Class 8 Maths Chapter 6 Practice Question Answers - Cubes and Cube Roots

Q1: You are told that 1,331 is a perfect cube. Can you guess without factorisation what is its cube root? Similarly, guess the cube roots of 4913, 12167, 32768.
Sol:
By grouping the digits, we get 1 and 331
Since, the unit digit of cube is 1, the unit digit of cube root is 1.
Therefore, we get 1 as the unit digit of the cube root of 1331.
The cube of 1 matches with the number of the second groups.
Therefore, the ten’s digit of our cube root is taken as the unit place of the smallest number.
We know that the unit’s digit of the cube of a number having digit as unit’s place 1 is 1.
Therefore, ∛1331 = 11
By grouping the digits, we get 4 and 913
We know that, since the unit digit of the cube is 3, the unit digit of the cube root is 7.
Therefore, we get 7 as unit digit of the cube root of 4913.
We know 13 = 1 and 23 = 8 , 1 > 4 > 8.
Thus, 1 is taken as ten-digit of the cube root. Therefore, ∛4913 = 17
By grouping the digits, we get 12 and 167.
Since the unit digit of the cube is 7, the unit digit of the cube root is 3.
Therefore, 3 is the unit digit of the cube root of 12167
We know 23 = 8 and 33 = 27, 8 > 12 > 27.
Thus, 2 is taken as the tenth digit of the cube root.
Therefore, ∛12167= 23
By grouping the digits, we get 32 and 768.
Since, the unit digit of the cube is 8, the unit digit of the cube root is 2.
Therefore, 2 is the unit digit of the cube root of 32768.
We know 33 = 27 and 43 = 64 , 27 > 32 > 64.
Thus, 3 is taken as ten-digit of the cube root.
Therefore, ∛32768 = 32

Q2: Find the cube root of 13824 by prime factorisation method.
Sol:
First let us prime factorise 13824:
13824 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3 × 3
= 2³ × 2³ × 2³ × 3³
∛13824 = 2 × 2 × 2 × 3
= 24

Q3: Find the smallest number by which 128 must be divided to obtain a perfect cube.
Sol: The prime factorisation of 128 gives:
128 = 2 × 2 × 2 × 2 × 2 × 2 × 2
Now, if we group the factors in triplets of equal factors,
128 = (2 × 2 × 2) × (2 × 2 × 2) × 2
Here, 2 cannot be grouped into triples of equal factors.
Therefore, we will divide 128 by 2 to get a perfect cube

Q4: Find the cube of 3.5.
Sol: 
3.5³ = 3.5 x 3.5 x 3.5

= 12.25 x 3.5

= 42.875

Q5: Is 392 a perfect cube? If not, find the smallest natural number by which 392 should be multiplied so that the product is a perfect cube.
Sol: The prime factorisation of 392 gives:
392 = 2 x 2 x 2 x 7 x 7
Since, we can see, number 7 cannot be paired in a group of three. Therefore, 392 is not a perfect cube.
To make it a perfect cube, we have to multiply the 7 by the original number.
Thus,
2 x 2 x 2 x 7 x 7 x 7 = 2744, which is a perfect cube, such as 23 x 73 or 143.
Hence, the smallest natural number which should be multiplied to 392 to make a perfect cube is 7.

Q6: Parikshit makes a cuboid of plasticine of sides 5 cm, 2 cm, 5 cm. How many such cuboids will he need to form a cube?
Sol:
Given, side of the cube is 5 cm, 2 cm and 5 cm.
Therefore, volume of cube = 5×2×5 = 50
The prime factorisation of 50 = 2×5×5
Here, 2, 5 and 5 cannot be grouped into triples of equal factors.
Therefore, we will multiply 50 by 2×2×5 = 20 to get perfect square.
Hence, 20 cuboid is needed.

Q7: Find the cube root of 17576 through estimation.
Sol:
The given number is 17576.
Step 1: Form groups of three digits starting from the rightmost digit.

• Here, one group has three digits i.e., 576 whereas the other group has only two digits i.e.,17.

Step 2: Take 576.

• The digit 6 is at its one’s place.
• We will take here the one’s place of the required cube root as 6.

Step 3: Take the other group, i.e., 17.

• Cube of number 2 is 8 and cube of number 3 is 27.
  17 lies between 8 and 27.
  The smaller number between 2 and 3 is 2.
  The one’s place of 2 is 2 itself.
  Now, take digit 2 as ten’s place of the required cube root.

Hence, the cube root of 17576 is;
∛17576 = 26

Q8: Find the smallest number by which each of the following numbers must be multiplied to obtain a perfect cube.
(a) 72
(b) 675
Sol:
(a) For 72:
Prime factorization of 72 = 2³ x 3²
To make it a perfect cube, we need one more factor of 2 and one more factor of 3.
So, the smallest number by which 72 must be multiplied is 2 x 3 = 6.
(b) For 675:
Prime factorization of 675 = 3³ x 5²
To make it a perfect cube, we need one more factor of 3.
So, the smallest number by which 675 must be multiplied is 3.
Therefore:
(a) The smallest number for 72 is 6.
(b) The smallest number for 675 is 3.

Q9: Find the cube root of 8000.
Sol: The cube root of a number is a value that, when multiplied by itself three times, gives the original number. To find the cube root of 8000, we can calculate:
Cube root of 8000 ≈ 20
So, the cube root of 8000 is approximately 20.

Q10: Find the cube root of each of the following numbers by prime factorisation method.
(a) 91125
(b) 110592
Sol:
(a) For 91125:
Prime factorization of 91125 = 3² x 5³ x 11²
Now, we group the factors in sets of three:
91125 = (3² x 5³ x 11²) = (3 x 5 x 11) x (3 x 5 x 11) x (3 x 5 x 11)
So, the cube root of 91125 is 3 x 5 x 11 = 165.
(b) For 110592:
Prime factorization of 110592 = 2⁸ x 3⁵
Now, we group the factors in sets of three:
110592 = (2³ x 3³) x (2³ x 3³) x (2²)
So, the cube root of 110592 is 2³ x 3³ = 24.
Therefore:
(a) The cube root of 91125 is 165.
(b) The cube root of 110592 is 24.