In this blog post, we will explore Class 8 Mathematics Chapter 6, Squares and Square Roots. We will go through all the exercises in detail to help you understand how to solve problems related to squares and square roots.
Introduction to Squares and Square Roots
Before we dive into the exercises, let’s first understand the core concepts of Squares and Square Roots.
- Square of a number: The square of a number is the result of multiplying the number by itself. Mathematically, it is represented as n2n^2n2, where nnn is the number.
- Example: 52=5×5=255^2 = 5 \times 5 = 2552=5×5=25
- Square Root of a number: The square root of a number is a value that, when multiplied by itself, gives the original number. It is represented as n\sqrt{n}n.
- Example: 25=5\sqrt{25} = 525=5, because 5×5=255 \times 5 = 255×5=25
Let’s explore the exercises now.
Exercise 6.1: Square of a Number
Q1: Find the square of the following numbers.
- a) 12
Solution:
122=12×12=14412^2 = 12 \times 12 = 144122=12×12=144 - b) 7
Solution:
72=7×7=497^2 = 7 \times 7 = 4972=7×7=49 - c) 15
Solution:
152=15×15=22515^2 = 15 \times 15 = 225152=15×15=225 - d) 9
Solution:
92=9×9=819^2 = 9 \times 9 = 8192=9×9=81
Exercise 6.2: Square Roots
Q2: Find the square root of the following numbers.
- a) 64
Solution:
64=8\sqrt{64} = 864=8
(because 8×8=648 \times 8 = 648×8=64) - b) 81
Solution:
81=9\sqrt{81} = 981=9
(because 9×9=819 \times 9 = 819×9=81) - c) 100
Solution:
100=10\sqrt{100} = 10100=10
(because 10×10=10010 \times 10 = 10010×10=100) - d) 121
Solution:
121=11\sqrt{121} = 11121=11
(because 11×11=12111 \times 11 = 12111×11=121)
Exercise 6.3: Identifying Perfect Squares
Q3: Identify whether the following numbers are perfect squares or not.
- a) 36
Solution:
Yes, 36 is a perfect square because 36=6\sqrt{36} = 636=6 and 6 is an integer. - b) 50
Solution:
No, 50 is not a perfect square because its square root is not an integer. - c) 121
Solution:
Yes, 121 is a perfect square because 121=11\sqrt{121} = 11121=11, which is an integer. - d) 25
Solution:
Yes, 25 is a perfect square because 25=5\sqrt{25} = 525=5, which is an integer.
Exercise 6.4: Square Roots of Large Numbers
Q4: Find the square roots of the following large numbers using prime factorization.
- a) 196
Solution:
First, perform the prime factorization of 196:196÷2=9898÷2=4949÷7=77÷7=1196 \div 2 = 98 \\ 98 \div 2 = 49 \\ 49 \div 7 = 7 \\ 7 \div 7 = 1196÷2=9898÷2=4949÷7=77÷7=1So, the prime factorization of 196 is 2×2×7×72 \times 2 \times 7 \times 72×2×7×7.
Pairing the common factors:196=2×7=14\sqrt{196} = 2 \times 7 = 14196=2×7=14
- b) 256
Solution:
Prime factorization of 256:256÷2=128128÷2=6464÷2=3232÷2=1616÷2=88÷2=44÷2=22÷2=1256 \div 2 = 128 \\ 128 \div 2 = 64 \\ 64 \div 2 = 32 \\ 32 \div 2 = 16 \\ 16 \div 2 = 8 \\ 8 \div 2 = 4 \\ 4 \div 2 = 2 \\ 2 \div 2 = 1256÷2=128128÷2=6464÷2=3232÷2=1616÷2=88÷2=44÷2=22÷2=1So, the prime factorization of 256 is 282^828.256=24=16\sqrt{256} = 2^4 = 16256=24=16
Exercise 6.5: Estimation of Square Roots
Q5: Estimate the square root of the following numbers between two perfect squares.
