Class 8 RD Sharma Solutions - Chapter 3 Squares and Square Roots - Exercise 3.5

In Class 8 Mathematics, understanding squares and square roots is crucial as it forms the foundation for algebra and various other mathematical concepts. Chapter 3 of RD Sharma's textbook delves into the squares and square roots exploring their properties, applications, and methods of calculation. Exercise 3.5 focuses on applying these concepts through practice problems helping the students reinforce their understanding and problem-solving skills.

Squares and Square Roots

Squares: A square of a number is the result of multiplying the number by itself. For example: a square of 4 is 4×4=16. Squares are fundamental in geometry and algebra and are used in various calculations.

Square Roots: The square root of a number is a value that when multiplied by itself gives the original number. For instance, the square root of 16 is 4 because 4×4=16. The Square roots are essential for solving equations and understanding geometric properties.

Question 1. Find the square root of each of the following by long division method:

(i) 12544  

Solution:

Here, by using long division method,

👁 Image

We found that the square root of 12544 is 112.

(ii) 97344 

Solution:

Here, by using long division method 

👁 Image

We found that the square root of 97344 is 312. 

(iii) 286225   

Solution:

Here, by using long division method, 

👁 Image

We found that the square root of 286225 is 535.

(iv) 390625

Solution:

Here, by using long division method,

👁 Image

We found that the square root of 390625 is 625. 

(v) 363609

Solution:

Here,  by using long division method,

👁 Image

We found that the square root of 363609 is 603.

(vi) 974169

Solution:

Here, by using long division method,

👁 Image

We found that the square root of 974169 is 987.

(vii) 120409 

Solution:

Here, by using long division method,

👁 Image

We found that the square root of 120409 is 347.

(viii) 1471369

Solution:

Here, by using long division method,

👁 Image

We found that the square root of 1471369 is 1213.

(ix) 291600

Solution:  

Here, by using long division method,

👁 Image

We found that the square root of 291600 is 540.

(x) 9653449

Solution:

Here, by using long division method,

👁 Image

We found that the square root of 9653449 is 3107.

(xi) 1745041

Solution:

Here, by using long division method,

👁 Image

We found that the square root of 1745041 is 1321.

(xii) 4008004 

Solution:

Here, by using long division method,

👁 Image

We found that the square root of 4008004 is 2002.

(xiii) 20657025

Solution: 

Here, by using long division method,

👁 Image

We found that the square root of 20657025 is 4545.

(xiv) 152547201

Solution:

Here, by using long division method,

👁 Image

We found that the square root of 152547201 is 12351.

(xv) 20421361

Solution: 

Here, by using long division method,

👁 Image

We found that the square root of 20421361 is 4519.

(xvi) 62504836

Solution:-  

Here, by using long division method,

👁 Image

We found that the square root of 62504836 is 7906.

(xvii) 82264900

Solution:

Here, using long division method,

👁 Image

We found that the square root of 82264900 is 9070.

(xviii) 3226694416

Solution:

Here, by using long division method,

👁 Image

We found that the square root of 3226694416 is 56804.

(xix) 6407522209

Solution:

Here, by using long division method,

👁 Image

We found that the square root of 6407522209 is 80047.

(xx) 3915380329

Solution: 

Here, by using long division method,

👁 Image

We found that the square root of 3915380329 is 62573.

Question 2. Find the least number which must be subtracted from the following numbers to make them a perfect square:

(i) 2361

Solution:

Using long division method,

👁 Image

Here, we observed that 2361 is 57 more than the square of 48.

So, we have to subtract 57 from 2361 to make it a perfect square.

(ii) 194491

Solution:

Using long division method,

👁 Image

Here, we observed that 194491 is 10 more than the square of 441.

So, we have to subtract 10 from 194491 to make it a perfect square.

(iii) 26535

Solution:

using long division method,

👁 Image

Here, we observed that 26535 is 291 more than the square of 162.

So, we have to subtract 291 from 26535 to make it a perfect square.

(iv) 161605

Solution:

Using long division method,

👁 Image

Here, we observed that 161605 is 1 more than the square of 402.

So, we have to subtract 1 from 161605 to make it a perfect square.

(v) 4401624

Solution:

Using long division method,

👁 Image

Here, we observed that 4401624 is 20 more than the square of 2098.

So, we have to subtract 20 from 4401624 to make it a perfect square.

Question 3. Find the least number which must be added to the following numbers to make them a perfect square:

(i) 5607

Solution:

Here, by using long division method,

👁 Image

Now, we observe that 5607 is 18 less than square of 75.

Therefore, we must add 18 to 5607 (18 + 5607 = 5625) to get a perfect square of 75 which is 5625. 

(ii) 4931

Solution:

Here, by using long division method,

👁 Image

Now, we observe that 4931 is 110 less than square of 71.

Therefore, we must add 110 to 4931 (110 + 4931 = 5041) to get a perfect square of 71 which is 5041. 

(iii) 4515600

Solution:

Here, by using long division method,

👁 Image

Now, we observe that 4515600 is 25 less than square of 2125.

Therefore, we must add 18 to 4515600 (25 + 4515600 = 4515625) to get a perfect square of 2125 which is 4515625. 

(iv) 37460

Solution:

Here, by using long division method,

👁 Image

Now, we observe that 37460 is 176 less than square of 194.

