## RD Sharma Solutions Class 8 Chapter 3 Exercise 3.5

**Q-1. Find the square root of each of the following by long division method:**

**(i) 12544Â **

**(ii) 97344Â **

**(iii) 286225**

**(iv) 390625**

**(v) 363609**

**(vi) 974169**

**(vii) 120409**

**(viii) 1471369**

**(ix) 291600**

**(x) 9653449**

**(xi) 1745041**

**(xii) 4008004**

**(xiii) 20657025**

**(xiv) 152547201**

**(xv) 20421361**

**(xvi) 62504836**

**(xvii) 82264900Â **

**(xviii) 3226694416 **

**(xix) 6407522209Â **

**(xx) 3915380329**

**Solution:**

**(i)Â **To find: Square root of a numberÂ 12544 using long division method

Proceed the long division method steps to find the solution of the given number.

Therefore, the square root of a number 12544 is 112.

**(ii)Â **To find: Square root of a numberÂ 97344 using long division method

Proceed the long division method steps to find the solution of the given number.

Hence, the square root ofÂ a number 97344 is 312.

**(iii)** To find: Square root of a number 286225 using long division method

Proceed the long division method steps to find the solution of the given number.

Therefore, the square root of a number 286225 is 535.

**(iv)Â **To find: Square root of a number 390625 using long division method

Proceed the long division method steps to find the solution of the given number.

Hence, the square root of a number 390625 is 625.

**(v)Â **To find: Square root of a number 363609 using long division method

Proceed the long division method steps to find the solution of the given number.

Therefore, the square root ofÂ a number 363609 is 603.

**(vi)Â **To find: Square root of a number 974169 using long division method

Proceed the long division method steps to find the solution of the given number.

Therefore, the square root of a number 974169 is 987.

**(vii)Â **To find: Square root of a number 120409 using long division method

Proceed the long division method steps to find the solution of the given number.

Hence, the square root of a number 120409 is 347.

**(viii)Â **To find: Square root of a numberÂ 1471369 using long division method

Proceed the long division method steps to find the solution of the given number.

Therefore, the square root ofÂ a number 1471369 is 1213.

**(ix)Â **To find: Square root of a number 291600 using long division method

Proceed the long division method steps to find the solution of the given number.

Therefore, the square root ofÂ a number 291600 is 540

**(x)Â **To find: Square root of a number 9653449 using long division method

Proceed the long division method steps to find the solution of the given number.

Therefore, the square root of a number 9653449 is 3107.

**(xi)Â **To find: Square root of a number 1745041 using long division method

Proceed the long division method steps to find the solution of the given number.

Therefore, the square root ofÂ a number 1745041 is 1321.

**(xii)Â **To find: Square root of a number 4008004 using long division method

Proceed the long division method steps to find the solution of the given number.

Therefore, the square rootÂ a number 4008004 is 2002

**(xiii)Â **To find: Square root of a number 20657025 using long division method

Proceed the long division method steps to find the solution of the given number.

Therefore, the square root of a number 20657025 is 4545

**(xiv)Â **To find: Square root of a numberÂ 152547201 using long division method

Proceed the long division method steps to find the solution of the given number.

Therefore, the square root of a number 152547201 is 12351.

**(xv)Â **To find: Square root of a number 20421361 using long division method

Proceed the long division method steps to find the solution of the given number.

Therefore, the square root of a number 20421361 is 4519.

**(xvi)Â **To find: Square root of a number 62504836 using long division method

Proceed the long division method steps to find the solution of the given number.

Therefore, the square root of a number 6250486 is 7906.

**(xvii)Â **To find: Square root of a number 82264900 using long division method

Proceed the long division method steps to find the solution of the given number.

Therefore, the square root ofÂ a number 82264900 is 9070.

**(xviii)Â **To find: Square root of a number 3226694416 using long division method

Proceed the long division method steps to find the solution of the given number.

Therefore, the square root of a number 3226694416 is 56804.

**(xix)Â **To find: Square root of a number 6407522209 using long division method

Proceed the long division method steps to find the solution of the given number.

