Why is "Working out $6^2$ as $6 \times 2 = 12$." a common mistake in Power and Roots?

Why it happens: The index 2 is mistaken for a number you multiply the base by. How to avoid it: An index counts how many times the base is used. 6^2 = 6 × 6 = 36.

Why is "Working out $4^3$ as $4 \times 3 = 12$." a common mistake in Power and Roots?

Why it happens: The index 3 is treated as a multiplier instead of a count of factors. How to avoid it: Cubing uses the base three times: 4^3 = 4 × 4 × 4 = 64.

Why is "Thinking $\sqrt{16}$ means $16 \div 2$ and writing the answer as 8." a common mistake in Power and Roots?

Why it happens: A root feels like it should 'halve' the number in some way. How to avoid it: A square root asks which number squared gives 16. That is 4, since 4^2 = 16.

Why is "Writing $10^4$ as $1000$ instead of $10\,000$." a common mistake in Power and Roots?

Why it happens: The index is counted as zeros but one zero is dropped, or the index itself is miscounted. How to avoid it: The index gives the number of zeros exactly: 10^4 has four zeros, so 10^4 = 10\,000.

Why is "Doing the addition before the index in a calculation like $2 + 3^2$, giving 25." a common mistake in Power and Roots?

Why it happens: The calculation is read strictly left to right instead of following BIDMAS. How to avoid it: Indices come before addition. Do 3^2 = 9 first, then 2 + 9 = 11.

Square numbers as dot squares

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Powers and Roots — Cambridge Lower Secondary Maths, Grade 6

Squaring, cubing and finding roots are short ways of writing repeated multiplication. This guide shows you square and cube numbers, how to find their roots, how index notation works, and why powers of 10 make big numbers easy.

At a glance

A power tells you how many times to multiply a number by itself — $4^2$ means $4 \times 4$ .
Squaring a number means multiplying it by itself: $6^2 = 36$ .
Cubing a number means using it three times: $4^3 = 4 \times 4 \times 4 = 64$ .
A square root undoes squaring — $\sqrt{36} = 6$ because $6^2 = 36$ .
A cube root undoes cubing — $\sqrt[3]{64} = 4$ because $4^3 = 64$ .
Index notation: in $5^3$ the 5 is the base and the 3 is the index (power).
Powers of 10 ( $10$ , $100$ , $1000$ ) match the number of zeros to the index.
Learning the square numbers up to $15^2$ by heart speeds up almost everything.

What you’ll learn

Mapped to the Cambridge Lower Secondary Mathematics curriculum framework.

Stage 7 — Recognise square numbers and cube numbers and use index notation to write powers.
Stage 7 — Find the square root and cube root of numbers that give a whole-number answer.
Stage 7 — Use powers of 10 to describe and write large numbers.
Stage 7 — Use the correct order of operations so that indices are worked out at the right stage.

Square numbers

A square number is what you get when you multiply a whole number by itself.

A square number is the answer when you multiply a whole number by itself. $5 \times 5 = 25$ , so 25 is a square number. We write this as $5^2$ , said "five squared".

The name "square" is not random. If you arrange 25 dots into a grid, they make a perfect square that is 5 dots wide and 5 dots tall.

Cube numbers

A cube number is what you get when you multiply a whole number by itself three times.

A cube number is the answer when a whole number is used three times in a multiplication. $4 \times 4 \times 4 = 64$ , so 64 is a cube number, written $4^3$ and said "four cubed".

Just like squares form a flat square, cube numbers form a solid cube. small blocks stack into a cube that is 3 blocks wide, 3 deep and 3 tall.

Square roots and cube roots

A root works backwards — it asks which number was squared or cubed.

Squaring and cubing have opposites that undo them, called roots.

The square root of a number asks: "what was squared to get this?" Since $6^2 = 36$ , the square root of 36 is 6, written $\sqrt{36} = 6$ .

Index notation

Index notation is a short way to write a number multiplied by itself many times.

Writing $2 \times 2 \times 2 \times 2 \times 2$ is slow and easy to miscount. Index notation packs it up neatly as $2^5$ .

Powers of 10

Powers of 10 are the backbone of place value — the index matches the zeros.

Powers of 10 deserve their own spotlight because our whole number system is built on them.

Look at the pattern: $10^1 = 10$ , $10^2 = 100$ , , . The follow the 1.

Where you'll use this next

Powers and roots sit underneath area, algebra and big-number work.

Getting comfortable with powers and roots now opens up a lot of topics ahead:

Area and volume are full of squares and cubes — the area of a square is a side squared, and the volume of a cube is an edge cubed.
Algebra uses index notation constantly, for example simplifying $x \times x \times x$ to $x^3$ .
Prime factorisation is written far more neatly using powers, such as .

