Simplify:
$| (25)^{3} - (75)^{3} + (50)^{3} |$

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Solution

${(25)}^{3} + {(- 75)}^{3} + {(50)}^{3}$ is of the form $a^{3} + b^{3} + c^{3}$

$a^{3} + b^{3} + c^{3} = (a + b + c) (a^{2} + b^{2} + c^{2} - a b - b c - c a) + 3 a b c$

where $a = 25, b = - 75$ and $c = 50$

${(25)}^{3} + {(- 75)}^{3} + {(50)}^{3}$
$= (25 - 75 + 50) ({(25)}^{2} + {(- 75)}^{2} + {(50)}^{2} - (25) (- 75) - (- 75) (50) - (50) (25)) + 3 (25) (- 75) (50)$
$= 0 \times ({(25)}^{2} + {(- 75)}^{2} + {(50)}^{2} - (25) (- 75) - (- 75) (50) - (50) (25)) - 281250$
$= - 281250$

$| (25)^{3} - (75)^{3} + (50)^{3} | = | - 281250 | = 281250$

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