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

This guide is written for the OCR GCSE (9-1) Mathematics specification, Version 2.1 (August 2026). It covers specification sections 3.01 (powers and roots) and 3.02 (standard form). Most of it applies to both Foundation and Higher tier. Fractional indices, fractional powers and estimating powers and roots are in the Higher tier column and are labelled “Higher tier only”. The specification says that content with no reference to calculator use may be assessed on either calculator or non-calculator papers, so you need to be able to do all of it by hand. Paper 2 (Foundation) and Paper 5 (Higher) do not permit a calculator.

Related pages: the OCR GCSE Mathematics hub, the printable checklist, the revision notes, the practice questions and the free 10-minute diagnostics. For the wider number topic, see Number Operations and Integers.

What this unit covers

Ref	What you must be able to do	Who
3.01a	Use positive integer indices; use negative integer indices as reciprocals	Positive: both. Negative: both
3.01a	Use fractional indices to represent roots and combinations of powers and roots	Higher tier only
3.01b	Calculate positive integer powers and exact roots; recognise simple powers of 2, 3, 4 and 5	Both
3.01b	Calculate with integer powers (including negative) and with roots	Both
3.01b	Calculate fractional powers; estimate powers and roots	Higher tier only
3.01c	Know and apply a^m × a^n = a^(m+n), a^m ÷ a^n = a^(m-n), (a^m)^n = a^(mn)	Both
3.02a	Interpret, order and convert numbers to and from standard form	Both
3.02b	Calculate with standard form using a calculator, then add, subtract, multiply and divide without one	Both

Index notation and positive powers (3.01a, 3.01b)

An index (or power) says how many copies of the base are multiplied together.

2⁴ = 2 × 2 × 2 × 2 = 16
base 2, index 4

Learn these so you can spot them quickly.

Index	1	2	3	4	5	6	7	8
Base 2	2	4	8	16	32	64	128	256
Base 3	3	9	27	81	243
Base 4	4	16	64	256
Base 5	5	25	125	625

Roots undo powers. The square root √144 = 12 because 12² = 144. The cube root ∛343 = 7 because 7³ = 343. An exact root is a whole number or fraction, so recognising the powers above is what makes roots quick.

Worked example 1. Work out √225 − ∛125 + 2⁵ by hand.

√225 = 15       (15 × 15 = 225)
∛125 = 5        (5 × 5 × 5 = 125)
2⁵   = 32
15 − 5 + 32 = 42

Answer: 42.

Negative indices (3.01a, 3.01b)

A negative index means “one over”. Write a⁻ⁿ = 1/aⁿ. The sign tells you to take the reciprocal. It does not make the answer negative.

3⁻² = 1/3² = 1/9

For a fraction, the reciprocal flips it and the index becomes positive.

Worked example 2. Evaluate (2/3)⁻³.

(2/3)⁻³ = (3/2)³
        = 3³/2³
        = 27/8

Answer: 27/8 (or 3 3/8).

A power of zero gives 1 for any non-zero base. This follows from the division law below: 5³ ÷ 5³ = 5⁰, and anything divided by itself is 1.

The laws of indices (3.01c)

These apply when the base is the same.

Law	Example
a^m × a^n = a^(m+n)	4³ × 4⁵ = 4⁸
a^m ÷ a^n = a^(m-n)	7⁹ ÷ 7⁴ = 7⁵
(a^m)^n = a^(mn)	(2³)⁴ = 2¹²

Add when multiplying, subtract when dividing, multiply when raising a power to a power. Negative indices follow the same rules: 6⁵ × 6⁻² = 6³.

In algebra, deal with the number part first, then each letter on its own.

Worked example 3. Simplify 6x⁵ × 3x⁻².

numbers: 6 × 3 = 18
x: x⁵ × x⁻² = x^(5 + (−2)) = x³

Answer: 18x³.

Worked example 4. Simplify (5x³y)² ÷ (10x⁴y⁻¹).

numerator: (5x³y)² = 25x⁶y²
divide:    25x⁶y² ÷ 10x⁴y⁻¹
numbers:   25 ÷ 10 = 5/2
x:         x^(6−4) = x²
y:         y^(2−(−1)) = y³

Answer: (5/2)x²y³, which can also be written 5x²y³/2.

Notice that the bracket (5x³y)² squares the 5 as well as the letters. That is the most common slip.

Worked example 5. Find n if 3ⁿ = 1/27.

1/27 = 1/3³ = 3⁻³
so n = −3

Answer: n = −3.

Fractional indices (3.01a, 3.01b) – Higher tier only

A fractional index is a root. In a^(m/n), the bottom number n names the root and the top number m is the power.

a^(1/n) = the nth root of a
a^(m/n) = (the nth root of a)^m
a^(−m/n) = 1 / a^(m/n)

Work out the root before applying the power, so the numbers stay manageable.

Worked example 6. Evaluate (a) 16^(1/4) (b) 8^(−1/3) (c) (27/8)^(2/3) (d) 32^(−2/5). Higher tier only.

(a) 16^(1/4) = fourth root of 16 = 2           (2⁴ = 16)
(b) 8^(−1/3) = 1 / 8^(1/3) = 1/2               (∛8 = 2)
(c) (27/8)^(2/3) = (∛(27/8))² = (3/2)² = 9/4
(d) 32^(−2/5) = 1 / (⁵√32)² = 1/2² = 1/4

Answers: 2, 1/2, 9/4, 1/4.

