Cambridge IGCSE Mathematics 0580: Powers, roots and indices I – Study Guide

This study guide teaches two subtopics of Topic 1 Number in the Cambridge IGCSE Mathematics 0580 syllabus for 2025, 2026 and 2027: 1.3 Powers and roots and 1.7 Indices I. Section 1.3 is the same for Core and Extended candidates. In section 1.7, positive, zero and negative integer indices and the rules of indices are Core; fractional indices are Extended only. Core candidates sit Paper 1 (non-calculator) and Paper 3 (calculator); Extended candidates sit Paper 2 (non-calculator) and Paper 4 (calculator), so you need these skills both with and without a calculator.

Use it with the 0580 course hub and the printable 0580 checklist. The wider topic is covered in the Number study guide and the Number revision notes. When you finish, test yourself with the powers, roots and indices practice questions. To find your weak spots across the whole syllabus, try the Core diagnostic or the Extended diagnostic.

What this unit covers

0580 section	What you must be able to do	Tier
1.3	Calculate with squares, square roots, cubes, cube roots and other powers and roots of numbers	Core and Extended
1.3 (note)	Recall the squares from 1² to 15² and their roots; recall the cubes of 1, 2, 3, 4, 5 and 10 and their roots	Core and Extended
1.7 item 1	Understand and use indices that are positive, zero or negative integers	Core and Extended
1.7 item 1	Understand and use fractional indices	Extended only
1.7 item 2	Understand and use the rules of indices	Core and Extended

Indices I works with numbers. Simplifying algebraic terms such as (5x³)² and solving equations such as 2ˣ = 32 belong to section 2.4, Indices II, in the algebra topic.

The language of powers

In 3⁴, the number 3 is the base and 4 is the index (also called the power or exponent). The index tells you how many times the base is multiplied by itself:

3^4 = 3 × 3 × 3 × 3 = 81

A square is a power of 2 and a cube is a power of 3. A root undoes a power. The square root of 49 is 7 because 7² = 49. The cube root of 64 is 4 because 4³ = 64.

1.3 Powers and roots

Squares and cubes you must recall

The syllabus expects these from memory, forwards and backwards.

n	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15
n²	1	4	9	16	25	36	49	64	81	100	121	144	169	196	225

n	1	2	3	4	5	10
n³	1	8	27	64	125	1000

“Backwards” matters as much as forwards. On a non-calculator paper you may be asked for √196 or ∛125 with no other help, so you need to see 196 and think “14”.

Square roots and cube roots

The symbol √ means the positive square root. So √81 = 9, not ±9. (Both 9 and −9 square to 81, but √81 names only the positive one.)

Cube roots behave differently with negative numbers. A negative number cubed stays negative, so a negative number has a cube root:

(-5)^3 = -125,  so  ∛(-125) = -5

A negative number has no square root among the real numbers, because any number squared is zero or positive.

Powers and roots of negatives, decimals and fractions

The same rules apply to any number, but take care with signs and place value.

• An even power of a negative number is positive: (−2)⁴ = 16. An odd power stays negative: (−3)³ = −27.
• Brackets matter: (−2)⁴ = 16 but −2⁴ = −(2⁴) = −16.
• Square a decimal by squaring the digits and counting decimal places: 0.3² = 0.09 (two decimal places), not 0.9.
• Square or cube a fraction by applying the power to the numerator and the denominator: (2/5)³ = 8/125.
• Take a root of a fraction the same way: √(9/64) = 3/8.
• Take a root of a decimal by writing it as a fraction first: √0.81 = √(81/100) = 9/10 = 0.9.

Worked example 1 (non-calculator)

(a) Work out √(13² − 12²).

13^2 = 169,  12^2 = 144
169 - 144 = 25
√25 = 5

Work out the inside of the root first. √(13² − 12²) is not 13 − 12.

(b) Work out (√49 + ∛64)².

√49 = 7,  ∛64 = 4
(7 + 4)^2 = 11^2 = 121

Other powers and roots

The syllabus also lists “other powers and roots of numbers”. These include fourth and fifth powers (2⁵ = 32) and fourth and fifth roots (⁴√16 = 2, because 2⁴ = 16). Small ones can be done by hand. For anything larger, use your calculator on Paper 3 or Paper 4.

On a scientific calculator, use the power key (usually labelled xʸ or ^) and the root key (often ˣ√ or ⁿ√, sometimes a shift function). Put brackets around a negative base, and around the whole expression under a root.

Worked example 2 (calculator)

(a) Work out the fourth root of (2.3⁵ + 41). Give your answer correct to 3 significant figures.

2.3^5 = 64.36343
64.36343 + 41 = 105.36343
fourth root of 105.36343 = 3.203852...

Answer: 3.20 (3 s.f.). Keep the full value of 2.3⁵ in the calculator rather than rounding it part-way. Early rounding can change the last significant figure of the answer. Write 3.20, not 3.2: the zero is the third significant figure.

(b) Work out 1.6⁵. Write down all the figures on your calculator display.

1.6^5 = 10.48576

1.7 Indices I

Zero and negative indices

Dividing powers of the same base explains what zero and negative indices mean.

5^3 ÷ 5^3 = 125 ÷ 125 = 1,   and by subtracting indices  5^(3-3) = 5^0
so 5^0 = 1

5^2 ÷ 5^5 = 25 ÷ 3125 = 1/125,   and  5^(2-5) = 5^-3
so 5^-3 = 1/5^3 = 1/125

In general, for any non-zero base a:

• a⁰ = 1
• a⁻ⁿ = 1/aⁿ (a negative index means “one over”, that is, the reciprocal)

A negative index never makes the answer negative. 4⁻³ = 1/4³ = 1/64, which is a small positive number.

