Class 8 Maths Square and Square Roots Worksheet - Free Printable
Educational worksheet: Class 8 Maths Square and Square Roots Worksheet. Download and print for classroom or home learning activities.
PNG
606×591
50.1 KB
Free · Personal Use
Quality Assured by Worksheets Library Team
Reviewed for educational accuracy and age-appropriateness
ID: #776860
⭐
Show Answer Key & Explanations
Step-by-step solution for: Class 8 Maths Square and Square Roots Worksheet
▼
Show Answer Key & Explanations
Step-by-step solution for: Class 8 Maths Square and Square Roots Worksheet
Let’s solve each problem one by one, carefully and step by step.
---
1. Fill in the blanks
(i) The digit at unit’s place of square of 239 = ?
We only care about the last digit of 239 → that’s 9.
Square of 9 is 81 → last digit is 1.
✔ Answer: 1
---
(ii) 29 × 31 = □ – 1
Notice: 29 = 30 - 1, 31 = 30 + 1
So, (30 - 1)(30 + 1) = 30² - 1² = 900 - 1
But the question says “= □ – 1”, so □ must be 900
✔ Answer: 900
---
(iii) 17 × 23 = □ – 3²
17 = 20 - 3, 23 = 20 + 3
So, (20 - 3)(20 + 3) = 20² - 3² = 400 - 9
Question says “= □ – 3²” → so □ = 400
✔ Answer: 400
---
(iv) (151)² – (150)² = ?
Use identity: a² - b² = (a - b)(a + b)
Here, a = 151, b = 150
→ (151 - 150)(151 + 150) = (1)(301) = 301
✔ Answer: 301
---
(v) The sum of first five odd numbers = ?
First five odd numbers: 1, 3, 5, 7, 9
Sum = 1+3=4; 4+5=9; 9+7=16; 16+9=25
Or use formula: Sum of first n odd numbers = n² → 5² = 25
✔ Answer: 25
---
(vi) If 6ˣ = 1296 then x = ?
Let’s write powers of 6:
6¹ = 6
6² = 36
6³ = 216
6⁴ = 1296 ← Yes!
✔ Answer: 4
---
2. Write the perfect square numbers between 100 and 150.
Perfect squares:
10² = 100 → not *between* (if strictly between, exclude 100)
11² = 121
12² = 144
13² = 169 → too big
So between 100 and 150: 121, 144
(If including 100, add it — but usually “between” means exclusive. Let’s check context — since 100 is boundary, better to include if not specified. But problem says “between 100 and 150”, so likely 101 to 149. So 121 and 144.)
✔ Answer: 121, 144
---
3. Write 17² as sum of two consecutive integers.
17² = 289
We need two consecutive integers: let them be n and n+1
n + (n+1) = 289 → 2n + 1 = 289 → 2n = 288 → n = 144
So numbers are 144 and 145
Check: 144 + 145 = 289 ✔
✔ Answer: 144 and 145
---
4. Find the Pythagorean triplet whose smallest number is 10.
Pythagorean triplet: a² + b² = c², with a < b < c
Smallest number is 10 → try multiples of known triplets.
Known triplet: 3-4-5 → multiply by 2 → 6-8-10 → smallest is 6, not 10.
Multiply 3-4-5 by something else? Try 5-12-13 → smallest is 5.
Wait — we want smallest = 10.
Try scaling 5-12-13 by 2 → 10-24-26
Check: 10² + 24² = 100 + 576 = 676 = 26² ✔
✔ Answer: 10, 24, 26
---
5. Find the smallest number by which 192 must be multiplied to make the product perfect square.
Factorize 192:
192 ÷ 2 = 96
96 ÷ 2 = 48
48 ÷ 2 = 24
24 ÷ 2 = 12
12 ÷ 2 = 6
6 ÷ 2 = 3
3 ÷ 3 = 1
So 192 = 2⁶ × 3¹
For perfect square, all exponents must be even.
2⁶ is already even exponent.
3¹ → needs another 3 to become 3².
So multiply by 3
✔ Answer: 3
---
6. Find the square root of the following
(i) √10609
Try estimating: 100² = 10000, 103² = ?
103² = (100+3)² = 10000 + 600 + 9 = 10609 ✔
✔ Answer: 103
---
(ii) √33.64
Note: 5.8² = ?
5.8 × 5.8 = (5 + 0.8)² = 25 + 2×5×0.8 + 0.64 = 25 + 8 + 0.64 = 33.64 ✔
✔ Answer: 5.8
---
(iii) √0.4489
Try 0.67² = ?
