Class 8 Maths important questions for chapter 6-Squares and Square Roots are available here with solutions. We have also provided the solutions for these problems based on **CBSE latest syllabus** and in reference with** NCERT book**. Therefore, it will be easy for students to revise the chapter with this material and score in the final examination.

## Important Questions with Solutions For Class 8 Maths Chapter 6 (Squares and Square Roots)

**Q.1: How many numbers lie between squares of the following numbers?**

**(i) 12 and 13**

**(ii) 25 and 26**

**(iii) 99 and 100**

Solution: As we know, between n2 and (n+1)2, the number of non–perfect square numbers are 2n.

(i) Between 122 and 132 there are 2×12 = 24 natural numbers.

(ii) Between 252 and 262 there are 2×25 = 50 natural numbers.

(iii) Between 992 and 1002 there are 2×99 =198 natural numbers.

**Q.2: Write a Pythagorean triplet whose one member is:**

**(i) 6**

**(ii) 14**

**(iii) 16**

**(iv) 18**

Solution:

We know, for any natural number m, 2m, m2–1, m2+1 is a Pythagorean triplet.

(i) 2m = 6

⇒ m = 6/2 = 3

m2–1= 32 – 1 = 9–1 = 8

m2+1= 32+1 = 9+1 = 10

Therefore, (6, 8, 10) is a Pythagorean triplet.

(ii) 2m = 14

⇒ m = 14/2 = 7

m2–1= 72–1 = 49–1 = 48

m2+1 = 72+1 = 49+1 = 50

Therefore, (14, 48, 50) is not a Pythagorean triplet.

(iii) 2m = 16

⇒ m = 16/2 = 8

m2–1 = 82–1 = 64–1 = 63

m2+ 1 = 82+1 = 64+1 = 65

Therefore, (16, 63, 65) is a Pythagorean triplet.

(iv) 2m = 18

⇒ m = 18/2 = 9

m2–1 = 92–1 = 81–1 = 80

m2+1 = 92+1 = 81+1 = 82

Therefore, (18, 80, 82) is a Pythagorean triplet.

**Q.3: (n+1)2-n2 = ?**

Solution:

(n+1)2-n2

= (n2 + 2n + 1) – n2

= 2n + 1

**Q.4: Show that 121 is the sum of 11 odd natural numbers.**

Solution: As 121 = 112

We know that the sum of first n odd natural numbers is n2.

Therefore, 121 = sum of first 11 odd natural numbers

= 1 + 3 + 5+ 7 + 9 + 11 +13 + 15 + 17 + 19 + 21

**Q.5: Show that the sum of two consecutive natural numbers is 132.**

Solution:

Let 2n + 1 = 13

So, n = 6

So, ( 2n + 1)2 = 4n2 + 4n + 1

= (2n2 + 2n) + (2n2 + 2n + 1)

Substitute n = 6,

(13)2 = ( 2 x 62 + 2 x 6) + (2 x 62 + 2 x 6 + 1)

= (72 + 12) + (72 + 12 + 1)

= 84 + 85

**Q.6: Use the identity and find the square of 189.**

**(a – b)2 = a2 – 2ab + b2**

Solution: 189 = (200 – 11)2

= 40000 – 2 x 200 x 11 + 112

= 40000 – 4400 + 121

= 35721

**Q.7: What would be the square root of 625 using the identity (a +b)2 = a2 + b2 + 2ab?**

Solution: (625)2

= (600 + 25)2

= 6002 + 2 x 600 x 25 +252

= 360000 + 30000 + 625

= 390625

**Q.8: Find the square roots of 100 and 169 by the method of repeated subtraction.**

Solution:

Let us find the square root of 100 first.

- 100 – 1 = 99
- 99 – 3 = 96
- 96 – 5 = 91
- 91 – 7 = 84
- 84 – 9 = 75
- 75 – 11 = 64
- 64 – 13 = 51
- 51 – 15 = 36
- 36 – 17 = 19
- 19 – 19 = 0

Here, we have performed a subtraction ten times.

Therefore, √100 = 10

Now, the square root of 169:

- 169 – 1 = 168
- 168 – 3 = 165
- 165 – 5 = 160
- 160 – 7 = 153
- 153 – 9 = 144
- 144 – 11 = 133
- 133 – 13 = 120
- 120 – 15 = 105
- 105 – 17 = 88
- 88 – 19 = 69
- 69 – 21 = 48
- 48 – 23 = 25
- 25 – 25 = 0

Here, we have performed subtraction thirteen times.

