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Show that the following integers are cubes of negative integers. Also, find the integer whose cube is the given integer:
(i) $-5832$
(ii) $-2744000$
To find:
We have to show that the given numbers are cubes of negative integers and find the integer whose cube is the given integer.
Solution:
(i) $-5832=-(2\times2\times2\times3\times3\times3\times3\times3\times3)$
$=-[(2^3)\times(3^3)\times(3^3)]$
$=-(2\times3\times3)^3$
$=-(18)^3$
All the factors of 5832 can be grouped in triplets of equal factors completely.
Therefore,
$-5832$ is a perfect cube of a negative integer. It is the cube of $-18$.
(ii) $-2744000=-(2\times2\times2\times2\times2\times2\times5\times5\times5\times7\times7\times7)$
$=-[(2^3)\times(2^3)\times(5^3)\times(7^3)]$
$=-(2\times2\times5\times7)^3$
$=-(140)^3$
All the factors of 2744000 can be grouped in triplets of equal factors completely.
Therefore,
$-2744000$ is a perfect cube of a negative integer. It is the cube of $-140$.