# Cube and Cube Root

## Exercise 7.1

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

(i) 216

**Answer:** 216 = 2 × 2 × 2 × 27

= 2 × 2 × 2 × 3 × 3 × 3

`3sqrt(216)` = 2 × 3 = 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 xx 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 xx 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`