- a) 45
Solution:
The perfect squares near 45 are 36 and 49.36=6and49=7\sqrt{36} = 6 \quad \text{and} \quad \sqrt{49} = 736=6and49=7Hence, 45\sqrt{45}45 will be between 6 and 7. - b) 58
Solution:
The perfect squares near 58 are 49 and 64.49=7and64=8\sqrt{49} = 7 \quad \text{and} \quad \sqrt{64} = 849=7and64=8Hence, 58\sqrt{58}58 will be between 7 and 8.
Exercise 6.6: Word Problems
Q6: Solve the following word problems related to squares and square roots.
- a) The area of a square park is 225 m². Find the length of one side of the park.
Solution:
The area of a square is given by the formula A=s2A = s^2A=s2, where sss is the side of the square.
So,225=s2⇒s=225=15225 = s^2 \quad \Rightarrow \quad s = \sqrt{225} = 15225=s2⇒s=225=15The length of one side of the park is 15 meters. - b) The length of a square garden is 144 meters. Find the area of the garden.
Solution:
The area of a square garden is given by A=s2A = s^2A=s2.
So,A=1442=144×144=20736 m2A = 144^2 = 144 \times 144 = 20736 \, \text{m}^2A=1442=144×144=20736m2The area of the garden is 20736 square meters.
Additonal questions
1. Square of a Number
Q1: Find the square of 18.
Solution:
182=18×18=32418^2 = 18 \times 18 = 324182=18×18=324
Q2: Find the square of 25.
Solution:
252=25×25=62525^2 = 25 \times 25 = 625252=25×25=625
Q3: Find the square of 14.
Solution:
142=14×14=19614^2 = 14 \times 14 = 196142=14×14=196
Q4: Find the square of 22.
Solution:
222=22×22=48422^2 = 22 \times 22 = 484222=22×22=484
2. Square Roots
Q5: Find the square root of 144.
Solution:
144=12\sqrt{144} = 12144=12
Q6: Find the square root of 169.
Solution:
169=13\sqrt{169} = 13169=13
Q7: Find the square root of 256.
Solution:
256=16\sqrt{256} = 16256=16
Q8: Find the square root of 324.
Solution:
324=18\sqrt{324} = 18324=18
Q9: Find the square root of 576.
Solution:
576=24\sqrt{576} = 24576=24
Q10: Find the square root of 900.
Solution:
900=30\sqrt{900} = 30900=30
3. Identifying Perfect Squares
Q11: Is 50 a perfect square?
Solution:
No, 50\sqrt{50}50 is not an integer.
Q12: Is 100 a perfect square?
Solution:
Yes, 100=10\sqrt{100} = 10100=10, which is an integer.
Q13: Is 145 a perfect square?
Solution:
No, 145\sqrt{145}145 is not an integer.
Q14: Is 81 a perfect square?
Solution:
Yes, 81=9\sqrt{81} = 981=9, which is an integer.
Q15: Is 36 a perfect square?
Solution:
Yes, 36=6\sqrt{36} = 636=6, which is an integer.
4. Square Roots Using Prime Factorization
Q16: Find the square root of 81 using prime factorization.
Solution:
Prime factorization of 81:
81=3×3×3×381 = 3 \times 3 \times 3 \times 381=3×3×3×3
Pairing the factors:
81=3×3=9\sqrt{81} = 3 \times 3 = 981=3×3=9
Q17: Find the square root of 225 using prime factorization.
Solution:
Prime factorization of 225:
225=3×3×5×5225 = 3 \times 3 \times 5 \times 5225=3×3×5×5
Pairing the factors:
225=3×5=15\sqrt{225} = 3 \times 5 = 15225=3×5=15
Q18: Find the square root of 484 using prime factorization.
Solution:
Prime factorization of 484:
484=2×2×11×11484 = 2 \times 2 \times 11 \times 11484=2×2×11×11
Pairing the factors:
484=2×11=22\sqrt{484} = 2 \times 11 = 22484=2×11=22
Q19: Find the square root of 900 using prime factorization.