0.67 × 0.67 = (67/100)² = 4489 / 10000 = 0.4489 ✔
✔ Answer: 0.67
---
(iv) √(289/361)
√289 = 17, √361 = 19 → so √(289/361) = 17/19
✔ Answer: 17/19
---
(v) √(1 7/9)
Convert mixed number: 1 7/9 = 16/9
√(16/9) = 4/3 = 1 1/3
✔ Answer: 4/3 or 1 1/3
---
7. Find:
(i) √55 × √220
√55 × √220 = √(55 × 220)
55 × 220 = 55 × (22 × 10) = (55 × 22) × 10
Better: 55 × 220 = 55 × (4 × 55) = 4 × 55²
Because 220 = 4 × 55
So √(4 × 55²) = √4 × (55²) = 2 × 55 = 110
✔ Answer: 110
---
(ii) √0.25 × √0.09
√0.25 = 0.5, √0.09 = 0.3
0.5 × 0.3 = 0.15
✔ Answer: 0.15
---
8. Find the smallest number of four digits which is a perfect square.
Smallest 4-digit number: 1000
Find smallest integer whose square ≥ 1000
√1000 ≈ 31.62 → so try 32² = 1024
31² = 961 (3-digit)
So 32² = 1024 is smallest 4-digit perfect square
✔ Answer: 1024
---
9. Find the smallest square number which is divisible by 4, 12 and 16.
First find LCM of 4, 12, 16.
Prime factors:
4 = 2²
12 = 2² × 3
16 = 2⁴
LCM = highest powers: 2⁴ × 3 = 16 × 3 = 48
Now, 48 is not a perfect square. Make it a perfect square by multiplying to make all exponents even.
48 = 2⁴ × 3¹ → need another 3 → 48 × 3 = 144
144 = 12² → perfect square
And divisible by 4, 12, 16?
144 ÷ 4 = 36 ✔
144 ÷ 12 = 12 ✔
144 ÷ 16 = 9 ✔
✔ Answer: 144
---
10. Find x if x/16 = √(9/16)
√(9/16) = 3/4
So x/16 = 3/4
Multiply both sides by 16: x = (3/4) × 16 = 3 × 4 = 12
✔ Answer: 12
---
11. Area of square field is 4225 m². Girl cycles along boundary twice. How much distance?
Area = side² = 4225 → side = √4225
What’s √4225? 65² = 4225 (since 60²=3600, 70²=4900, 65²=(60+5)²=3600+600+25=4225)
Perimeter = 4 × side = 4 × 65 = 260 m
She covers boundary twice → 2 × 260 = 520 m
✔ Answer: 520 meters
---
12. Arrange 1024 students such that each row has as many students as number of rows. Find number of students in each row.
Let number of rows = x → students per row = x
Total students = x × x = x² = 1024
x = √1024 = 32 (since 30²=900, 32²=1024)
✔ Answer: 32
---
Final Answer:
1. (i) 1
(ii) 900
(iii) 400
(iv) 301
(v) 25
(vi) 4
2. 121, 144
3. 144 and 145
4. 10, 24, 26
5. 3
6. (i) 103
(ii) 5.8
(iii) 0.67
(iv) 17/19
(v) 4/3 or 1 1/3
7. (i) 110
(ii) 0.15
8. 1024
9. 144
10. 12
11. 520 meters
12. 32
---
1. Fill in the blanks
(i) The digit at unit’s place of square of 239 = ?
We only care about the last digit of 239 → that’s 9.
Square of 9 is 81 → last digit is 1.
✔ Answer: 1
---
(ii) 29 × 31 = □ – 1
Notice: 29 = 30 - 1, 31 = 30 + 1
So, (30 - 1)(30 + 1) = 30² - 1² = 900 - 1
But the question says “= □ – 1”, so □ must be 900
✔ Answer: 900
---
(iii) 17 × 23 = □ – 3²
17 = 20 - 3, 23 = 20 + 3
So, (20 - 3)(20 + 3) = 20² - 3² = 400 - 9
Question says “= □ – 3²” → so □ = 400
✔ Answer: 400
---
(iv) (151)² – (150)² = ?
Use identity: a² - b² = (a - b)(a + b)
Here, a = 151, b = 150
→ (151 - 150)(151 + 150) = (1)(301) = 301
✔ Answer: 301
---
(v) The sum of first five odd numbers = ?
First five odd numbers: 1, 3, 5, 7, 9
Sum = 1+3=4; 4+5=9; 9+7=16; 16+9=25
Or use formula: Sum of first n odd numbers = n² → 5² = 25
✔ Answer: 25
---
(vi) If 6ˣ = 1296 then x = ?
Let’s write powers of 6:
6¹ = 6
6² = 36
6³ = 216
6⁴ = 1296 ← Yes!
✔ Answer: 4
---
2. Write the perfect square numbers between 100 and 150.
Perfect squares:
10² = 100 → not *between* (if strictly between, exclude 100)
11² = 121
12² = 144
13² = 169 → too big
So between 100 and 150: 121, 144
(If including 100, add it — but usually “between” means exclusive. Let’s check context — since 100 is boundary, better to include if not specified. But problem says “between 100 and 150”, so likely 101 to 149. So 121 and 144.)
✔ Answer: 121, 144
---
3. Write 17² as sum of two consecutive integers.
17² = 289
We need two consecutive integers: let them be n and n+1
n + (n+1) = 289 → 2n + 1 = 289 → 2n = 288 → n = 144
So numbers are 144 and 145
Check: 144 + 145 = 289 ✔
✔ Answer: 144 and 145
---
4. Find the Pythagorean triplet whose smallest number is 10.
Pythagorean triplet: a² + b² = c², with a < b < c
Smallest number is 10 → try multiples of known triplets.
