Squares, Cubes & Square Roots

Squares

The square of a number is the result of multiplying that number by itself. It is written as the number raised to the power of 2, for example, $a^2$ means $a \times a$.

Calculating squares of integers is straightforward: multiply the integer by itself.

For example, the square of 5 is $5^2 = 5 \times 5 = 25$.

For example, $(-3)^2 = (-3) \times (-3) = 9$, showing that squaring a negative number results in a positive number.

Some common square numbers are:

• $1^2 = 1$
• $2^2 = 4$
• $3^2 = 9$
• $4^2 = 16$
• $5^2 = 25$
• $6^2 = 36$
• $7^2 = 49$
• $8^2 = 64$
• $9^2 = 81$
• $10^2 = 100$

Knowing these common squares helps with mental maths and recognising patterns.

For instance, if you want to find the square of 12, you calculate $12 \times 12 = 144$.

Cubes

The cube of a number is the result of multiplying that number by itself twice, or raised to the power of 3. It is written as $a^3$, meaning $a \times a \times a$.

Calculating cubes of integers involves multiplying the integer three times.

For example, the cube of 3 is $3^3 = 3 \times 3 \times 3 = 27$.

Some common cube numbers are:

• $1^3 = 1$
• $2^3 = 8$
• $3^3 = 27$
• $4^3 = 64$
• $5^3 = 125$
• $6^3 = 216$
• $7^3 = 343$
• $8^3 = 512$
• $9^3 = 729$
• $10^3 = 1000$

These cubes are useful in geometry, especially when dealing with volumes of cubes and cuboids.

For example, the cube of 4 is $4 \times 4 \times 4 = 64$.

Square Roots

The square root of a number is the value that, when multiplied by itself, gives the original number. It is the inverse operation of squaring.

The square root of a number $x$ is written as $\sqrt{x}$.

Note that the square root symbol $\sqrt{}$ represents the principal (positive) square root. For example, both 7 and -7 squared give 49, but $\sqrt{49} = 7$.

For example, since $7^2 = 49$, the square root of 49 is $\sqrt{49} = 7$.

Square roots are usually found for perfect squares (numbers like 1, 4, 9, 16, 25, etc.).

If a number is not a perfect square, its square root is an irrational number and often approximated.

For example, $\sqrt{16} = 4$ because $4 \times 4 = 16$.