Class VIII Mathematics worksheet on squares and square roots with various problems and exercises.

I'll solve each problem step by step, checking my calculations carefully.

Problem 1: Find the square root of 6400
- 6400 = 64 × 100 = 8² × 10² = (8 × 10)² = 80²
- √6400 = 80 ✓

Problem 2: Is 90 a perfect square?
- Perfect squares near 90: 9² = 81 and 10² = 100
- 90 is between 81 and 100, not equal to any perfect square
- Answer: No, 90 is not a perfect square ✓

Problem 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.

Let me factorize 2352:
- 2352 ÷ 2 = 1176
- 1176 ÷ 2 = 588
- 588 ÷ 2 = 294
- 294 ÷ 2 = 147
- 147 ÷ 3 = 49
- 49 = 7²

So, 2352 = 2⁴ × 3¹ × 7²

For a perfect square, all exponents must be even. The exponent of 3 is odd (1).
To make it a perfect square, multiply by 3.
Smallest multiple = 2352 × 3 = 7056

Check: 7056 = 2⁴ × 3² × 7² = (2² × 3 × 7)² = (4 × 3 × 7)² = 84²
√7056 = 84 ✓

Problem 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 = 7²

So, 9408 = 2⁶ × 3¹ × 7²

The exponent of 3 is odd (1). To get a perfect square, divide by 3.
Quotient = 9408 ÷ 3 = 3136

Check: 3136 = 2⁶ × 7² = (2³ × 7)² = (8 × 7)² = 56²
√3136 = 56 ✓

Problem 5: Without doing any calculation, find the numbers which are surely not perfect square.

Numbers ending in 2, 3, 7, or 8 cannot be perfect squares.
- i) 153 - ends in 3 → NOT a perfect square
- ii) 257 - ends in 7 → NOT a perfect square
- iii) 408 - ends in 8 → NOT a perfect square
- iv) 441 - ends in 1 → Could be a perfect square (21² = 441)

Answer: 153, 257, and 408 are surely not perfect squares ✓

Problem 6: Find the square root using prime factorization method

i) 400 = 2⁴ × 5² = (2² × 5)² = 20² → √400 = 20 ✓

ii) 9604 = 2² × 7⁴ = (2 × 7²)² = (2 × 49)² = 98² → √9604 = 98 ✓

iii) 8100 = 2² × 3 × 5² = (2 × 3² × 5)² = (2 × 9 × 5)² = 90² → √8100 = 90 ✓

iv) 1764 = 2² × 3² × 7² = (2 × 3 × 7)² = 42² → √1764 = 42 ✓

v) 5929 = 7² × 11² = (7 × 11)² = 77² → √5929 = 77 ✓

vi) 9216 = 2¹⁰ × 3² = (2⁵ × 3)² = (32 × 3)² = 96² → √9216 = 96 ✓

Problem 7: Find the smallest whole number to multiply to get a perfect square, and find the square root

i) 252 = 2² × 3² × 7¹ → Multiply by 7 → 252 × 7 = 1764 = 42² → √1764 = 42 ✓

ii) 2925 = 3² × 5² × 13¹ → Multiply by 13 → 2925 × 13 = 38025 = 195² → √38025 = 195 ✓

iii) 396 = 2² × 3² × 11¹ → Multiply by 11 → 396 × 11 = 4356 = 66² → √4356 = 66 ✓

iv) 2028 = 2² × 3¹ × 13² → Multiply by 3 → 2028 × 3 = 6084 = 78² → √6084 = 78 ✓

v) 1458 = 2¹ × 3⁶ → Multiply by 2 → 1458 × 2 = 2916 = 54² → √2916 = 54 ✓

vi) 768 = 2⁸ × 3¹ → Multiply by 3 → 768 × 3 = 2304 = 48² → √2304 = 48 ✓

Problem 8: Find the smallest whole number to divide to get a perfect square, and find the square root

i) 252 = 2² × 3² × 7¹ → Divide by 7 → 252 ÷ 7 = 36 = 6² → 36 = 6 ✓

ii) 180 = 2² × 3² × 5¹ → Divide by 5 → 180 ÷ 5 = 36 = 6² → √36 = 6 ✓

iii) 1008 = 2 × 3² × 7¹ → Divide by 7 → 1008 ÷ 7 = 144 = 12² → √144 = 12 ✓

iv) 2028 = 2² × 3¹ × 13² → Divide by 3 → 2028 ÷ 3 = 676 = 26² → √676 = 26 ✓

v) 1458 = 2¹ × 3⁶ → Divide by 2 → 1458 ÷ 2 = 729 = 27² → √729 = 27 ✓

vi) 768 = 2⁸ × 3¹ → Divide by 3 → 768 ÷ 3 = 256 = 16² → √256 = 16 ✓

Problem 9: Students donated Rs 2401. Each student donated as many rupees as the number of students. Find the number of students.

If there are n students and each donates n rupees, then total = n × n = n²
n² = 2401
n = √2401

2401 = 7⁴ = (7²)² = 49²
So, n = 49

Number of students = 49 ✓

Problem 10: 2118 plants to be planted with equal rows and columns. Find the number of rows and plants in each row.

Wait, let me check if 2118 is a perfect square first.
√2118 ≈ 46.02...

Let me verify: 46² = 2116 and 47² = 2209
2118 is NOT a perfect square!

Let me re-read the problem... "2118 plants" - this might be a typo in the question. Let me check if it should be 2116 instead.

2116 = 46², which would give 46 rows and 46 plants per row.

However, since the problem states 2118, I need to work with what's given. But 2118 cannot form a perfect square arrangement.

Actually, looking at this more carefully - perhaps there's an error in the original problem statement. Let me assume it should be 2116 (which is 46²).

But wait, let me double-check my calculation for 2118:
46² = 2116
47² = 2209

2118 is definitely not a perfect square. This appears to be an error in the problem. However, if we're meant to find the closest perfect square or if there's a typo and it should be 2116:

If it's 2116: Number of rows = 46, Plants per row = 46

Given the context of the worksheet being about perfect squares, I believe there's a typo and it should be 2116.

Final Answer:

1. √6400 = 80

2. No, 90 is not a perfect square

3. No, 2352 is not a perfect square. Smallest multiple = 7056, √7056 = 84

4. Divide by 3, Quotient = 3136, √3136 = 56

5. Numbers that are surely not perfect squares: 153, 257, 408

6.
i) √400 = 20
ii) √9604 = 98
iii) √8100 = 90
iv) √1764 = 42
v) √5929 = 77
vi) √9216 = 96

7.
i) Multiply by 7, √1764 = 42
ii) Multiply by 13, 38025 = 195
iii) Multiply by 11, √4356 = 66
iv) Multiply by 3, √6084 = 78
v) Multiply by 2, √2916 = 54
vi) Multiply by 3, √2304 = 48

8.
i) Divide by 7, √36 = 6
ii) Divide by 5, √36 = 6
iii) Divide by 7, √144 = 12
iv) Divide by 3, √676 = 26
v) Divide by 2, √729 = 27
vi) Divide by 3, √256 = 16

9. Number of students = 49

10. Note: 2118 is not a perfect square. Assuming typo for 2116: Number of rows = 46, Plants per row = 46