Solution:
Prime factorization of 900:
900=2×2×3×3×5×5900 = 2 \times 2 \times 3 \times 3 \times 5 \times 5900=2×2×3×3×5×5
Pairing the factors:
900=2×3×5=30\sqrt{900} = 2 \times 3 \times 5 = 30900=2×3×5=30
5. Estimating Square Roots
Q20: Estimate 50\sqrt{50}50.
Solution:
The perfect squares near 50 are 49 and 64.
Hence, 50\sqrt{50}50 lies between 7 and 8.
Q21: Estimate 130\sqrt{130}130.
Solution:
The perfect squares near 130 are 121 and 144.
Hence, 130\sqrt{130}130 lies between 11 and 12.
Q22: Estimate 200\sqrt{200}200.
Solution:
The perfect squares near 200 are 196 and 225.
Hence, 200\sqrt{200}200 lies between 14 and 15.
Q23: Estimate 85\sqrt{85}85.
Solution:
The perfect squares near 85 are 81 and 100.
Hence, 85\sqrt{85}85 lies between 9 and 10.
Q24: Estimate 60\sqrt{60}60.
Solution:
The perfect squares near 60 are 49 and 64.
Hence, 60\sqrt{60}60 lies between 7 and 8.
6. Word Problems
Q25: A square garden has an area of 256 square meters. Find the length of one side.
Solution:
Area of the square = s2s^2s2.
s2=256s^2 = 256s2=256
s=256=16s = \sqrt{256} = 16s=256=16
The length of one side is 16 meters.
Q26: A square room has a side of 12 meters. Find its area.
Solution:
Area = s2s^2s2.
Area=122=12×12=144 m2Area = 12^2 = 12 \times 12 = 144 \, \text{m}^2Area=122=12×12=144m2
Q27: The area of a square is 121 cm². Find the length of one side.
Solution:
s2=121s^2 = 121s2=121
s=121=11s = \sqrt{121} = 11s=121=11
The length of one side is 11 cm.
Q28: A square park has a side length of 30 meters. What is its perimeter?
Solution:
Perimeter of a square = 4×s4 \times s4×s.
P=4×30=120P = 4 \times 30 = 120P=4×30=120 meters.
Q29: A square field has a side of 16 meters. Find its area and perimeter.
Solution:
Area = s2=162=256 m2s^2 = 16^2 = 256 \, \text{m}^2s2=162=256m2
Perimeter = 4×s=4×16=644 \times s = 4 \times 16 = 644×s=4×16=64 meters.
7. Additional Practice Questions
Q30: Find the square of 8.
Solution:
82=8×8=648^2 = 8 \times 8 = 6482=8×8=64
Q31: Find the square root of 121.
Solution:
121=11\sqrt{121} = 11121=11
Q32: Find the square of 13.
Solution:
132=13×13=16913^2 = 13 \times 13 = 169132=13×13=169
Q33: Find the square of 11.
Solution:
112=11×11=12111^2 = 11 \times 11 = 121112=11×11=121
Q34: Is 81 a perfect square?
Solution:
Yes, 81=9\sqrt{81} = 981=9, which is an integer.
Q35: Find the square root of 625.
Solution:
625=25\sqrt{625} = 25625=25
Q36: What is the square root of 1?
Solution:
1=1\sqrt{1} = 11=1
Q37: Is 1024 a perfect square?
Solution:
Yes, 1024=32\sqrt{1024} = 321024=32.
Q38: Find the square of 23.
Solution:
232=23×23=52923^2 = 23 \times 23 = 529232=23×23=529
Q39: Find the square of 17.
Solution:
172=17×17=28917^2 = 17 \times 17 = 289172=17×17=289
Q40: Find the square root of 169.
Solution:
169=13\sqrt{169} = 13169=13
Here are 20 more questions on Squares and Square Roots, along with their solutions:
1. Square of a Number
Q1: Find the square of 21.
