# Squares and cubes

**What are squares and cubes?**

The square of a number is multiplying the number by itself. Assume $$x$$ to be a number, the square of $$x$$ is $$x \times x$$ or $$x^{2}$$.

The cube of a number is the number multiplied by itself twice. The cube of a number $$x$$ is $$x \times x \times x$$ or $$x^{3}$$.

Suppose a boy plays chess with his friend. He knows the length of the side of the chessboard and wants to calculate its area. How can he calculate this?

This can be done using the concept of squaring.

Multiplying the length of the side of the chessboard with itself, we get, $$\left(\text{side} \right)^{2}$$, that is the area of the chessboard (it is in the shape of a square).

**E1.1A: Identify and use square numbers**

**Square of a number**

When a number is multiplied by itself, the result obtained is known as the square. If $$a$$ is an integer, then its square is written as $$a \times a = a^{2}$$.

For an integer $$a$$, $$a^{2}$$ is read ‘square of $$a$$.’

In $$a^{2}$$, $$2$$ is known as the exponent of the integer $$a$$, whereas $$a$$ is the base of $$a^{2}$$. Some examples of square are $$21^{2}= 21 \times 21$$, $$32^{2}= 32 \times 32$$, and $$48^{2}= 48 \times 48$$.

The result obtained on squaring a whole number is always a whole number. Only two numbers whose square is the same as the original number are $$0$$ and $$1$$.

**Worked examples**

**Example 1:** Calculate the value of $$24^{2}$$.

**Step 1: Expand the given number.**

$$24^{2}= 24 \times 24$$

**Step 2: Multiply accordingly**

$$24^{2}= 24 \times 24 = 576$$

**Example 2:** Find the square of 54.

**Step 1: Write the square of the given number.**

$$54^2$$

**Step 2: Expand the square.**

$$54^{2}= 54 \times 54$$

**Step 2: Multiply both the numbers present on the right-hand side.**

$$54^{2}= 2916$$

Therefore, the final answer is $$2916$$.

**E1.1B: Identify and use cube numbers**

**Cube of a number**

When a number is multiplied by itself twice, the result obtained is denoted as the cube of the number. If $$a$$ is an integer, then the cube is written as, $$a \times a \times a = a^{3}$$.

For an integer $$a$$, $$a^{3}$$ is read, ‘cube of $$a$$.’

In $$a^{3}$$, $$3$$ is known as the exponent of the integer $$a$$, and $$a$$ is the base of $$a^{3}$$. Some examples of cube are, $$10^{3}= 10 \times 10 \times 10$$, $$41^{3}= 41 \times 41 \times 41$$, $$24^{3}= 24 \times 24 \times 24 $$.

The result obtained cubing a whole number is always a whole number. Only two numbers whose cube is the same as the original number are $$0$$ and $$1$$.

**Worked examples**

**Example 1**: Calculate the value of $$12^{3}$$.

**Step 1: Expand the given number.**

$$12^{3}= 12 \times 12 \times 12 $$

**Step 2: Multiply the numbers present on the right-hand side.**

$$12^{3}=1728$$

Therefore, the final answer is $$1728$$.

**Example 2**: Find the cube of $$26$$.

**Step 1: Write down the required operation to be performed.**

$$26^{3}$$

**Step 2: Expand the given number.**

$$26^{3}=26 \times 26 \times 26 $$

**Step 3: Multiply the numbers present on the right-hand side.**

$$26^{3}=17576$$

Therefore, the final answer is $$17576$$.

**Example 3**: Find the cube of $$-16$$.

**Step 1: Write down the required operation to be performed.**

$$-16^{3}$$

**Step 2: Expand the given number.**

$$-16^{3}=-16 \times -16 \times -16 $$

**Step 3: Multiply the numbers present on the right-hand side.**

$$-16^{3}=-4096$$

Therefore, the final answer is $$-4096$$.