OCR GCSE Mathematics: Powers, roots, laws of indices and standard form (J560) – Revision Notes

These notes condense the full study guide for the OCR GCSE (9-1) Mathematics specification, Version 2.1 (August 2026), sections 3.01 and 3.02. Fractional indices, fractional powers and estimating powers and roots are Higher tier only and are labelled. Everything else is assessed at both tiers. Use the OCR GCSE Mathematics hub and the checklist to track progress, and the free 10-minute diagnostics to find weak spots. When you are ready, move on to the practice questions.

Values to memorise

Item	Values
2¹ to 2⁸	2, 4, 8, 16, 32, 64, 128, 256
3¹ to 3⁵	3, 9, 27, 81, 243
4¹ to 4⁴	4, 16, 64, 256
5¹ to 5⁴	5, 25, 125, 625
Squares to 15	1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225
Cubes to 5 and 10	1, 8, 27, 64, 125 and 1000

Index rules at a glance

Rule	Meaning	Example	Tier
aⁿ	n copies of a multiplied	2⁴ = 16	Both
a⁻ⁿ = 1/aⁿ	Reciprocal, not negative	3⁻² = 1/9	Both
(a/b)⁻ⁿ = (b/a)ⁿ	Flip the fraction	(2/3)⁻³ = 27/8	Both
a⁰ = 1	Any non-zero base	9⁰ = 1	Both
a^m × a^n = a^(m+n)	Add	4³ × 4⁵ = 4⁸	Both
a^m ÷ a^n = a^(m-n)	Subtract	7⁹ ÷ 7⁴ = 7⁵	Both
(a^m)^n = a^(mn)	Multiply	(2³)⁴ = 2¹²	Both
a^(1/n) = nth root of a	Denominator is the root	16^(1/4) = 2	Higher tier only
a^(m/n) = (nth root of a)^m	Root, then power	32^(2/5) = 4	Higher tier only
a^(−m/n) = 1/a^(m/n)	Reciprocal of the above	8^(−1/3) = 1/2	Higher tier only

The three laws need the same base. Different bases means no combining: 2³ × 3² stays as 8 × 9 = 72.

Method: simplifying an algebraic power expression

1. Expand any bracket, applying the power to the number and every letter.
2. Handle the numbers on their own (multiply, divide or reduce).
3. Handle each letter separately using the laws.
4. Write negative indices as fractions if the question asks for positive powers.

Reminder: (2a³)⁴ = 16a¹². The 2 is raised to the 4th power as well.

Roots

Root	Meaning	Example
√a	Square root	√144 = 12
∛a	Cube root	∛343 = 7
ⁿ√a	nth root	⁴√81 = 3

A root of a perfect square or cube is exact. Calculating with roots means working each root out first: √49 + ∛64 = 7 + 4 = 11.

Method: fractional powers (Higher tier only)

1. Split a^(m/n) into root first, power second.
2. Take the nth root of the base.
3. Raise the result to the power m.
4. If the index is negative, take the reciprocal at the end.

Reminder: 8^(−1/3) = 1/(∛8) = 1/2. Reminder: (27/8)^(2/3) = (3/2)² = 9/4.

Combining powers and roots (Higher tier only): √x = x^(1/2), ∛x = x^(1/3), 1/√x = x^(−1/2), (8x⁶)^(1/3) = 2x².

Method: estimating a root (Higher tier only)

1. Find the perfect squares (or cubes) either side of the number.
2. Square (or cube) the midpoint between the two roots.
3. If the number is below that, round down. If above, round up.

Reminder: √200 sits between 14 and 15. 14.5² = 210.25 is above 200, so the answer is 14. To estimate a power, round the base: 4.1³ is close to 64.

Standard form

A × 10ⁿ with 1 ≤ A < 10 and n a whole number.

Number	Standard form	Reason
1320	1.32 × 10³	Point moves 3 places left
0.00943	9.43 × 10⁻³	Point moves 3 places right
45 000 000	4.5 × 10⁷	7 places left
0.000602	6.02 × 10⁻⁴	4 places right
37.5 × 10⁴	not standard	A is above 10

Large numbers have a positive index. Numbers between 0 and 1 have a negative index.

Ordering: compare the powers of 10 first, then the A values when the powers match. Put every number in standard form before you compare.

Method: calculating in standard form by hand

Operation	What to do	Then
Multiply	Multiply the A values, add the indices	Tidy A into 1 to 10
Divide	Divide the A values, subtract the indices	Tidy A into 1 to 10
Add or subtract	Rewrite to the same power of 10, combine the A values	Tidy A into 1 to 10

Reminders:

• (3.2 × 10⁵) × (4 × 10⁻⁸) = 12.8 × 10⁻³ = 1.28 × 10⁻².
• (9.6 × 10⁷) ÷ (1.6 × 10⁻³) = 6 × 10¹⁰.
• 4.5 × 10⁶ + 3.2 × 10⁵ = 4.5 × 10⁶ + 0.32 × 10⁶ = 4.82 × 10⁶.