For a fraction, a negative index flips it:

(3/4)^-2 = (4/3)^2 = 16/9

Powers of 10 with negative indices give decimals: 10⁻³ = 1/1000 = 0.001. This links directly to standard form, section 1.8, which the Number study guide covers.

The rules of indices

These rules work only when the bases are the same.

Rule	In words	Example
aᵐ × aⁿ = aᵐ⁺ⁿ	Multiplying: add the indices	6³ × 6⁻⁵ = 6⁻²
aᵐ ÷ aⁿ = aᵐ⁻ⁿ	Dividing: subtract the indices	2⁴ ÷ 2⁷ = 2⁻³
(aᵐ)ⁿ = aᵐⁿ	Power of a power: multiply the indices	(3²)⁴ = 3⁸
a⁰ = 1	Any non-zero base to the power 0	9⁰ = 1
a⁻ⁿ = 1/aⁿ	Negative index: reciprocal	2⁻³ = 1/8

Worked example 3 (non-calculator)

(a) Find the value of 3⁵ × 3⁻² ÷ 3⁴.

3^5 × 3^-2 = 3^(5 + (-2)) = 3^3
3^3 ÷ 3^4 = 3^(3 - 4) = 3^-1
3^-1 = 1/3

Answer: 1/3.

(b) Find the value of (2⁻²)³ × 2⁸.

(2^-2)^3 = 2^(-2 × 3) = 2^-6
2^-6 × 2^8 = 2^2 = 4

Answer: 4.

(c) Find the value of 6³ × 6⁻⁵ ÷ 6⁻⁴.

6^3 × 6^-5 = 6^-2
6^-2 ÷ 6^-4 = 6^(-2 - (-4)) = 6^2 = 36

Subtracting a negative index adds: −2 − (−4) = 2. This is where many sign errors happen.

Different bases: rewrite first

If the bases differ, write each number as a power of a common base before using the rules.

Worked example 4 (non-calculator). Write 4³ × 2⁻⁵ as a power of 2, and find its value.

4 = 2^2, so 4^3 = (2^2)^3 = 2^6
2^6 × 2^-5 = 2^1 = 2

You may also be asked to write a number as a power of a given base: 0.125 = 1/8 = 1/2³ = 2⁻³.

Fractional indices (Extended only)

Extended candidates extend the index rules to fractions. A unit fraction index is a root:

• a^(1/2) = √a and a^(1/3) = ∛a; in general, a^(1/n) is the nth root of a.
• a^(m/n) = (nth root of a)^m. Take the root first, then the power. The numbers stay smaller that way.
• A negative fractional index combines both ideas: take the reciprocal as well.

Write fractional indices with brackets when typing, for example 64^(1/3), so the index is clear.

Worked example 5 (Extended only, non-calculator)

(a) 64^(1/3) = ∛64 = 4.

(b) 32^(3/5): the fifth root of 32 is 2, then 2³ = 8.

(c) 125^(−2/3):

cube root of 125 = 5
5^2 = 25
negative index -> reciprocal: 1/25

(d) (27/8)^(−1/3):

(27/8)^(1/3) = 3/2
reciprocal: 2/3

(e) (16/81)^(3/4):

fourth root of 16/81 = 2/3
(2/3)^3 = 8/27

Combining fractional indices with the rules

Worked example 6 (Extended only, non-calculator). Find the value of 9^(1/2) × 27^(2/3).

Method 1, evaluate each part:

9^(1/2) = 3
27^(2/3) = (∛27)^2 = 3^2 = 9
3 × 9 = 27

Method 2, use base 3:

9^(1/2) = (3^2)^(1/2) = 3^1
27^(2/3) = (3^3)^(2/3) = 3^2
3^1 × 3^2 = 3^3 = 27

Both give 27. Method 2 is the one to use when a question asks you to write an answer “as a power of 3”.

Calculator or no calculator

Everything on this page can appear on a non-calculator paper (Paper 1 or Paper 2), so practise the recall facts and the index rules by hand. On the calculator papers (Paper 3 or Paper 4), questions on other powers and roots often ask you to write all the figures on the display and then round. Write the full display first. It can earn credit even if your rounding slips.

Common errors

• Writing 3⁴ as 3 × 4 = 12. The index counts repeated multiplication: 3⁴ = 81.
• Making a negative index give a negative answer: 2⁻³ = 1/8, not −8 and not −6.
• Writing 7⁰ = 0 or 7⁰ = 7. Any non-zero number to the power 0 is 1.
• Multiplying the bases when multiplying powers: 3² × 3⁴ = 3⁶, not 9⁶.
• Adding indices for a power of a power: (2³)² = 2⁶, not 2⁵.
• Using the index rules when the bases differ, such as 2³ × 3² = 6⁵. Convert to a common base first, or just evaluate: 8 × 9 = 72.
• Squaring a decimal by place value incorrectly: 0.4² = 0.16, not 1.6 or 0.8.
• Treating √(a² + b²) as a + b.
• (Extended only) Applying the power before the root for large bases, which makes the arithmetic much harder, or forgetting to take the reciprocal when the fractional index is negative.

Where to go next

Condense this page into a few lines of index rules, then work through the powers, roots and indices practice questions, which cover every learning outcome here with full mark schemes. For mixed questions across the rest of Topic 1, use the Number practice questions and, for Extended candidates, the Number (Extended) practice questions.

Official syllabus

Cambridge IGCSE Mathematics 0580 syllabus for 2025, 2026 and 2027, Cambridge International Education. Subject content sections C1.3 and E1.3 (Powers and roots) and C1.7 and E1.7 (Indices I).