Therefore, the square root of a number 6407522209 is 80047

(xx)Â To find: Square root of a number 3915380329 using long division method

Proceed the long division method steps to find the solution of the given number.

Therefore, the square root of a number 3915380329 is 625763.

**Â **

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

**(i) 2361Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â (ii) 194491Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â (iii) 26535Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â (iv) 16160Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â (v) 4401624**

Solution.

(i) Use the long division method to find the least number

It is observed that 2361 is 57 more thanÂ 47^{2}.

If you subtract 57 from 2361, you will get a perfect square.

i.e., 2361 – 57 = 2304

The number obtained is a perfect square number.

Therefore, 57 is the least number that makes to get a perfect square.

(ii) Use the long division method to find the least number

It is observed that 194491 is 10 more thanÂ 441^{2}.

If you subtract 10 from 194491, you will get a perfect square.

i.e., 194491 – 10 = 194481

The number obtained is a perfect square number.

Therefore, 10 is the least number that makes to get a perfect square.

(iii) Use the long division method to find the least number

It is observed that 26535 is 291 more than 162^{2}.

If you subtract 291 from 26535, you will get a perfect square.

i.e., 26535 – 291 = 26244

The number obtained is a perfect square number.

Therefore, 291 is the least number that makes to get a perfect square.

(iv)Â Use the long division method to find the least number

It is observed that 16160 is 31 more than 127^{2}.

If you subtract 31 from 16160, you will get a perfect square.

i.e., 16160 – 31= 16129

The number obtained is a perfect square number.

Therefore, 31 is the least number that makes to get a perfect square.

(v) Use the long division method to find the least number

It is observed that 4401624 is 20 more than 2098^{2}.

If you subtract 20 from 4401624, you will get a perfect square.

i.e., 4401624 – 20= 4401604

The number obtained is a perfect square number.

Therefore, 20 is the least number that makes to get a perfect square.

**Q-3. Find the least number which must be addedÂ from the following numbers to make them a perfect square:**

**(i) 5607Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â (ii) 4931Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â (iii) 4515600Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â (iv) 37460Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â (v) 506900**

**Â **

Solution.

(i) Use the long division method to find the least number

It is observed that 5607 is 18 less than 75^{2}.

If you add 18 with 5607, you will get a perfect square.

i.e., 5607Â + 18Â =5625

The number obtained is a perfect square number.

Therefore, 18 is the least number that makes to get a perfect square.

(ii) Use the long division method to find the least number

It is observed that 4931 is 110 less than 71^{2}.

If you add 110 with 4931, you will get a perfect square.

i.e., 4931 + 110= 5041

The number obtained is a perfect square number.

Therefore, 110 is the least number that makes to get a perfect square.

(iii) Use the long division method to find the least number

It is observed that 4515600 is 25 less than 2125^{2}.

If you add 25 with 4515600, you will get a perfect square.

i.e., 4515600 +25= 4515625

The number obtained is a perfect square number.

Therefore, 25 is the least number that makes to get a perfect square.

(iv) Use the long division method to find the least number

It is observed that 37460 is 176 less than 194^{2}.

If you add 176 with 37460, you will get a perfect square.

i.e., 37460 +176 = 37636

The number obtained is a perfect square number.

Therefore, 176 is the least number that makes to get a perfect square.

(v) Use the long division method to find the least number

It is observed that 506900 is 44 less than 712^{2}.

If you add 44 with 506900, you will get a perfect square.

i.e.,506900 +44 = 506944

The number obtained is a perfect square number.

Therefore, 44 is the least number that makes to get a perfect square.

**Q-4. Find the greatest number of 5 digits which is a perfect square.**

**Solution:**

We know that the greatest number with five digits is 99999.

To make a perfect square number with five digits, first, find the smallest number that should be subtracted from 99999.

To find the smallest number, use the long division method to find the square root of 99999.

If you subtract, 143 from 99999, you will get a perfect square number.

i.e., 99999 â€“ 143 = 99856

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

**Q-5. Find the least number of 4 digits which is a perfect square.**

**Solution:**

We know that the least number with four digits is 1000.

