You sat 77 questions across eleven strands. In this strand — indices, roots and standard form — you were secure at step 1 and the first gap opened at step 2. Step 2 is the age 11–12 rung. That is the lowest break of all eleven strands, which is why this guide exists before any of the others.
A break that low is almost never a sign that the hard material is too hard. It is a sign that one early idea — what a power actually means — got learned as a rule instead of as a picture, and everything stacked on top of it has been wobbling ever since.
It maps onto four syllabus points, and every one of them is rated high non-calculator risk:
| Code | Topic | Where it bites |
|---|---|---|
| E1.3 | Powers and roots | Squares, cubes and their roots by recall |
| E1.7 | Indices I | The three laws, zero, negative, fractional |
| E1.8 | Standard form | Writing it, and multiplying and dividing by hand |
| E1.9 | Estimation | Rounding to 1 significant figure to check an answer |
Paper 2 is non-calculator: 2 hours, 100 marks, 50% of your grade. So nothing in this guide is done with a calculator, and nowhere will you be told to reach for one. Everything is by hand, every line shown.
Indices also feed upward. E2.4 Indices II (indices with algebra), surds, and every standard-form calculation in physics and chemistry all sit on this brick. One loose brick is quietly taxing several other topics at once.
Number sense and the four operations: solid on all seven rungs. No gap anywhere. Your arithmetic mechanics work. You can add, subtract, multiply, divide, handle negatives in ordinary sums, and hold a calculation together in your head.
That matters more than it sounds. The usual pattern in a weak maths student is shaky arithmetic underneath everything, which makes every topic hard and slow to fix. Yours is the opposite: a good foundation with two loose bricks sitting on it — indices, and algebraic manipulation. That is a much easier problem, and a much faster one.
One more thing the check picked up: sign errors recurred across several different topics. Negative indices are exactly where sign errors do the most damage, so they get their own warnings in section 4.
Each idea appears four times, and it gets less help each time. That is deliberate — it is the fastest way to learn a method you do not yet own.
Sections 1 to 7 rebuild from below the break upward. Section 8 is a mixed set with the topics deliberately unlabelled and out of order, because in an exam nobody tells you which idea applies.
Start here, below the break. This is the idea that step 2 tested, and it is almost certainly the thing that went wrong — because there is one specific mistake nearly everybody makes at this rung.
The syllabus names these explicitly, and Paper 2 assumes them. Recall, not working out.
| n | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
|---|---|---|---|---|---|---|---|---|
| n2 | 1 | 4 | 9 | 16 | 25 | 36 | 49 | 64 |
| n | 9 | 10 | 11 | 12 | 13 | 14 | 15 | |
| n2 | 81 | 100 | 121 | 144 | 169 | 196 | 225 |
| n | 1 | 2 | 3 | 4 | 5 | 10 |
|---|---|---|---|---|---|---|
| n3 | 1 | 8 | 27 | 64 | 125 | 1000 |
A root is not a new operation to learn. It is the reverse of a power, in exactly the way that dividing is the reverse of multiplying.
So every square you know gives you a square root for free. That table in section 1 is doing double duty. √169 is not something you compute; it is something you recognise, because you already know 13².
Paper 2 will ask you to estimate a root like √50. You cannot work it out exactly by hand, and you are not meant to. You trap it between the two squares either side of it.
There are three laws. You are not going to memorise them, because memorised rules are the thing that collapsed at step 2. You are going to see why each one has to be true, from what a power means. Once you have seen it, you can rebuild any of them in ten seconds in an exam.
Three as in one bracket and four in the other. Push them together and you have seven as. That is all “add the indices” means: you are counting how many copies there are in total.
Two of the as on the top cancel against the two on the bottom. Five copies, take away two, leaves three.
Two lots of three as is six as. Not three plus two.
This is where the check flagged sign errors recurring across topics, and this is where they cost the most marks. Slow down here.
