Question
Write true $(T)$ or false $(F)$ for the following statement:
$8640$ is not a perfect cube.
$8640$ is not a perfect cube.
✓
Answer
On factorising 8640 into prime factors, we got:
$8640 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 3 \times 5$
On grouping the factors in triples of equal factors, we get:
$8640 = \{2 \times 2 \times 2\} \times \{2 \times 2 \times 2\} \times \{3 \times 3 \times 3\} \times 5$
It is evident that the prime factors of $8640$ cannot be grouped into triples of equal factors such that no factor is left over. Therefore, $8640$ is not a perfect cube.
$8640 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 3 \times 5$
On grouping the factors in triples of equal factors, we get:
$8640 = \{2 \times 2 \times 2\} \times \{2 \times 2 \times 2\} \times \{3 \times 3 \times 3\} \times 5$
It is evident that the prime factors of $8640$ cannot be grouped into triples of equal factors such that no factor is left over. Therefore, $8640$ is not a perfect cube.
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