CG Board Class 8 Mathematics Chapter 1 Solutions: Square and Cube (वर्ग एवं घन) – 2026-27

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CG SCERT (Chhattisgarh Board) Class 8 Mathematics Solutions | 2026-27 Official Edition

Complete 100% textbook-grounded CG Board Class 8 Mathematics Chapter 1 Solutions: Square and Cube (वर्ग एवं घन) in Bilingual Format (English & Hindi Medium).

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Question 1

Pair the prime factors of the following numbers…


📝 Question:

Pair the prime factors of the following numbers and state whether they are perfect squares or not.**

💡 Solution:

(i) 164**

Prime factorization of 164 = 2 × 2 × 41. Here, 41 cannot be paired. Therefore, 164 is not a perfect square.

  • (ii) 121

Prime factorization of 121 = 11 × 11. All prime factors form complete pairs. Therefore, 121 is a perfect square.

  • (iii) 289

Prime factorization of 289 = 17 × 17. All prime factors form complete pairs. Therefore, 289 is a perfect square.

  • (iv) 729

Prime factorization of 729 = (3 × 3) × (3 × 3) × (3 × 3). All factors form complete pairs. Therefore, 729 is a perfect square.

  • (v) 1100

Prime factorization of 1100 = (2 × 2) × (5 × 5) × 11. Factor 11 remains unpaired. Therefore, 1100 is not a perfect square.

Question 2

Give reasons why the given numbers are not…


📝 Question:

Give reasons why the given numbers are not perfect squares

💡 Solution:

(i) 12000:** Ends with 3 zeroes (an odd number of zeroes). Hence, not a perfect square. )

  • (iii) 790: Ends with 1 zero (odd number of zeroes). Hence, not a perfect square. )
  • (v) 165000: Ends with 3 zeroes (odd number of zeroes). Hence, not a perfect square. )
  • (vii) 1078: Unit digit is 8. Perfect squares cannot end in 8.

Question 3

Find out the numbers whose squares are even…


📝 Question:

Find out the numbers whose squares are even numbers and whose squares are odd numbers

💡 Solution:

(i) 14: Even number ⇒ Square is Even*.

  • (iii) 179: Odd number ⇒ Square is *Odd.
  • (v) 608: Even number ⇒ Square is *Even.
  • (vii) 1079: Odd number ⇒ Square is *Odd.
  • (ix) 1225: Odd number ⇒ Square is *Odd.

Question 4

Look at the given pattern and fill in…


📝 Question:

Look at the given pattern and fill in the blanks

💡 Solution:

(i) 125: 125 = 5 × 5 × 5 = 53 ⇒ Perfect cube*.

  • (iii) 729: 729 = (3 × 3 × 3) × (3 × 3 × 3) = 93 ⇒ *Perfect cube.
  • (v) 22000: 22000 = (2 × 2 × 2) × 2 × (5 × 5 × 5) × 11 ⇒ *Not a perfect cube.

Question 2

Find out the smallest number which when multiplied…


📝 Question:

Find out the smallest number which when multiplied by 256 will make the product a perfect cube

💡 Solution:

  • Prime factorization of 256 = (2 × 2 × 2) × (2 × 2 × 2) × 2 × 2.
  • To complete the triplet for factor 2, one more 2 is required.
  • Therefore, the required smallest multiplier is 2.
  • × (2 × 2 × 2) × 2 × 2 2 2 2 )

Question 3

Find out the smallest number which when multiplied…


📝 Question:

Find out the smallest number which when multiplied by 1352 will make the product a perfect cube

💡 Solution:

  • Prime factorization of 1352 = (2 × 2 × 2) × 13 × 13.
  • To form a complete triplet for factor 13, one more 13 is needed.
  • Therefore, the required smallest multiplier is 13.
  • × 13 × 13 13 13 13 )

Question 4

Find out the smallest number which when divided…


📝 Question:

Find out the smallest number which when divided into 8019 will make the quotient a perfect cube

💡 Solution:

  • Prime factorization of 8019 = (3 × 3 × 3) × (3 × 3 × 3) × 11.
  • Here, 11 is an extra factor without a triplet. Dividing by 11 leaves (33 × 33) = 729 = 93, which is a perfect cube.
  • Therefore, the required smallest divisor is 11.
  • × (3 × 3 × 3) × 11 11 11 11 )

Question 5

Find out the smallest number which when multiplied…


📝 Question:

Find out the smallest number which when multiplied by the numbers which are not cubes in Question 1 give cube numbers

💡 Solution:

(ii) For 800:**

  • 800 = (2 × 2 × 2) × 2 × 2 × 5 × 5.
  • Needed to complete triplets: 2 × 5 = 10.
  • (v) For 22000:
  • 22000 = (2 × 2 × 2) × 2 × (5 × 5 × 5) × 11.
  • Needed to complete triplets: 2 × 2 × 11 × 11 = 484.
  • (vi) For 832:
  • 832 = (2 × 2 × 2) × (2 × 2 × 2) × 13.
  • Needed to complete triplets: 13 × 13 = 169.
  • (i) 361: 361 = 19 × 19 ⇒ √(361) = 19.
  • (ii) 400: 400 = (2 × 2) × (2 × 2) × (5 × 5) ⇒ √(400) = 2 × 2 × 5 = 20.
  • (iii) 784: 784 = (2 × 2) × (2 × 2) × (7 × 7) ⇒ √(784) = 2 × 2 × 7 = 28.
  • (iv) 1024: 1024 = (2 × 2)5 ⇒ √(1024) = 25 = 32.
  • (v) 2304: 2304 = (2 × 2)4 × (3 × 3) ⇒ √(2304) = 24 × 3 = 48.
  • (vi) 7056: 7056 = (2 × 2)2 × (3 × 3) × (7 × 7) ⇒ √(7056) = 2 × 2 × 3 × 7 = 84.

