CBSE Class 8 Mathematics Worksheet - Square and Square Roots ... - Free Printable

Here is the complete solution to all 10 problems from the "Squares and Square Roots" worksheet for Class VIII.

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1. Find the square root of 6400

We can factorize 6400:

6400 = 64 × 100
√6400 = √(64 × 100) = √64 × √100 = 8 × 10 = 80

✔ Answer: 80

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2. Is 90 a perfect square?

Check prime factorization:

90 = 2 × 3² × 5

In a perfect square, every prime factor must appear an even number of times.

Here, 2 and 5 appear only once → not a perfect square

✔ Answer: No

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3. Is 2352 a perfect square? If not, find the smallest multiple of 2352 which is a perfect square. Find the square root of the new number.

Factorize 2352:

2352 ÷ 2 = 1176
1176 ÷ 2 = 588
588 ÷ 2 = 294
294 ÷ 2 = 147
147 ÷ 3 = 49
49 ÷ 7 = 7
7 ÷ 7 = 1

So, 2352 = 2⁴ × 3¹ × 7²

To make it a perfect square, all exponents must be even. Here, exponent of 3 is 1 → we need to multiply by 3.

Smallest multiple = 2352 × 3 = 7056

Now, √7056 = √(2⁴ × 3² × 7²) = 2² × 3 × 7 = 4 × 3 × 7 = 84

✔ Answer: Not a perfect square. Smallest multiple = 7056. Square root = 84

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4. Find the smallest number by which 9408 must be divided so that the quotient is a perfect square. Find the square root of the quotient.

Factorize 9408:

9408 ÷ 2 = 4704
4704 ÷ 2 = 2352
2352 ÷ 2 = 1176
1176 ÷ 2 = 588
588 ÷ 2 = 294
294 ÷ 2 = 147
147 ÷ 3 = 49
49 ÷ 7 = 7
7 ÷ 7 = 1

So, 9408 = 2⁶ × 3¹ × 7²

To make quotient a perfect square, remove the odd-powered primes → divide by 3.

Quotient = 9408 ÷ 3 = 3136

√3136 = √(2⁶ × 7²) = 2³ × 7 = 8 × 7 = 56

✔ Answer: Divide by 3. Quotient = 3136. Square root = 56

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5. Without doing any calculation, find the numbers which are surely not perfect squares.

i) 153 → ends in 3 → not possible (perfect squares end in 0,1,4,5,6,9)

ii) 257 → ends in 7 → not possible

iii) 408 → ends in 8 → not possible

iv) 441 → ends in 1 → possible (e.g., 21²=441)

✔ Answer: i) 153, ii) 257, iii) 408

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6. Find the square root by prime factorisation method.

#### i) 400

400 = 2⁴ × 5²
√400 = 2² × 5 = 4 × 5 = 20

#### ii) 9604

9604 ÷ 2 = 4802
4802 ÷ 2 = 2401
2401 = 7⁴ (since 7×7=49, 49×49=2401)

So, 9604 = 2² × 7⁴
√9604 = 2 × 7² = 2 × 49 = 98

#### iii) 8100

8100 = 81 × 100 = 9² × 10² = (9×10)² = 90² → 90

Prime factors: 2² × 3⁴ × 5² → √ = 2 × 3² × 5 = 2×9×5 = 90

#### iv) 1764

1764 ÷ 2 = 882
882 ÷ 2 = 441
441 = 21² = 3² × 7²

So, 1764 = 2² × 3² × 7²
√1764 = 2 × 3 × 7 = 42

#### v) 5929

5929 ÷ 7 = 847
847 ÷ 7 = 121
121 = 11²

So, 5929 = 7² × 11²
√5929 = 7 × 11 = 77

#### vi) 9216

9216 ÷ 2 = 4608
÷2=2304, ÷2=1152, ÷2=576, ÷2=288, ÷2=144, ÷2=72, ÷2=36, ÷2=18, ÷2=9, ÷3=3, ÷3=1

Total: 2¹⁰ × 3²
√9216 = 2⁵ × 3 = 32 × 3 = 96

✔ Answers:
i) 20
ii) 98
iii) 90
iv) 42
v) 77
vi) 96

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7. For each number, find smallest whole number to multiply to get perfect square. Also find square root.