Therefore, we must add 176 to 37460 (176 + 37460 = 37636) to get a perfect square of 194 which is 37636. 

(v) 506900

Solution:

Here, by using long division method,

👁 Image

Now, we observe that 506900 is 44 less than square of 712.

Therefore, we must add 44 to 506900 (44 + 506900 = 506944) to get a perfect square of 712 which is 506944. 

Question 4. Find the greatest number of 5 digits which is a perfect square.

Solution:

We know that 99999 is the greatest 5-digit number.

Now, to get the greatest 5 digits square number, we have to find the smallest

number that can be subtracted to obtain a perfect square.

So, By using long division method,

👁 Image

Now, the remainder here is 143.

So, 143 must be subtracted from 99999 to get a perfect square.

So, 99999 - 143 = 99856.

Therefore, 99856 is the greatest 5 digits which is a perfect square.

Question 5. Find the least number of 4 digits which is a perfect square.

Solution: 

We know that 1000 is the smallest 4-digit number.

Now, in order to obtain the least 4 digits perfect square number, we have to find the 

smallest number that can be added to obtain a perfect square. 

So, By using long division method,

👁 Image

Now, we have to add 24 to 1000 to get a perfect square number.

So, 24 + 1000 = 1024

Therefore, 1024 is the least perfect square number with 4 digits which is a square of 32.

Question 6. Find the least number of six digits which is a perfect square.

Solution:

We know that 100000 is the smallest 6 digit number.

Now, in order to obtain the least 6 digits perfect square number, we have to find 

the smallest number that can be added to obtain a perfect square.

So, By using long division method,

👁 Image

Now, we have to add 489 to 100000 to get a perfect square number.

So, 489 + 100000 = 100489

Therefore, 100489 is the least perfect square number with 6 digits which is a square of 317.

Question 7. Find the greatest number of 4 digits which is a perfect square.

Solution:

We know that 9999 is the greatest 4 digit number.

Now, to get the greatest 4 digits square number, we have to find the smallest number

that can be subtracted to obtain a perfect square.

So, By using long division method,

👁 Image

Now, the remainder here is 198.

So, 198 must be subtracted from 9999 to get a perfect square.

So, 9999 - 198 = 9801.

Therefore, 9801 is the greatest 4 digits which is a perfect square number.

Question 8. A General arranges his soldiers in rows to form a perfect square. He finds that in doing so, 60 soldiers are left out. If the total number of soldiers be 8160, find the number of soldiers in each row?

Solution:

Given, Total number of Soldiers = 8160

Number of left out Soldiers = 60

So, number of soldiers arranged in rows to form a perfect square is 8100 (8160 - 60 = 8100)

So, the number of soldiers in each row = square root of 8100

So, √8100 = 90

Therefore, there are 90 soldiers in each row.

Question 9. The area of a square field is 60025 m². A man cycles along its boundary at 18 Km/hr. In how much time will he return at the starting point?

Solution:

Given, area of square field = 60025 m²

Speed of the cyclist = 18 km/h

= 18 × (1000/60 × 60)

= 5 m/s² 

Area of square field = 60025 m²

or, (side)² = 60025

or, side = √60025

or, side = 245 m 

We know that, Total length of boundary = perimeter 

= 4 * side (Perimeter of square = 4 * side of square) 

 = 4 × 245

= 980 m

Hence, Time taken to return to the starting point = 980/5 (time = distance / speed)

= 196 seconds

= 3 minutes 16 seconds

Question 10. The cost of leveling and turning a square lawn at Rs 2.50 per m² is Rs13322.50. Find the cost of fencing it at Rs 5 per meter.

Solution:

Given, The cost of leveling and turning a square lawn = Rs 2.50 per m²

Total cost of leveling and turning a square lawn = Rs13322.50

Now, Area of square lawn = total cost of leveling and turning a square lawn / the cost of leveling and turning a square lawn

= 13322.50 / 2.50

= 5329 m²

So, the length of side of square = √area of square (As area of square = side²)

= √5329 = 73 m

So, Total length for fencing = Perimeter (As perimeter of square = 4 * side)

= 4 * 73 = 292 m

Therefore, cost of fencing the square lawn = 5 * 292 

= Rs. 1460

Question 11. Find the greatest number of three digits which is a perfect square.

Solution:

We know that 999 is the greatest 3-digit number.

Now, to get the greatest 3 digits square number, we have to find the smallest number 

that can be subtracted to obtain a perfect square.

So, By using long division method,

👁 Image

Now, the remainder here is 38.

So, 38 must be subtracted from 999 to get a perfect square.

So, 999 - 38 = 961.

961 = 31²

Therefore, 961 is the greatest 3 digits which is a perfect square number.

Question 12. Find the smallest number which must be added to 2300 so that it becomes a perfect square.

Solution:

We will first find the square root of the given number by using the long division method,

👁 Image

Now, we come to know that 2300 is 4 less than square of 48.

So, 48² = 2304

Hence, 4 must be added to 2300 in order to get a perfect square.

Read More:

• Squares and Square Roots
• Square and Square Roots Practice Questions

Conclusion

The Mastering squares and square roots is essential for the solving a wide range of the mathematical problems. Exercise 3.5 in RD Sharma’s Chapter 3 provides the valuable practice in these areas reinforcing the student’s ability to work with these fundamental concepts. Understanding and applying these skills will enhance proficiency in the algebra and prepare students for the more complex mathematical topics.