Cubes and Cube Roots
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Check which of the following are perfect cubes.

(i) 2700 (ii) 16000 (iii) 64000 (iv) 900

(v) 125000 (vi) 36000 (vii) 21600 (viii) 10000

(ix) 27000000 (x) 1000
What pattern do you observe in these perfect cubes?

(i) We have 2700 = 2 x 2 x 3 x 3 x 3 x 5 x 5
We do  not get complete triples of prime factors, i.e. 2 x 2 and 5 x 5 are left over.
(i) We have 2700 = 2 x 2 x 3 x 3 x 3 x 5 x 5We do  not get complete
∴ 2700 is not a perfect cube.
(ii) we  have 1600 = 2 x 2 x 2 x 2 x 2 x 2 x 5 x 5
After grouping id 3's we get 5 x 5 which is ungrouped in triples.
(i) We have 2700 = 2 x 2 x 3 x 3 x 3 x 5 x 5We do  not get complete
∴ 1600 is not a perfect cube.
(iii) We have 64000 = 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 5 x 5 x 5
Since, we get groups of triples.
(i) We have 2700 = 2 x 2 x 3 x 3 x 3 x 5 x 5We do  not get complete
∴ 64000 is a perfect cube.
(iv) We have 900 = 2 x 2 x 3 x 3 x 5 x 5
which are ungrouped in triples.

(i) We have 2700 = 2 x 2 x 3 x 3 x 3 x 5 x 5We do  not get complete
∴ 900 is not a perfect cube.
(v) We have 125000 = 2  x 2 x 2 x 5 x 5 x 5 x 5 x 5 x 5
As we get all the prime factors in the group of triples.
(i) We have 2700 = 2 x 2 x 3 x 3 x 3 x 5 x 5We do  not get complete
∴ 125000 is a perfect cube
(vi) we have 36000 = 2 x 2 x 2 x 2 x 2 x 3 x 3 x 5 x 5 x 5
While grouping the prime factors of 36000 in triples, we are left over with 2 x 2 and 3 x 3
(i) We have 2700 = 2 x 2 x 3 x 3 x 3 x 5 x 5We do  not get complete
∴ 36000 is not a perfect cube
(vii) We have 21600 = 2 x 2 x 2 x 2 x 2 x 3 x 3 x 3 x 5 x 5
While grouping the prime factors of 21600 in triples, we are left with 2 x 2 and 5 x 5.
(i) We have 2700 = 2 x 2 x 3 x 3 x 3 x 5 x 5We do  not get complete
∴ 21600 is not a perfect cube
(viii) we have 10000 = 2 x 2 x 2 x 2 x 5 x 5 x 5 x 5
While grouping the prime factors into triples, we are left over with 2 and 5.

(i) We have 2700 = 2 x 2 x 3 x 3 x 3 x 5 x 5We do  not get complete
∴ 10000 is not a perfect cube
(ix) we have 27000000 = 2 x 2 x 2 x 2 x 2 x 2 x 3 x 3 x 3 x 5 x 5 x 5 x 5 x 5 x 5

(i) We have 2700 = 2 x 2 x 3 x 3 x 3 x 5 x 5We do  not get complete
Since, all the prime factors of 27000000 appear in groups of triples.
∴ 27000000 is a perfect cube.
(x) We have 1000 = 2 x 2 x 2 x 5 x 5 x 5

(i) We have 2700 = 2 x 2 x 3 x 3 x 3 x 5 x 5We do  not get complete
∴ 1000 is a perfect cube.
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