Standard Form Worked ExamplesIGCSE Maths 0580

The sign of the index is where this mark goes. The June 2025 examiner report found that "The most common error was for candidates to replace the power of –3 with +3" when writing a small decimal, and the March 2026 report found the reverse slip on a large number: "The most frequent error was to give a power of –5 rather than 5." A negative index means a small number, not a negative one — $4.3 \times 10^{-5}$ is positive and less than $1$.

Two marks routinely lost here: forgetting to re-adjust when the number part reaches $10$ or more, and mishandling $5 - (-3)$. Subtracting a negative index adds.

Addition and subtraction need a common index; multiplication and division do not. The June 2025 examiner report observed that "Successful candidates usually changed one of the original values so that it had the same power as the other", while "Less able candidates tried to process the number and the power separately" — adding $3.2 + 4.5$ to get $7.7 \times 10^{6}$ is that error — check by converting to ordinary numbers: $3\,200\,000 + 450\,000 = 3\,650\,000$.

Raising a power to a power multiplies the indices; only multiplying two terms adds them. Mixing these up turns $10^{12}$ into $10^{7}$. On squaring a standard-form number, the March 2025 examiner report described errors of "forgetting to square the 4 and adding 2 to the power of 10 instead of doubling" — here, cube the $2.5$ as well as the power of ten.

Three index rules in one question: multiply the index when raising to a power, add when multiplying, subtract when dividing. $10^{0} = 1$ trips students up — it is not zero. Sense-check by converting: $0.000016 \times 600000 \div 0.0008 = 12000$. ✓

In (b) the number part comes out below $1$, so the index goes down by one rather than up. Part (b) is a ratio of two masses, so it has no units. Sense-check both answers: the June 2025 examiner report advised that "In context questions such as this one, candidates need to consider whether their answers are realistic."

Decide the direction before calculating: image to real divides, real to image multiplies. Most lost marks come from the unit change — $0.024\text{ mm}$ is $2.4 \times 10^{-5}\text{ m}$, not $2.4 \times 10^{-2}\text{ m}$. A quick sense check: a real cell is tiny, so a large negative index is expected.

The diagram decides the operation: same side of the Sun means subtract, opposite sides means add. On a non-calculator subtraction like (a), the March 2026 examiner report found that "Quite a few candidates made arithmetic errors" once the powers were matched, so check $2.28 - 1.5$ carefully. Leaving (a) as $0.78 \times 10^{8}$ loses the accuracy mark because it is not in standard form. In (b), $260\text{ s}$ is not $2.6$ minutes — convert using $60$ seconds in a minute.

In (a), write every step — the March 2026 examiner report advised that "When candidates are required to show a given result, they should show every step in their method". Convert $1.2\text{ cm}$ to metres before dividing — mixing cm and m makes the answer $100$ times too big. In (c), four times as many squares on the same area means each side halves; dividing the side by $4$ is the common error, and would fit $16$ times as many transistors.

The March 2026 examiner report found that most candidates "did not realise they needed to amend the terms to allow cube rooting" to work. Taking the cube root of $2.7$ and dividing $40$ by $3$ leads nowhere without a calculator. Check the direction of the adjustment: making the number part $10$ times bigger means the index must go down by $1$.

Check the size of the answer: a tiny mass per molecule must mean a huge number of molecules. The June 2025 examiner report found that "A large minority chose multiplication" on this type, and advised that "In context questions such as this one, candidates need to consider whether their answers are realistic." Forgetting the kg to g conversion gives an answer $1000$ times too small.