Class 8 Maths

Cube and Cube Root

Exercise 7.1

Question 1: Which of the following numbers are not perfect cubes?

(i) 216

Answer: `216 = 2 xx 2 xx 2 xx 27`
`= 2 xx 2 xx 2 xx 3 xx 3 xx 3`

`3sqrt(216)=2xx3=6`

As number of 2s and 3s is 3 in the factorization so it is a perfect cube.

(ii) 128

Answer: `128 = 2 xx 2 xx 2 xx 2 xx 2 xx 2 xx 2`
Number of 2s is 7 and 7 is not divisible by three so 128 is not a perfect cube

(iii) 1000

Answer: `1000 = 2 xx 2 xx 2 xx 5 xx 5 xx 5`
Number of 2s and 5s is 3 each so 1000 is a perfect cube.


(iv) 100

Answer: `100 = 2 xx 2 xx 5 xx 5`
Number of 2s and 5s is 2 each and not 3 so 100 is not a perfect cube.

(v) 46656

Answer: `46656 = 2 xx 2 xx 2 xx 5832`
`= 2 xx 2 xx 2 xx 2 xx 2 xx 2 xx 729`
`= 2 xx 2 xx 2 xx 2 xx 2 xx 2 xx 3 xx 3 xx 3 xx 3 xx 3 xx 3`
Number of 2s and 3s is 6 each and 6 is divisible by 3 so 46656 is a perfect cube

Question 2: Find the smallest number by which each of the following numbers must be multiplied to obtain a perfect cube.

(i) 243

Answer: `243 = 3 xx 3 xx 3 xx 3 xx 3`
Number of 3s is 5, so we need to another 3 in the factorization to make 243 a perfect cube. 243 multiplied by 3 will be a perfect cube.

(ii) 256

Answer: `256 = 2 x 2 xx 2 xx 2 xx 2 xx 2 xx 2 xx 2`
Number of 2s is 8 so 256 needs to be multiplied by 2 to become a perfect cube.

(iii) 72

Answer: `72 = 2 xx 2 xx 2 xx 3 xx 3`
Number of 2s is 3 and that of 3s is 2, so 72 needs to be multiplied by 3 to become a perfect cube.


(iv) 675

Answer: `675 = 5 xx 5 xx 27 = 5 xx 5 xx 3 xx 3 xx 3`
675 needs to be multiplied by 5 to become a perfect cube.

(v) 100

Answer: `100 = 10 x 10`
100 needs to be multiplied by 10 to become a perfect cube.

Question 3: Find the smallest number by which each of the following numbers must be divided to obtain a perfect cube.

(i) 81

Answer: `81 = 3 xx 3 xx 3 xx 3`
81 needs to be divided by 3 to become a perfect cube.

(ii) 128

Answer: `128 = 2 xx 2 xx 2 xx 2 xx 2 xx 2 xx 2`
128 needs to be divided by 2 to become a perfect cube.

(iii) 135

Answer: `135 = 5 xx 3 xx 3 xx 3`
135 needs to be divide by 5 to become a perfect cube.


(iv) 192

Answer: `192 = 2 xx 2 xx 2 xx 2 xx 2 xx 2 xx 3`
192 needs to be divided by 3 to become a perfect cube.

(v) 704

Answer: `704 = 2 xx 2 xx 2 xx 2 xx 2 xx 2 xx 11`
704 needs to be divided by 11 to become a perfect cube.

Question 4: 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?

Answer: To find the answer we need to calculate the LCM of 5 and 2. The cube with each sides measuring equal to the LCM will give the dimensions of required cube.
LCM of 5 and 2 = 10
Volume of cuboid `= 5 xx 5 xx 2=50` cubic cms
Volume of bigger cube `=10^3=1000` cubic cms
Number of cuboids required to make the bigger cube

`=text(Volume of bigger cube)/text(Volume of cuboid)`

`=(1000)/(50)=20`


Ex 7.1

Ex 7.2