Question

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Answer 1

Question:-

Examine, if 2160 is a perfect cube. If not find the smallest number by which it must be divided so that the quotient is a perfect cube. Also find the cube root of new cube number.

Answer:-

Taking LCM of 2160 we get,

5 |2160

3 |432

2 |144

2 |72

2 |36

2 |18

3 |9

3 |3

|1

⟶ 2160 = (2 × 2 × 2) × (3 × 3 × 3) × (5 × 2)

We observe that,

2 , 3 exist in a group of three but 5 × 2 does not.

So, 2160 is not a perfect cube.

So, 2160 should be divided by 10 , to get a quotient which is a perfect cube.

⟶ 2160/10 = 216.

Now,

LCM of 216 is ,

3 | 216

3 |72

3 |24

2 |8

2 |4

2 | 2

| 1

⟶ 216 = (3 × 3 × 3) × (2 × 2 × 2)

⟶ ³√216 = ³√(3)³ × (2)³

⟶ ³√216 = ³√(3 × 2)³

⟶ ³√216 = 6

∴

• The smallest number by which 2160 is divided to get a perfect cube is 10.

• Cube root of the new cube number is 6.