Question

Which of the following numbers are not perfect cubes: 

1. 216 
2. 128 
3.  1000 
4.  100 
5.  46656

Answer 1

(i) \(216\)

Prime factors of \(216 = 2 × 2 × 2 × 3 × 3 × 3\)
Here all factors are in groups of \(3\)’s (in triplets)
Therefore, \(216\) is a perfect cube number.

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(ii) \(128\)

Prime factors of \(128 = 2× 2 × 2 × 2× 2 × 2× 2\)
Here one factor \(2\) does not appear in a \(3\)’s group
Therefore, \(128\) is not a perfect cube.

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(iii) \(1000\)

Prime factors of \(1000 = 2\times2\times2\times5\times5\times5\)
Here all factors appear in \(3\)’s group.
Therefore, \(1000\) is a perfect cube.

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(iv) \(100\)

Prime factors of \( 100 = 2 \times 2 \times 5 \times 5\)
Here all factors do not appear in \(3\)’s group.
Therefore, \(100\) is not a perfect cube.

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(v) \(46656\)

Prime factors of \(46656 =2 \times2\times2\times2\times2\times2\times3\times3\times3\times3\times3\times3\)
Here all factors appear in \(3\)’s group.
Therefore, \(46656\) is a perfect cube.