With a calculator, bracket carefully, keep the full unrounded value until the end, and give the answer in standard form if the question asks.

Small worked reminders

Finding an unknown index. Solve 2ⁿ = 1/32. Since 32 = 2⁵, 1/32 = 2⁻⁵, so n = −5.

Mixed laws. Write 2⁷ ÷ 2⁴ × 2⁻¹ as one power: 7 − 4 − 1 = 2, so 2² = 4.

Algebra with a bracket. Simplify (5x³y)² ÷ (10x⁴y⁻¹). The bracket gives 25x⁶y². Then 25 ÷ 10 = 5/2, x^(6−4) = x² and y^(2−(−1)) = y³. The answer is 5x²y³/2.

Unknown base (Higher tier only). Solve x^(−1/2) = 1/5. This says 1/√x = 1/5, so √x = 5 and x = 25.

Ordering. Order 3.2 × 10⁵, 4.1 × 10⁻², 2.9 × 10⁵ and 8 × 10⁴. The smallest power of 10 comes first, then the 10⁴ number, then the two 10⁵ numbers by their A values: 4.1 × 10⁻², 8 × 10⁴, 2.9 × 10⁵, 3.2 × 10⁵.

Context question. 2.4 × 10¹³ bytes shared over 8 × 10⁹ drives. Divide the A values (2.4 ÷ 8 = 0.3) and subtract the indices (13 − 9 = 4). That gives 0.3 × 10⁴, which is 3 × 10³ after tidying.

Exam routine for this topic

1. Check the tier. Fractional indices, fractional powers and estimating are Higher tier content, so Foundation students concentrate on integer powers, laws and standard form.
2. Decide whether the paper allows a calculator. On a hand question, show every step: the matched powers, the reciprocal or the root-then-power split.
3. Simplify before you substitute. Combine powers of the same base before working out any numbers.
4. Finish by checking the form of the answer: A between 1 and 10 for standard form, a fraction when the question wants one, positive indices if the question says so.

Must-know distinctions

• Negative index vs negative number: 2⁻³ = 1/8, which is positive.
• Index vs root: the index 1/2 means square root. The index −1/2 means one over the square root.
• Multiplying powers vs raising a power: x² × x³ = x⁵ but (x²)³ = x⁶.
• Adding indices vs adding numbers: you add indices when multiplying standard form, not when adding.
• Exact root vs estimate: an estimate asks for the nearest whole number, so you do not need decimals.

Quick self-test

1. Work out 2⁶ and 5³.
2. Write 4⁻³ as a fraction.
3. Simplify 9a⁷ ÷ 3a⁻².
4. Evaluate (3/5)⁻².
5. Write 0.0000715 in standard form.
6. Work out (5 × 10⁴) × (6 × 10⁻⁹) in standard form.
7. Work out 7.5 × 10³ − 6 × 10² in standard form.
8. Higher tier only: evaluate 125^(2/3).
9. Higher tier only: evaluate 16^(−3/4).
10. Higher tier only: estimate ∛30 to the nearest whole number.

Answers

1. 2⁶ = 64 and 5³ = 125.
2. 4⁻³ = 1/4³ = 1/64.
3. 9 ÷ 3 = 3 and a^(7−(−2)) = a⁹, so 3a⁹.
4. (3/5)⁻² = (5/3)² = 25/9.
5. The point moves 5 places right: 7.15 × 10⁻⁵.
6. 5 × 6 = 30 and 10⁴ × 10⁻⁹ = 10⁻⁵, giving 30 × 10⁻⁵ = 3 × 10⁻⁴.
7. 7.5 × 10³ − 0.6 × 10³ = 6.9 × 10³.
8. (∛125)² = 5² = 25.
9. 1/(⁴√16)³ = 1/2³ = 1/8.
10. 3³ = 27 and 4³ = 64. The midpoint 3.5³ = 42.875 is above 30, so 3.

Where marks are usually lost

• Reading 5⁻² as −25 or −10 when it is 1/25.
• Forgetting to apply a bracket power to the coefficient, writing (3x²)³ as 3x⁶.
• Combining different bases with the laws of indices.
• Leaving an answer such as 12.8 × 10⁻³, which is not in standard form.
• Moving the decimal point the right number of places in the wrong direction, so n has the wrong sign.
• Adding standard form numbers without first matching the powers of 10.
• Giving a decimal to a question that asks for a fraction or exact value.
• Cubing before rooting with a fractional index, which makes the numbers large and error-prone.
• Skipping the working on a hand calculation. Showing the matched powers or the reciprocal step lets an examiner see your method.

Official syllabus

OCR GCSE (9-1) Mathematics J560, Version 2.1 (August 2026), for first assessment in 2017, published by OCR. Sections 3.01 and 3.02.