Question 2

A group of boys bought 256 mangoes and…


📝 Question:

A group of boys bought 256 mangoes and distributed them among themselves. If each boy got the number of mangoes equal to the number of boys in the group, find out the number of boys in the group

💡 Solution:

  • Let the number of boys in the group = x.
  • Total mangoes = x × x = x2 = 256.
  • x = √(256) = √((2 × 2) × (2 × 2) × (2 × 2) × (2 × 2)) = 2 × 2 × 2 × 2 = 16.
  • Therefore, the number of boys in the group is 16.
  • = 16)

### Exercise 1.4 / 1.4

Question 1

Find out the square root of the following…


📝 Question:

Find out the square root of the following by division method

💡 Solution:

(i) 529:** Pair (5)(29) ⇒ 2 × 2 = 4, remainder 129; 43 × 3 = 129 ⇒ √(529) = 23.

  • (ii) 1369: Pair (13)(69) ⇒ 3 × 3 = 9, remainder 469; 67 × 7 = 469 ⇒ √(1369) = 37.
  • (iii) 1024: Pair (10)(24) ⇒ 3 × 3 = 9, remainder 124; 62 × 2 = 124 ⇒ √(1024) = 32.
  • (iv) 5776: Pair (57)(76) ⇒ 7 × 7 = 49, remainder 876; 146 × 6 = 876 ⇒ √(5776) = 76.
  • (v) 900: Pair (9)(00) ⇒ 3 × 3 = 9, quotient = 30.
  • (vi) 7921: Pair (79)(21) ⇒ 8 × 8 = 64, remainder 1521; 169 × 9 = 1521 ⇒ √(7921) = 89.
  • (vii) 50625: Pair (5)(06)(25) ⇒ 2 × 2 = 4, remainder 106; 42 × 2 = 84, remainder 2225; 445 × 5 = 2225 ⇒ √(50625) = 225.
  • (viii) 363609: Pair (36)(36)(09) ⇒ 6 × 6 = 36, remainder 36; 120 × 0 = 0, remainder 3609; 1203 × 3 = 3609 ⇒ √(363609) = 603.

Question 2

In a cinema hall the owner wants to…


📝 Question:

In a cinema hall the owner wants to organize the chairs in such a way that the number of rows and columns of seats are equal. If there are total 1849 seats, find out the number of rows and columns

💡 Solution:

  • Let rows = columns = x.
  • Total seats = x2 = 1849 ⇒ x = √(1849).
  • Division method on 1849 ⇒ 4 × 4 = 16, remainder 249; 83 × 3 = 249 ⇒ x = 43.
  • Number of rows/columns = 43.
  • = 43)

Question 3

The area of a square garden is 1444…


📝 Question:

The area of a square garden is 1444 square meters, so find out the length and breadth of that garden

💡 Solution:

  • Area of square garden = side2 = 1444 m2.
  • Side = √(1444) = 38 m (via division method: 3 × 3 = 9, remainder 544; 68 × 8 = 544).
  • Length = Breadth = 38 m.
  • = 38 )

### Exercise 1.5 / 1.5

Question 1

Find out the square root of the following…


📝 Question:

Find out the square root of the following numbers

💡 Solution:

(i) 7.29:** √(7.29) = 2.7 (2 × 2 = 4, remainder 329; 47 × 7 = 329).

  • (ii) 16.81: √(16.81) = 4.1 (4 × 4 = 16, remainder 81; 81 × 1 = 81).
  • (iii) 9.3025: √(9.3025) = 3.05 (3 × 3 = 9, remainder 30; 60 × 0 = 0, remainder 3025; 605 × 5 = 3025).

Question 2

Find out the square root of the following…


📝 Question:

Find out the square root of the following up to two places of decimal

💡 Solution:

(i) 0.9:**

  • √(0.900000) = 0.948… ≈ 0.95.
  • (ii) 5:
  • √(5.000000) = 2.236… ≈ 2.24.
  • (iii) 7:
  • √(7.000000) = 2.645… ≈ 2.64.
  • (i) 125: √(125) = √(5 × 5 × 5) = 5.
  • (ii) 343: √(343) = √(7 × 7 × 7) = 7.
  • (iii) 1331: √(1331) = √(11 × 11 × 11) = 11.
  • (iv) 2197: √(2197) = √(13 × 13 × 13) = 13.
  • (v) 9261: √(9261) = √((3 × 3 × 3) × (7 × 7 × 7)) = 3 × 7 = 21.
  • (vi) 166375: √(166375) = √((5 × 5 × 5) × (11 × 11 × 11)) = 5 × 11 = 55.
  • (vii) 4913: √(4913) = √(17 × 17 × 17) = 17.
  • (viii) 42875: √(42875) = √((5 × 5 × 5) × (7 × 7 × 7)) = 5 × 7 = 35.

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