Table of Contents
IGCSE Mathematics – Squares, square roots, cubes, cube roots, and other powers
What you need to know: You must be able to recall squares of numbers 1–15 and their square roots. You must also recall cubes of 1, 2, 3, 4, 5, and 10. You will also calculate other powers and roots.
1. Squares #
To square a number, you multiply it by itself. For example, $5^2 = 5 \times 5 = 25$. The small raised number (2) is called a power or index.
$$a^2 = a \times a$$
Squares to Memorise (1 to 15) #
| Number | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ | $11$ | $12$ | $13$ | $14$ | $15$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Square | $1$ | $4$ | $9$ | $16$ | $25$ | $36$ | $49$ | $64$ | $81$ | $100$ | $121$ | $144$ | $169$ | $196$ | $225$ |
Exam Tip: You must know these from memory. A common exam question is: “Write down the value of $\sqrt{169}$.” If you know your squares table, you instantly know the answer is 13.
2. Square Roots #
The square root is the opposite of squaring. If $5^2 = 25$, then $\sqrt{25} = 5$. The symbol $\sqrt{\phantom{x}}$ means “square root of”.
$$\sqrt{a} = \text{the number that, when multiplied by itself, gives } a$$
Example: Write down the value of $\sqrt{169}$
Ask yourself: “Which number squared gives 169?”
From your squares table: $13^2 = 169$
Answer: $\sqrt{169} = 13$
3. Cubes #
To cube a number, you multiply it by itself three times.
$$a^3 = a \times a \times a$$
Cubes to Memorise #
| Number | $1$ | $2$ | $3$ | $4$ | $5$ | $10$ |
|---|---|---|---|---|---|---|
| Cube | $1$ | $8$ | $27$ | $64$ | $125$ | $1000$ |
Example: Work out $4^3$
$4^3 = 4 \times 4 \times 4$
$= 16 \times 4$
Answer: $= 64$
4. Cube Roots #
The cube root is the opposite of cubing. The symbol is $\sqrt[3]{\phantom{x}}$.
$$\sqrt[3]{a} = \text{the number that, when cubed, gives } a$$
Example: Work out $\sqrt[3]{8}$
Ask: “Which number cubed gives 8?”
$2^3 = 2 \times 2 \times 2 = 8$ ✓
Answer: $\sqrt[3]{8} = 2$
5. Other Powers and Roots #
Powers Greater Than 3 #
You can raise any number to any power. The power (index) tells you how many times to multiply the number by itself.
Example: Work out $2^5$
$2^5 = 2 \times 2 \times 2 \times 2 \times 2$
Answer: $= 32$
Other Roots #
Just as $\sqrt{\phantom{x}}$ means square root and $\sqrt[3]{\phantom{x}}$ means cube root, you can have any root. The small number in the root sign tells you which root.
Example: Work out $\sqrt[4]{16}$
Ask: “Which number to the power 4 gives 16?”
$2^4 = 2 \times 2 \times 2 \times 2 = 16$ ✓
Answer: $\sqrt[4]{16} = 2$
6. Combined Calculations #
In exams, you may be asked to combine powers and roots in one calculation. Always follow the order of operations (BIDMAS/BODMAS) — work out powers and roots before multiplying or adding.
Example (from syllabus): Work out $5^2 \times \sqrt[3]{8}$
Step 1: Work out $5^2 = 25$
Step 2: Work out $\sqrt[3]{8} = 2$
Step 3: Multiply: $25 \times 2$
Answer: $= 50$
Exam Tip: When a calculation has both a power and a root, work each one out separately first, then combine them. Never rush — write out each step clearly.
Summary #
Squaring & Square Roots #
$a^2 = a \times a$
$\sqrt{a}$ = reverse of squaring
Must memorise: $1^2$ to $15^2$
Cubing & Cube Roots #
$a^3 = a \times a \times a$
$\sqrt[3]{a}$ = reverse of cubing
Must memorise: cubes of 1, 2, 3, 4, 5, 10
Remember: Powers and roots are opposites of each other. If you know your squares and cubes tables, you can always find the roots quickly.