To make a perfect square number with four digits, first, find the number that should be added with 1000.

To find the smallest number, use the long division method to find the square root of 1000.

If you add, 24 with 1000, you will get a perfect square number.

i.e., 1000+24 = 1024

Therefore, the least number with 4 digits is 1024 whichÂ is a perfect square number

**Q-6. Find the least number of six digits which is a perfect square.**

**Solution:**

We know that the least number with six digits is 100000.

To make a perfect square number with four digits, first, find the number that should be added with 100000.

To find the smallest number, use the long division method to find the square root of 100000.

If you add, 489 with 100000, you will get a perfect square number.

i.e., 100000 + 489 = 100489

Therefore, the least number with 6 digits 100489 Â is a perfect square number

**Q-7. Find the greatest number of 4 digits which is a perfect square. **

**Solution:**

We know that the greatest number with 4 digits is 9999.

To make a perfect square number with 4 digits, first, find the smallest number that should be subtracted from 9999.

To find the smallest number, use the long division method to find the square root of 9999.

If you subtract, 198 from 9999, you will get a perfect square number.

i.e., 9999 â€” 198 = 9801

Therefore, the greatest number with four digits is 9801, which is a perfect square number

**Q-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 data:Â Total number of soldiers is 8160

Out of that, 60 soldiers are left out.

Therefore, the remaining soldiers = 8160 â€” 60 = 8100

So, the number of soldiers in each row to form a perfect square should be the square root of 8100.

Find the square root of 8100 by the long division method as shown below:

Therefore, the total number of soldiers in each row to form a perfect square is 90.

**Â **

**Q-9. The area of a square field is 60025 m ^{2}. A man cycles along its boundary at 18 km/hr. In how much time will he return at the starting point?**

**Solution:**

Given: The area of the square field is 60025 m^{2 }

So, the length of the square field should be the square root of 60025.

Use the long division method to find the length of the square field.

From the long division method, we can say that the length of the square field is 245 m.

We know that the square has four sides.

Therefore, the number of boundaries of the field is 4.

So, the distance “s” covered by the man = 245 m x 4 = 980 m = 0.98 km

If the velocity “v” is 18 km/hr, the required time “t” is calculated using the given formula:

t = s/v

t = 0.98/18

t= 0.054 hr = 3 minutes, 16 seconds

Therefore, the man will return to the starting point of the square field is after 3 minutes and 16 seconds.

**Q-10. The cost of levelling and turfing a square lawn at Rs. 2.50 per m ^{2} is Rs. 13322.50. Find the cost of fencing it at Rs. 5 per metres.**

**Solution:**

To find the cost of fencing, first, we have to find the area of the square lawn.

So, the area of a square is the total cost divided by the cost of levelling and turfing per square metre

Area of a square = Â 13322.5/ 2.5= 5329 m^{2}

To find the length of one side of the square is equal to the square root of the area.

Use the long division method to find the length of the square lawn as shown below:

So, the length of one side of the square is 73 m

Since the square has four boundaries, the perimeter of the square is 73 Ã— 4 = 292 m

Therefore, the cost of fencing the square lawn at Rs. 5 per metre = 292 Ã— 5 = Rs. 1460.

**Q-11. Find the greatest number of three digits which is a perfect square.**

**Solution:**

We know that the greatest number with 3 digits is 999.

To make a perfect square number with 3 digits, first, find the smallest number that should be subtracted from 999.

To find the smallest number, use the long division method to find the square root of 999.

If you subtract, 38 from 999, you will get a perfect square number.

i.e., 999 â€” 38 = 961

Therefore, the greatest number with three digits is 961, which is a perfect square number

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

**Solution:**

To find the smallest number to become a perfect square when it is added to 2300, we use the long division method:

From the long division method, it is observed that 2300 is 4Â less than 48^{2}

i.e., 2300 + 4 =Â 2304

Therefore, 4 must be added to 2300 to get a perfect square.