Standard form writes any number as A × 10ⁿ, where 1 ≤ A < 10 and n is an integer. To convert, count how many places the decimal point moves to bring the number into that range: n is positive for large numbers and negative for small numbers. To multiply or divide, work with the A values and the powers of 10 separately using the index laws. To add or subtract, first rewrite both numbers with the same power of 10, then combine the A values.
Converting a single number to standard form is usually fine. The marks get lost once a calculation is involved. Students multiply or divide the A values correctly but forget the power of 10 needs the index laws applied too, or they add two standard form numbers as if the powers were the same when they are not. Every A value here is a rational number, a terminating decimal that stays exact through the working.
The other common loss is presentation: the final answer must have 1 ≤ A < 10. An arithmetically correct answer written as 34 × 10⁵ instead of 3.4 × 10⁶ is not in standard form, and O-Level markers will not award the mark for it.
Convert
Move the decimal point so 1 ≤ A < 10; count the moves as n
Multiply / divide
Combine the A values, combine the powers of 10 with index laws
Add / subtract
Rewrite both numbers with the same power of 10 first
Check
Is 1 ≤ A < 10? If not, adjust A and n together
To write 4 500 000 in standard form, move the decimal point left until only one non-zero digit sits before it: 4.5. The point moved 6 places, so 4 500 000 = 4.5 × 10⁶.
To write 0.000032 in standard form, move the decimal point right until the first digit is between 1 and 9: 3.2. The point moved 5 places, and because the original number is less than 1, the power is negative: 0.000032 = 3.2 × 10⁻⁵.
Quick check: for numbers 10 or larger, n is positive and equals the number of digits after the first one, counted before the decimal point. For numbers less than 1, n is negative and equals how many places the point moves right to reach the first non-zero digit.
Let A = 3.6 × 10⁵ and B = 4 × 10². Find A × B and A ÷ B, both in standard form.
Solution: A × B
3.6 × 4 = 14.4.10ᵐ × 10ⁿ = 10ᵐ⁺ⁿ: 10⁵ × 10² = 10⁷.14.4 × 10⁷. But 14.4 is not between 1 and 10, so this is not yet standard form.1.44 × 10, then combine the powers again: 14.4 × 10⁷ = 1.44 × 10 × 10⁷ = 1.44 × 10⁸.Solution: A ÷ B
3.6 ÷ 4 = 0.9.10ᵐ ÷ 10ⁿ = 10ᵐ⁻ⁿ: 10⁵ ÷ 10² = 10³.0.9 × 10³. But 0.9 is less than 1, so this is not yet standard form.9 × 10⁻¹, then combine the powers again: 0.9 × 10³ = 9 × 10⁻¹ × 10³ = 9 × 10².The step students get wrong
Getting the arithmetic right and stopping there. If the A value in your final line is not between 1 and 10, the calculation is not finished. Shift the decimal point in A back into range, and adjust the power of 10 by the same number of places in the opposite direction. This is the exact mistake behind writing 34 × 10⁵ as a final answer instead of 3.4 × 10⁶: the digits are correct, the form is not, and the mark is not awarded.
Find 5.2 × 10⁴ + 3.8 × 10³, giving your answer in standard form.
Solution
3.8 × 10³ = 0.38 × 10⁴ (moving the decimal point one place left makes A ten times smaller, so the power of 10 must rise by 1 to compensate).10⁴, so add the A values directly: 5.2 + 0.38 = 5.58.5.58 × 10⁴. Since 5.58 is between 1 and 10, this is already in standard form.Same idea for subtraction: align the powers of 10 first, subtract the A values, then check the result is still in the range 1 to 10. Never subtract or add the powers of 10 themselves; only the A values combine once the powers match.
The fx-97SG has a dedicated key for entering and reading standard form, usually labelled ×10ˣ (sometimes reached via SHIFT then a log-related key, depending on the model). Use it to enter numbers directly as A × 10ⁿ instead of typing out long strings of zeros, which cuts down on typing errors.
Set the calculator to standard form output when a question specifically asks for standard form, so the display matches what you need to write down. Even so, always write the final answer in the A × 10ⁿ form by hand and check 1 ≤ A < 10 yourself. The calculator's internal display format is not always what the exam wants written on paper, and relying on it without checking is a common source of lost presentation marks. For the same reason, get into the habit of a final calculator check on any topic with a mode setting, the same discipline covered in the degree and radian check guide.
Can n be zero in standard form?
Yes. Any number from 1 up to just under 10 is already in standard form with n = 0, for example 7.5 = 7.5 × 10⁰. It looks unusual but it is still correct.
What is the difference between standard form and index laws?
Standard form is a way of writing a number; index laws are the rules for combining powers of the same base, in this case powers of 10. You need index laws to multiply or divide numbers once they are already in standard form. See the index laws guide for the full set of rules, including negative and fractional indices.
Why does the power of 10 change when I move the A value's decimal point?
Moving the decimal point in A changes its size, so the power of 10 must change to compensate and keep the overall value the same. Moving the point one place left in A means A got 10 times smaller, so the power of 10 must increase by 1 to balance it, and vice versa.
Do I need standard form for very ordinary-sized numbers?
No. Standard form is most useful, and most often tested, for very large numbers (distances, populations, large sums of money) and very small numbers (measurements in science contexts), the same values often written with metric prefixes like kilo or micro instead. A number like 45 does not need to be written in standard form unless a question specifically asks for it.
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Mr. Gan works with students who want a systematic method that works every time, not shortcuts that fall apart the moment the powers of 10 do not match.
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