NCERT Solutions for Class 8 Maths Chapter 6 Exercise 6.2 in Hindi and English Medium updated for CBSE 2022-2023 exams.

## 8th Maths Exercise 6.2 Solution in Hindi and English Medium

### Class 8 Maths Chapter 6 Exercise 6.2 Solution

Class VIII Mathematics Ex. 6.2 Square and Square Roots free to use without any login or registration updated for CBSE session 2022-23. Class 8 NCERT book Maths Exercise 6.2 is helpful for state board students who are using NCERT like books for their course books. Here we will learn how to find the prime factorisation of the given number and then group the prime factors in such a way that in each pair, both factors are the same. In this way we will get the square root of a given number by prime factorisation.

Class: 8 | Mathematics |

Chapter: 6 | Exercise: 6.2 |

Chapter Name: | Square and Square Roots |

Medium: | Hindi and English Medium |

Content: | Text and Videos Format |

#### Properties of Perfect Squares

##### Property 1

A number ending in 2, 3, 7 or 8 is never a perfect square.

Example:

The numbers 82, 93, 187, 248 end in 2, 3, 7, 8 respectively. So, none of them is a perfect square.

##### Property 2

A number ending in an odd number of zeros is never a perfect square.

**Example:**

The numbers 160, 4000, 900000 end in one zero, three zeros and five zeros respectively. So, none of them is a perfect square.

##### Property 3

If a number when divided by 3 leaves a remainder 2, then it is not a perfect square.

Example:

170, 578, 617, 722, etc.

##### Property 4

If a number when divided by 4 leaves a remainder 2 or 3, then it is not a perfect square.

Example:

578, 654, 798, 1002, etc.

##### Property 5

The square of an even number is always even.

Example:

2Â² = 4, 4Â² = 16, 6Â² = 36, 8Â² = 64, etc.

##### Property 6

The square of an odd number is always odd.

Example:

1Â² = 1, 3Â² = 9, 5Â² = 25, 7Â² = 49, 9Â² = 81, etc.

##### Property 7

The square of a proper fraction is smaller than the fraction.

Example:

(2/3)Â² = (2/3 x 2/3) = 4/9 and 4/9 < 2/3, since (4 x 3) < (9 x 2)

##### Property 8

For every natural number n, we have:

(n + 1)Â² â€“nÂ² = (n + 1 + n) (n + 1 – n) = {(n + 1) + n)}.

{(n + 1)Â² â€“nÂ²} = {(n + 1) + n)}.

Example:

(i) {(36)Â² â€“ (35)Â²} = (36 + 35) = 71

##### Property 9

For every natural number n, we have:

sum of first n odd natural numbers = nÂ².

Example:

(i) (1 + 3 + 5 + 7 + 9) = sum of first 5 odd numbers = 5Â².

(ii) (1 + 3 + 5 + 7 + 9 + 11 + 13 + 15) = sum of first 8 odd numbers = 8Â².

##### Property 10

Between two consecutive square numbers n and (n + 1), there are 2n non-perfect square numbers.

Example:

Square numbers are 1Â², 2Â², 3Â², 4Â², ….., etc.

- Between 1Â² and 2Â², there are 2 x 1 = 2 non-perfect square numbers, viz., 2Â², 3Â².
- Between 4Â² and 5Â², there are 4 x 2 = 8 non-perfect square number, viz., 17, 18, 19, 20, 21, 22, 23, 24.

##### Property 11

The square of an even number is always an even number and the square of an odd number is always an odd number.

##### Property 12

- (i) The square of a natural number (except 1) is either a multiple of 3 or exceeds a multiple of 3 by 1.
- (ii) The square of a natural number (except 1) is either a multiple of 4 or exceeds a multiple of 4 by 1.

###### Property 13

The square of a natural number ending with 5 follows a definite pattern.

(25)Â² = (2 x 3) hundred + 25 = 625

(35)Â² = (3 x 4) hundred + 25 = 1225

(45)Â² = (4 x 5) hundred + 25 = 2025

###### Property 14

The sum of first n odd natural numbers is nÂ².

Examples:

First odd number = 1 = 1Â²

Sum of first two odd numbers = 1 + 3 = 4 = 2Â²

Sum of first three odd numbers = 1 + 3 + 5 = 9 = 3Â²

Sum of first four odd numbers = 1 + 3 + 5 + 7 = 16 = 4Â²

In other words, if a number is a prefect square, then it is always equal to the sum of consecutive odd numbers starting from 1.

###### Property 15

Some square numbers follow interesting patterns. Observe the following squares:

Example:

- (1)Â² = 1
- (11)Â² = 121
- (111)Â² = 12321
- (1111)Â² = 1234321
- (11111)Â² = 123454321

### Class 8 Maths Exercise 6.2 Important Questions

### What are the properties of a perfect square?

(i) A Square number can only end with digits 0, 1, 4, 5, 6, 9.

(ii) Number of zeroes at the end of a perfect square is always even.

(iii) Square of even numbers are always even,

(iv) Square of odd numbers are always odd.

(v) If a number has 1 or 9 in its unit place,

(vi) its square ends with 1.

(vii) If a number has 4 or 6 in its unit place,

### How do you solve a perfect square question?

Steps to Solving Equations by Completing the Square:

Rewrite the equation in the form xÂ² + bx = c.

Add to both sides the term needed to complete the square.

Factor the perfect square trinomial.

Solve the resulting equation by using the square root property.

### Perfect Square Trinomial Formula

An expression is said to a perfect square trinomial if it takes the form axÂ² + bx + c and satisfies the condition bÂ² = 4ac. The perfect square formula takes the following forms: (ax)Â² + 2abx + bÂ² = (ax + b)Â²

If we look at the results of these squares, we find that, they have a special property i.e., the sum of the digits of every number on the right hand side is a perfect square.

121 = 1 + 2 + 1 = 4 = 2Â²

12321= 1 + 2 + 3 + 2 + 1 = 9 = 3Â²

1234321 = 1 + 2 + 3 + 4 + 3 + 2 + 1 = 16 = 4Â²

123454321 = 1 + 2 + 3 + 4 + 5 + 4 + 3 + 2 + 1 = 25 = 5Â²

And so onâ€¦

(7)Â² = 49

(67)Â² = 4489

(667)Â² = 444889

(6667)Â² = 44448889

If we look at the results of these squares, we find that they follow another interesting property. Using this pattern, we can find the square of the numbers 66667, 666667, ….