The index laws still work with fractions. Combinations of powers and roots become single powers.

Worked example 7. Simplify (8x⁶)^(1/3). Then write 1/√x as a power of x. Higher tier only.

(8x⁶)^(1/3) = 8^(1/3) × (x⁶)^(1/3) = 2 × x² = 2x²
1/√x = 1/x^(1/2) = x^(−1/2)

Answers: 2x² and x^(−1/2).

Worked example 8. Solve y^(−1/2) = 1/5. Higher tier only.

y^(−1/2) = 1/√y, so 1/√y = 1/5
√y = 5
y = 25

Answer: y = 25.

Estimating powers and roots (3.01b) – Higher tier only

To estimate a root, find the perfect squares or cubes on either side, then decide which is closer.

Worked example 9. Estimate √200 to the nearest whole number. Higher tier only.

14² = 196 and 15² = 225, so √200 is between 14 and 15
the midpoint 14.5² = 210.25, which is above 200
so √200 is below 14.5, and rounds to 14

Answer: 14.

The same method works for cube roots. For ∛70: 4³ = 64 and 5³ = 125, and 4.5³ = 91.125 is above 70, so ∛70 is nearer 4. To estimate a power, round the base: 4.1³ is close to 4³ = 64.

Standard form: converting and interpreting (3.02a)

A number is in standard form when it is written A × 10ⁿ with 1 ≤ A < 10 and n a whole number (positive or negative).

• Large numbers have a positive n. 1320 = 1.32 × 10³.
• Numbers below 1 have a negative n. 0.00943 = 9.43 × 10⁻³.

To convert, count how many places the decimal point moves to get a number from 1 up to (but not including) 10. The count is n. Moving left gives a positive n. Moving right gives a negative n.

Worked example 10. Write 45 000 000 and 0.000602 in standard form.

45 000 000 → 4.5 (point moves 7 places left)   = 4.5 × 10⁷
0.000602   → 6.02 (point moves 4 places right) = 6.02 × 10⁻⁴

To convert back, move the point the number of places the index says. 3.7 × 10⁻² = 0.037.

Numbers like 37.5 × 10⁴ or 0.8 × 10⁵ are not in standard form because A is outside 1 to 10.

Ordering. Compare the powers of 10 first. Only compare the A values when the powers match.

Worked example 11. Put in order, smallest first: 3.2 × 10⁵, 4.1 × 10⁻², 2.9 × 10⁵, 8 × 10⁴.

10⁻² is the smallest power: 4.1 × 10⁻²
10⁴ next: 8 × 10⁴
10⁵ for the other two, compare 2.9 and 3.2: 2.9 × 10⁵, then 3.2 × 10⁵

Order: 4.1 × 10⁻², 8 × 10⁴, 2.9 × 10⁵, 3.2 × 10⁵.

Calculations in standard form (3.02b)

With a calculator. Enter each number carefully, using brackets round a whole product or quotient where needed. Write the result in standard form unless the question asks for something else. Round only at the end.

Without a calculator. Use the index laws.

Multiply: multiply the A values, add the indices, then tidy up.

Divide: divide the A values, subtract the indices, then tidy up.

Add or subtract: write both numbers with the same power of 10, combine the A values, then tidy up.

Worked example 12. Work out (3.2 × 10⁵) × (4 × 10⁻⁸). Give the answer in standard form.

3.2 × 4 = 12.8
10⁵ × 10⁻⁸ = 10⁻³
12.8 × 10⁻³ is not standard form (12.8 is too big)
12.8 × 10⁻³ = 1.28 × 10 × 10⁻³ = 1.28 × 10⁻²

Answer: 1.28 × 10⁻².

Worked example 13. Work out (9.6 × 10⁷) ÷ (1.6 × 10⁻³).

9.6 ÷ 1.6 = 6
10⁷ ÷ 10⁻³ = 10^(7−(−3)) = 10¹⁰

Answer: 6 × 10¹⁰.

Worked example 14. Work out 4.5 × 10⁶ + 3.2 × 10⁵.

rewrite 3.2 × 10⁵ = 0.32 × 10⁶
4.5 × 10⁶ + 0.32 × 10⁶ = 4.82 × 10⁶

Answer: 4.82 × 10⁶.

Worked example 15. A data centre stores 2.4 × 10¹³ bytes spread equally over 8 × 10⁹ drives. How many bytes does each drive hold?

(2.4 × 10¹³) ÷ (8 × 10⁹)
2.4 ÷ 8 = 0.3
10¹³ ÷ 10⁹ = 10⁴
0.3 × 10⁴ = 3 × 10³

Answer: 3 × 10³ bytes.

Common errors

• Reading 2⁻³ as −8 when it is 1/8.
• Forgetting that a bracket power applies to the number part too: (2a³)⁴ is 16a¹².
• Using the index laws when the bases differ: 2³ × 3² is not 6⁵.
• Taking the power before the root with fractional indices and getting huge numbers. Root first.
• Reading a⁻ⁿ as a root or a^(1/n) as a reciprocal.
• Writing A outside the range 1 to 10, such as 12.8 × 10⁻³.
• Counting the decimal point moves in the wrong direction, so the sign of n is wrong.
• Adding standard form numbers by adding the indices. Indices are added only when you multiply.

Next steps

Condense this with the revision notes, then test yourself with the practice questions. Tick off progress on the checklist and use the diagnostics to find weak spots.

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.