# Square Roots and Cube Roots

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 - 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

## Square Root

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.

## Cube Root

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

## Square Roots and Cube Roots Practice questions

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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