Quick recap

A square number is a whole number multiplied by itself, like $7^2 = 49$ .
A cube number is a whole number used three times, like $3^3 = 27$ .
A square root undoes squaring and a cube root undoes cubing.
In , the 4 is the base and the 3 is the index that counts the factors.

Sources: Cambridge Lower Secondary Mathematics curriculum framework (Stage 7). Last reviewed 2026-05-19.

Key Definitions and Keywords — Power and Roots

The important words for power and roots and what they mean — learn these so you can explain your thinking clearly.

Power
Key word
A way of writing repeated multiplication, made of a base and an index — for example $3^4$ means $3 \times 3 \times 3 \times 3$ .
Base
The number that is being multiplied repeatedly in a power.
Example
In $2^5$ the base is 2.
Index
Key word
The small raised number in a power that counts how many times the base is multiplied. Also called the power or exponent.
Example
In $2^5$ the index is 5, so 2 is used five times.
Square number
Key word
The result of multiplying a whole number by itself, such as $1, 4, 9, 16, 25$ .
Example
$36$ is a square number because $6 \times 6 = 36$ .
Cube number
Key word
The result of multiplying a whole number by itself three times, such as $1, 8, 27, 64$ .
Example
$27$ is a cube number because $3 \times 3 \times 3 = 27$ .
Squared
The name for the index 2 — it tells you to multiply the base by itself once.
Example
'Five squared' is $5^2 = 25$ .
Cubed
The name for the index 3 — it tells you to use the base three times in a multiplication.
Example
'Four cubed' is $4^3 = 64$ .
Square root
Key word
The number that was squared to give a particular value. It undoes squaring and uses the symbol $\sqrt{\phantom{x}}$ .
Example
$\sqrt{49}$ because .
Cube root
The number that was cubed to give a particular value. It undoes cubing and uses the symbol $\sqrt[3]{\phantom{x}}$ .
Example
Power of 10
Key word
A number made by multiplying 10 by itself, such as $10, 100, 1000$ . The index equals the number of zeros.
Index notation
A short way of writing repeated multiplication using a base and an index, such as $2^6$ instead of $2 \times 2 \times 2 \times 2 \times 2 \times 2$ .
Perfect square
A number whose square root is a whole number — every square number is a perfect square.
Example
$81$ is a perfect square since $\sqrt{81} = 9$ .

5

3

=

5

\times

5

\times

5

5^3 = 5 \times 5 \times 5

100 000 = 10^{5}

100\,000 = 10^5

=

7

\sqrt{49} = 7

3^2 = 9

Power and Roots — Cambridge Lower Secondary Mathematics — Stage 7 Revision Notes & Practice | Tutopiya

Power and Roots

Detailed Study Notes

Take this whole topic with you

Powers and Roots — Cambridge Lower Secondary Maths, Grade 6

At a glance

What you’ll learn

Square numbers

Cube numbers

Square roots and cube roots

Index notation

Powers of 10

Where you'll use this next

Quick recap

Step-by-step worked examples — Power and Roots

1Squaring a number

2Finding a square root

3Cubing a number

4Writing a number as a power of 10

5Using indices in a calculation

Key Definitions and Keywords — Power and Roots

Power

Base

Index

Square number

Cube number

Squared

Cubed

Square root

Cube root

Power of 10

Index notation

Perfect square

Common Mistakes and Misconceptions — Power and Roots

✕Working out $6^2$ as $6 \times 2 = 12$ .

✕Working out $4^3$ as $4 \times 3 = 12$ .

✕Thinking $\sqrt{16}$ means $16 \div 2$ and writing the answer as 8.

✕Writing $10^4$ as $1000$ instead of $10\,000$ .

✕Doing the addition before the index in a calculation like $2 + 3^2$ , giving 25.

Power and Roots

Detailed Study Notes

Take this whole topic with you

At a glance

What you’ll learn

Quick recap

1Squaring a number

2Finding a square root

3Cubing a number

4Writing a number as a power of 10

5Using indices in a calculation

Power

Base

Index

Square number

Cube number

Squared

Cubed

Square root

Cube root

Power of 10

Index notation

Perfect square

✕Working out 626^262 as 6×2=126 \times 2 = 126×2=12.

✕Working out 434^343 as 4×3=124 \times 3 = 124×3=12.

✕Thinking 16\sqrt{16}16​ means 16÷216 \div 216÷2 and writing the answer as 8.

✕Writing 10410^4104 as 100010001000 instead of 10 00010\,00010000.

✕Doing the addition before the index in a calculation like 2+322 + 3^22+32, giving 25.

✕Working out $6^2$ as $6 \times 2 = 12$ .

✕Working out $4^3$ as $4 \times 3 = 12$ .

✕Thinking $\sqrt{16}$ means $16 \div 2$ and writing the answer as 8.

✕Writing $10^4$ as $1000$ instead of $10\,000$ .

✕Doing the addition before the index in a calculation like $2 + 3^2$ , giving 25.