In this section we are going to calculate squares, square roots, cubes and cube roots.

** Squares: **If a number is given we multiply the number twice which results in a square of a number.

**Example 1: **

Find the square of 21

**Solution:** Here 21 is multiplied twice that is (*21*) $\times$ (*21*) = 441.

**Example 2:**

Find the square of x-3y

**Solution:** Multiply x-3y twice that is (*x - 3y*) $\times$ (*x - 3y*)

= x(* x - 3y*) - 3y (

= $x^2$ - 3xy -3xy + 9$y^2$

= $x^2$ - 6xy + 9$y^2$.

**Example3:**

Find the square of 2 - $\frac{4}{x}$

Solution: The square of 2 - $\frac{4}{x}$ is given by

(*2 - $\frac{4}{x}$*)(*2 - $\frac{4}{x}$*) = 2(*2 - $\frac{4}{x}$*) - $\frac{4}{x}$ (*2 - $\frac{4}{x}$*)

= 4 - $\frac{8}{x}$ - $\frac{8}{x}$ + $\frac{16}{x^2}$

= 4- $\frac{16}{x}$ + $\frac{16}{x^2}$.

We shall now discuss about the square roots

The square root of a number is written in the form of $\sqrt{n}$ where n denotes the number.

1. **Example 1:** $\sqrt{100}$ = 10 $\times$ 10 = $\sqrt{10^2}$. Here root and square cancels.

$\sqrt{100}$ =10.

2. **Example 2:** $\sqrt{121}$ = 11$\times$ 11 = $\sqrt{11^2}$. Here root and square cancels. $\sqrt{121}$ = 11.

3. **Example 3:** Find the square root of 1296.
Step1: start dividing 1296 by 2, we get like this

So 1296 = $\sqrt{2 \times 2 \times 2 \times 2 \times 9 \times 9}$

= 2 $\times$ 2 $\times$ 9 = 36

**Cube:** If we are given a number then multiplying it thrice gives the cube of a number.

The square root of a number is written in the form of $\sqrt[3]{n}$ where n denotes the number.

**Example 1: **Find the cube of the number 7

So 7 $\times$ 7 $\times$ 7 = 343

**Example 2:** Find the cube of the number 10

So 10 $\times$ 10 $\times$ 10 = 1000

**Example 3:** Find the cube of the number 14

14 $\times$ 14 $\times$ 14 = 2744

**We shall find out the cube roots of some of the numbers:**

**Example 1:** Find the cube root of 5832

**Solution:** Divide the number by 2

we get 2916

Again divide by 2

we get 1458

Again divide by 2

729

Again divide by 3

243

Again divide by 3

81

Again divide by 3

27

Again divide by 3

9

Again diviide by 3

3

Again divide by 3

5832 = $2^3 \times 3^3 \times 3^3$

5832 = $2 \times 3 \times 3$

5832 = 18

**Example 2:** Find the cube roots of 1728

**Solution:** As shown before start dividing the given number 1728 by 2

we get 864

once again by 2

432

once again by 2

216

once again by 2

108

once again by 2

54

once again by2

27

now by 3

9

Again by 3

3

Again by 3

1

So cube root of (*1728*) = cube root (*$2^3 \times 2^3 \times 3^3$*) = $2 \times 2 \times 3$ = 12

**Find the square roots of **

1) 841

2) 2704

3) 2401

**Find the cube roots of **

1) 2744

2) 32768

3) 729

**Find the squares of **

1) 3x-8

2) 5 - $\frac{4}{x}$

3) $\frac{8}{y}$ -5

**Find the cube of **

1) y - 6

2) 6x - 7

3) 3x - 9

4) 4 - $\frac{y}{2}$

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