Solution:
212=21×21=44121^2 = 21 \times 21 = 441212=21×21=441
Q2: Find the square of 32.
Solution:
322=32×32=102432^2 = 32 \times 32 = 1024322=32×32=1024
Q3: Find the square of 40.
Solution:
402=40×40=160040^2 = 40 \times 40 = 1600402=40×40=1600
Q4: Find the square of 50.
Solution:
502=50×50=250050^2 = 50 \times 50 = 2500502=50×50=2500
2. Square Roots
Q5: Find the square root of 121.
Solution:
121=11\sqrt{121} = 11121=11
Q6: Find the square root of 196.
Solution:
196=14\sqrt{196} = 14196=14
Q7: Find the square root of 784.
Solution:
784=28\sqrt{784} = 28784=28
Q8: Find the square root of 1024.
Solution:
1024=32\sqrt{1024} = 321024=32
Q9: Find the square root of 625.
Solution:
625=25\sqrt{625} = 25625=25
3. Identifying Perfect Squares
Q10: Is 72 a perfect square?
Solution:
No, 72\sqrt{72}72 is not an integer.
Q11: Is 100 a perfect square?
Solution:
Yes, 100=10\sqrt{100} = 10100=10, which is an integer.
Q12: Is 169 a perfect square?
Solution:
Yes, 169=13\sqrt{169} = 13169=13, which is an integer.
Q13: Is 50 a perfect square?
Solution:
No, 50\sqrt{50}50 is not an integer.
Q14: Is 256 a perfect square?
Solution:
Yes, 256=16\sqrt{256} = 16256=16, which is an integer.
4. Square Roots Using Prime Factorization
Q15: Find the square root of 144 using prime factorization.
Solution:
Prime factorization of 144:
144=2×2×2×2×3×3144 = 2 \times 2 \times 2 \times 2 \times 3 \times 3144=2×2×2×2×3×3
Pairing the factors:
144=2×2×3=12\sqrt{144} = 2 \times 2 \times 3 = 12144=2×2×3=12
Q16: Find the square root of 484 using prime factorization.
Solution:
Prime factorization of 484:
484=2×2×11×11484 = 2 \times 2 \times 11 \times 11484=2×2×11×11
Pairing the factors:
484=2×11=22\sqrt{484} = 2 \times 11 = 22484=2×11=22
Q17: Find the square root of 900 using prime factorization.
Solution:
Prime factorization of 900:
900=2×2×3×3×5×5900 = 2 \times 2 \times 3 \times 3 \times 5 \times 5900=2×2×3×3×5×5
Pairing the factors:
900=2×3×5=30\sqrt{900} = 2 \times 3 \times 5 = 30900=2×3×5=30
5. Estimating Square Roots
Q18: Estimate 75\sqrt{75}75.
Solution:
The perfect squares near 75 are 64 and 81.
Hence, 75\sqrt{75}75 lies between 8 and 9.
Q19: Estimate 200\sqrt{200}200.
Solution:
The perfect squares near 200 are 196 and 225.
Hence, 200\sqrt{200}200 lies between 14 and 15.
6. Word Problems
Q20: The area of a square playground is 256 square meters. Find the length of one side.
Solution:
Area of square = s2s^2s2.
s2=256s^2 = 256s2=256
s=256=16s = \sqrt{256} = 16s=256=16 meters.
Also Read: Class 8 Mathematics – Chapter 5: Understanding Quadrilaterals | Complete Guide with Solutions
For more updates Click Here.
As a seasoned content writer specialized in the fitness and health niche, Arun Bhagat has always wanted to promote wellness. After gaining proper certification as a gym trainer with in-depth knowledge of virtually all the information related to it, he exercised his flair for writing interesting, informative content to advise readers on their healthier lifestyle. His topics range from workout routines, nutrition, and mental health to strategies on how to be more fit in general. His writing is informative but inspiring for people to achieve their wellness goals as well. Arun is committed to equipping those he reaches with the insights and knowledge gained through fitness.