Therefore, √169 = 13

**Q.10: Find the square root of 729 using factorisation method.**

Solution:

729 = 3×3×3×3×3×3×1

⇒ 729 = (3×3)×(3×3)×(3×3)

⇒ 729 = (3×3×3)×(3×3×3)

⇒ 729 = (3×3×3)2

Therefore,

⇒ √729 = 3×3×3 = 27

**Q. 11: Find the smallest whole number by which 1008 should be multiplied so as to get a perfect square number. Also, find the square root of the square number so obtained.**

Solution:

Let us factorise the number 1008.

1008 = 2×2×2×2×3×3×7

= (2×2)×(2×2)×(3×3)×7

Here, 7 cannot be paired.

Therefore, we will multiply 1008 by 7 to get a perfect square.

New number so obtained = 1008×7 = 7056

Now, let us find the square root of 7056

7056 = 2×2×2×2×3×3×7×7

⇒ 7056 = (2×2)×(2×2)×(3×3)×(7×7)

⇒ 7056 = 22×22×32×72

⇒ 7056 = (2×2×3×7)2

Therefore;

⇒ √7056 = 2×2×3×7 = 84

**Q. 12: Find the smallest whole number by which 2800 should be divided so as to get a perfect square.**

Solution:

Let us first factorise the number 2800.

2800 = 2×2×2×2×5×5×7

= (2×2)×(2×2)×(5×5)×7

Here, 7 cannot be paired.

Therefore, we will divide 2800 by 7 to get a perfect square.

New number = 2800 ÷ 7 = 400

**Q. 13: 2025 plants are to be planted in a garden in such a way that each row contains as many plants as the number of rows. Find the number of rows and the number of plants in each row.**

Solution:

Let the number of rows be, x.

Therefore, the number of plants in each row = x.

Total many contributed by all the students = x×x = x2

Given, x2 = Rs.2025

x2 = 3×3×3×3×5×5

⇒ x2 = (3×3)×(3×3)×(5×5)

⇒ x2 = (3×3×5)×(3×3×5)

⇒ x2 = 45×45

⇒ x = √(45×45)

⇒ x = 45

Therefore,

Number of rows = 45

Number of plants in each rows = 45

**Q.14: Find the smallest square number that is divisible by each of the numbers 8, 15 and 20.**

Solution:

L.C.M of 8, 15 and 20 is (2×2×5×2×3) = 120.

120 = 2×2×3×5×2 = (2×2)×3×5×2

Here, 3, 5 and 2 cannot be paired.

Therefore, we need to multiply 120 by (3×5×2) i.e. 30 to get a perfect square.

Hence, the smallest squared number which is divisible by numbers 8, 15 and 20 = 120×30 = 3600

**Q.15: Find the square root of 7921 using long division method.**

Solution:

∴ √7921 = 89

**Q. 16: Find the square root of 42.25 using long division method.**

Solution:

∴ √42.25 = 6.5

**Q. 17: Find the least number which must be added to 1750 so as to get a perfect square. Also, find the square root of the obtained number.**

Solution:

Using long division method:

Here, (41)2 < 1750 > (42)2

We can say 1750 is ( 164 – 150 ) = 14 less than (42)2.

Therefore, if we add 14 to 1750, it will be a perfect square.

New number = 1750 + 14 = 1764

Therefore, the square root of 1764 is as follows:

∴√1764 = 42

### Class 8 Maths Chapter 6 Extra Questions

- In a right triangle ABC, ∠B = 90°. a. If AB = 6 cm, BC = 8 cm, find AC b. If AC = 13 cm, BC = 5 cm, find AB.
- Find the length of the side of a square whose area is 441 m2.
- There are 500 children in a school. For a P.T. drill, they have to stand in such a manner that the number of rows is equal to the number of columns. How many children would be left out in this arrangement?
- A gardener has 1000 plants. He wants to plant these in such a way that the number of rows and the number of columns remain the same. Find the minimum number of plants he needs more for this.
- Express 49 as the sum of 7 odd numbers.
- Find the square roots of the following decimal numbers

(i) 1056.25

(ii) 10020.01 - Mention the smallest number, which when multiplied by 5408 gives a perfect square.