Known triplet: 3-4-5 → multiply by 2 → 6-8-10 → smallest is 6, not 10.
Multiply 3-4-5 by something else? Try 5-12-13 → smallest is 5.
Wait — we want smallest = 10.
Try scaling 5-12-13 by 2 → 10-24-26
Check: 10² + 24² = 100 + 576 = 676 = 26² ✔
✔ Answer: 10, 24, 26
---
5. Find the smallest number by which 192 must be multiplied to make the product perfect square.
Factorize 192:
192 ÷ 2 = 96
96 ÷ 2 = 48
48 ÷ 2 = 24
24 ÷ 2 = 12
12 ÷ 2 = 6
6 ÷ 2 = 3
3 ÷ 3 = 1
So 192 = 2⁶ × 3¹
For perfect square, all exponents must be even.
2⁶ is already even exponent.
3¹ → needs another 3 to become 3².
So multiply by 3
✔ Answer: 3
---
6. Find the square root of the following
(i) √10609
Try estimating: 100² = 10000, 103² = ?
103² = (100+3)² = 10000 + 600 + 9 = 10609 ✔
✔ Answer: 103
---
(ii) √33.64
Note: 5.8² = ?
5.8 × 5.8 = (5 + 0.8)² = 25 + 2×5×0.8 + 0.64 = 25 + 8 + 0.64 = 33.64 ✔
✔ Answer: 5.8
---
(iii) √0.4489
Try 0.67² = ?
0.67 × 0.67 = (67/100)² = 4489 / 10000 = 0.4489 ✔
✔ Answer: 0.67
---
(iv) √(289/361)
√289 = 17, √361 = 19 → so √(289/361) = 17/19
✔ Answer: 17/19
---
(v) √(1 7/9)
Convert mixed number: 1 7/9 = 16/9
√(16/9) = 4/3 = 1 1/3
✔ Answer: 4/3 or 1 1/3
---
7. Find:
(i) √55 × √220
√55 × √220 = √(55 × 220)
55 × 220 = 55 × (22 × 10) = (55 × 22) × 10
Better: 55 × 220 = 55 × (4 × 55) = 4 × 55²
Because 220 = 4 × 55
So √(4 × 55²) = √4 × (55²) = 2 × 55 = 110
✔ Answer: 110
---
(ii) √0.25 × √0.09
√0.25 = 0.5, √0.09 = 0.3
0.5 × 0.3 = 0.15
✔ Answer: 0.15
---
8. Find the smallest number of four digits which is a perfect square.
Smallest 4-digit number: 1000
Find smallest integer whose square ≥ 1000
√1000 ≈ 31.62 → so try 32² = 1024
31² = 961 (3-digit)
So 32² = 1024 is smallest 4-digit perfect square
✔ Answer: 1024
---
9. Find the smallest square number which is divisible by 4, 12 and 16.
First find LCM of 4, 12, 16.
Prime factors:
4 = 2²
12 = 2² × 3
16 = 2⁴
LCM = highest powers: 2⁴ × 3 = 16 × 3 = 48
Now, 48 is not a perfect square. Make it a perfect square by multiplying to make all exponents even.
48 = 2⁴ × 3¹ → need another 3 → 48 × 3 = 144
144 = 12² → perfect square
And divisible by 4, 12, 16?
144 ÷ 4 = 36 ✔
144 ÷ 12 = 12 ✔
144 ÷ 16 = 9 ✔
✔ Answer: 144
---
10. Find x if x/16 = √(9/16)
√(9/16) = 3/4
So x/16 = 3/4
Multiply both sides by 16: x = (3/4) × 16 = 3 × 4 = 12
✔ Answer: 12
---
11. Area of square field is 4225 m². Girl cycles along boundary twice. How much distance?
Area = side² = 4225 → side = √4225
What’s √4225? 65² = 4225 (since 60²=3600, 70²=4900, 65²=(60+5)²=3600+600+25=4225)
Perimeter = 4 × side = 4 × 65 = 260 m
She covers boundary twice → 2 × 260 = 520 m
✔ Answer: 520 meters
---
12. Arrange 1024 students such that each row has as many students as number of rows. Find number of students in each row.
Let number of rows = x → students per row = x
Total students = x × x = x² = 1024
x = √1024 = 32 (since 30²=900, 32²=1024)
✔ Answer: 32
---
Final Answer:
1. (i) 1
(ii) 900
(iii) 400
(iv) 301
(v) 25
(vi) 4
2. 121, 144
3. 144 and 145
4. 10, 24, 26
5. 3
6. (i) 103
(ii) 5.8
(iii) 0.67
(iv) 17/19
(v) 4/3 or 1 1/3
7. (i) 110
(ii) 0.15
8. 1024
9. 144
10. 12
11. 520 meters
12. 32
Parent Tip: Review the logic above to help your child master the concept of square root worksheet grade 8.
