes
Do not give these marks away
- 01
Leaving 12 × 10³ as the final answer instead of 1.2 × 10⁴.
- 02
Adding indices when adding numbers, or adding a-values when the powers of 10 are different.
- 03
Subtracting indices as 6 − −2 = 4 instead of 6 − (−2) = 8 when dividing by a negative power.
(3 × 10⁵) × (4 × 10⁻²) in standard form is
A1.2 × 10⁴
B12 × 10³
C1.2 × 10³
D7 × 10³
Show the answer
1.2 × 10⁴. 3 × 4 = 12 and 5 + (−2) = 3, so 12 × 10³, which rewrites as 1.2 × 10⁴. 12 × 10³ is equal in value but not standard form. 1.2 × 10³ forgot to increase the power when 12 became 1.2. 7 × 10³ added 3 + 4 and 5 − 2.
Quick questions
If this is the bit you searched
How do you multiply numbers in standard form?
Multiply the a-values, add the powers of 10, then rewrite so that 1 ≤ a < 10. Example: (3 × 10⁵) × (4 × 10⁻²) = 1.2 × 10⁴.
How do you add numbers in standard form?
Make the powers of 10 the same, add the a-values, then restore standard form. You can convert to ordinary numbers first if the powers are close and the arithmetic is kind.
Why is 10.5 × 10³ not standard form?
The first factor must satisfy 1 ≤ a < 10. Rewrite as 1.05 × 10⁴. The same rewrite is needed after a calculation that produces 10.5 × 10³.
How do I enter standard form on a calculator?
Use the ×10ˣ or EXP button, not × 10 × n. Then still rewrite the display if it is not in standard form, because the paper asked for that form.