#### i) 252

252 = 2² × 3² × 7 → missing one 7 → multiply by 7

New number = 252 × 7 = 1764 → √1764 = 42

#### ii) 2925

2925 ÷ 5 = 585
585 ÷ 5 = 117
117 ÷ 3 = 39
39 ÷ 3 = 13

→ 2925 = 3² × 5² × 13 → multiply by 13

New number = 2925 × 13 = 38025 → √38025 = 3×5×13 = 195

#### iii) 396

396 = 2² × 3² × 11 → multiply by 11

New number = 396 × 11 = 4356 → √4356 = 2×3×11 = 66

#### iv) 2028

2028 ÷ 2 = 1014
÷2=507
507 ÷ 3 = 169
169 = 13²

→ 2028 = 2² × 3 × 13² → multiply by 3

New number = 2028 × 3 = 6084 → √6084 = 2×13×3 = 78

#### v) 1458

1458 ÷ 2 = 729
729 = 3⁶

→ 1458 = 2 × 3⁶ → multiply by 2

New number = 1458 × 2 = 2916 → √2916 = 3³ × 2 = 27×2 = 54

#### vi) 768

768 = 2⁸ × 3 → multiply by 3

New number = 768 × 3 = 2304 → √2304 = 2⁴ × 3 = 16×3 = 48

✔ Answers:
i) Multiply by 7 → √ = 42
ii) Multiply by 13 → √ = 195
iii) Multiply by 11 → √ = 66
iv) Multiply by 3 → √ = 78
v) Multiply by 2 → √ = 54
vi) Multiply by 3 → √ = 48

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8. For each number, find smallest whole number to divide to get perfect square. Also find square root.

#### i) 252

252 = 2² × 3² × 7 → divide by 7

Quotient = 36 → √36 = 6

#### ii) 180

180 = 2² × 3² × 5 → divide by 5

Quotient = 36 → √36 = 6

#### iii) 1008

1008 = 2⁴ × 3² × 7 → divide by 7

Quotient = 144 → √144 = 12

#### iv) 2028

2028 = 2² × 3 × 13² → divide by 3

Quotient = 676 → √676 = 2×13 = 26

#### v) 1458

1458 = 2 × 3⁶ → divide by 2

Quotient = 729 → √729 = 27

#### vi) 768

768 = 2⁸ × 3 → divide by 3

Quotient = 256 → √256 = 16

✔ Answers:
i) Divide by 7 → √ = 6
ii) Divide by 5 → √ = 6
iii) Divide by 7 → √ = 12
iv) Divide by 3 → √ = 26
v) Divide by 2 → √ = 27
vi) Divide by 3 → √ = 16

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9. Students donated Rs 2401. Each donated as many rupees as number of students. Find number of students.

Let number of students = x
Each donates x rupees → total donation = x × x = x² = 2401

x = √2401

Factorize 2401: 2401 ÷ 7 = 343, 343 ÷ 7 = 49, 49 ÷ 7 = 7, 7 ÷ 7 = 1 → 7⁴

√2401 = 7² = 49

✔ Answer: 49 students

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10. 2118 plants to be planted such that each row has as many plants as number of rows. Find number of rows and plants per row.

Let number of rows = x → plants per row = x → total plants = x² = 2118

But 2118 is not a perfect square! Let’s check:

√2118 ≈ 46.02 → 46² = 2116, 47² = 2209 → 2118 is not a perfect square.

This suggests there may be a typo in the problem. The closest perfect square is 2116 = 46².

If we assume the intended number was 2116, then:

Number of rows = 46, plants per row = 46

✔ Answer (assuming typo): 46 rows, 46 plants per row

*(Note: If the number is indeed 2118, no integer solution exists. But given context, likely meant 2116.)*

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## ✔ Final Summary of Answers:

1. 80
2. No
3. Not a perfect square; multiply by 3 → 7056; √ = 84
4. Divide by 3 → 3136; √ = 56
5. 153, 257, 408
6.
i) 20
ii) 98
iii) 90
iv) 42
v) 77
vi) 96
7.
i) ×7 → √42
ii) ×13 → √195
iii) ×11 → √66
iv) ×3 → √78
v) ×2 → √54
vi) ×3 → √48
8.
i) ÷7 → √6
ii) ÷5 → √6
iii) ÷7 → √12
iv) ÷3 → √26
v) ÷2 → √27
vi) ÷3 → √16
9. 49 students
10. 46 rows, 46 plants per row *(assuming typo: 2116 instead of 2118)*

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