“Zero copies of a multiplied together” is a sentence that means nothing, so the picture does not help here. The division law does.
Both lines describe the same thing, so a0 and 1 must be the same thing. That is the whole argument. 70 = 1. 10000 = 1. (−4)0 = 1.
Same trick, keep dividing.
A negative index is an instruction to flip, never an instruction to make the answer negative.
An index of 1⁄2 looks strange until you push the multiplying law at it.
So a½ is the thing that gives you a when multiplied by itself. That is the definition of a square root. Therefore:
The bottom of the fraction is the root. Now the top. Using the power-of-a-power law:
You are allowed to do them in either order — (n√a)m and n√(am) give the same answer. But by hand the difference is enormous.
| 163/4 — root first | 163/4 — power first |
|---|---|
| 4th root of 16 = 2 23 = 8 Two easy steps |
163 = 4096 4th root of 4096 = 8 You have to cube 16 and then root a four-digit number |
Standard form is a way of writing any number as
The condition on A is the whole game. A must have exactly one non-zero digit before the decimal point. 0.9 × 104 and 12 × 103 are both correct values and both wrong form — and in an exam the form is what is being marked.
Put the decimal point after the first non-zero digit, then count how many places it moved.
This is the part that actually gets examined, and it is easier than it looks because the two halves separate completely.
Numbers with numbers, powers of ten with powers of ten. Then check the front number, because it often lands outside 1 to 10 and has to be fixed.
Multiplying and dividing let the two halves separate. Adding and subtracting do not: you may only add the front numbers when the powers of 10 are the same. Rewrite one number so the powers match, add, then put the answer back into standard form.
E1.9. This is pure Paper 2 material, and it is the cheapest topic on the syllabus: no method to learn, just one rule applied honestly. It also rescues you elsewhere — when you are unsure whether an answer is roughly right, an estimate settles it in fifteen seconds.
The first significant figure is the first non-zero digit. Round to it and turn everything after it into zeros (or nothing, after a decimal point).
| Number | To 1 s.f. | Why |
|---|---|---|
| 4.87 | 5 | First digit is 4; the next digit is 8, so round up |
| 19.6 | 20 | First digit is 1; the next digit is 9, so round the 1 up to 2 |
| 0.0412 | 0.04 | First non-zero digit is 4; the leading zeros are placeholders, not significant |
| 612 | 600 | First digit 6; next digit 1, so it stays |
| 0.0489 | 0.05 | First non-zero digit 4; next digit 8, so round up |
| instruction | count from | example |
|---|---|---|
| decimal places (d.p.) | the decimal point | 7.4651 to 2 d.p. is 7.47; 3.8962 to 2 d.p. is 3.90 (write the 0) |
| significant figures (s.f.) | the first non-zero digit | 0.04056 to 2 s.f. is 0.041; 48 520 to 3 s.f. is 48 500 |
| the nearest 10, 100, 1000 | that place value | 5764 to the nearest 1000 is 6000 |
In every case look at the next digit: 5 or more rounds up. Keep the size of the number: 48 520 to 3 s.f. is 48 500, not 485. And when a rounded answer ends in 0 after the point, write the 0: 3.90 to 2 d.p., not 3.9.
Decide what the answer means before you round. People or things that must all be carried: round up. Complete items you can make: round down. Money: 2 decimal places. Lengths: 3 significant figures unless told otherwise.
Sixteen questions drawn from everything above, shuffled and unlabelled. Ordinary revision does one topic at a time, which quietly does the hardest part for you — deciding which idea applies. Here nobody tells you. Before you calculate anything, say to yourself which of these it is: a power, a root, an index law, a zero or negative index, a fractional index, standard form, or an estimate.
Five things, and they cover most of what an examiner can ask in this strand:
Come back to section 8 in a week without reading anything above it. If the score bar sits high on a cold run, this brick is set and the next repair guide can start.