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 - $\frac{y}{2}$

Cost per point, also known as CPP in short is a planning tool for the advertisers. CPP is ...

Cost per thousand, also known as CPM in short is the amount of money that would require to...

Parabolas can be classified by 2 kinds of factors: 1. By Concavity 2. By number of roots...

It's very crucial to check the shape of distribution of the random variable, which is not easy to examine with the help of frequency...

When every element undergoes multiplication or increased by a constant it is called as linear transformation. Linear...