Question

without actually calculating the cubes find the values of the following
(-12)^3+(7)^3 +(5)^3​

Answer 1

Answer:

$( - 12) {}^{3} + {7}^{3} + {5}^{3} \\ = - (7 + 5) {}^{3} + {7}^{3} + {5}^{3} \\ use \: \: \\ (x + y) {}^{3} = {x}^{3} + {y}^{3} + 3xy(x + y) \\ = - ({7}^{3} + {5}^{3} + 3 \times 7 \times 5(7 + 5)) + {7}^{3} + {5}^{3} \\ = - 3 \times 7 \times 5 \times 12 \\ = - 1260$

Answer 2

$\huge{ \underline{ \mathfrak{ \green{ \: Answer}}}}$

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$\boxed{\red{\bold{Assume:}}}$

$a = -12$

$b = 7$

$c = 5$

$\boxed{\red{\bold{Their\: sum}}}$

$a + b + c = -12 + 7 + 5$

$a + b + c = -12 + 12$

$a + b + c = 0$

$\boxed{\red{\bold{By\: identity\:}}}$

$a^3 + b^3 + c^3 - 3abc = \\ (a+b+c)(a^2 + b^2 + c^2 + 2ab + 2bc + 2ac)$

$\boxed{\red{\bold{Here\: a+b+c = 0:}}}$

$a^3 + b^3 + c^3 -3abc = \\ 0 × (a^2 + b^2 + c^2 + 2ab + 2bc + 2ac)$

$a^3 + b^3 + c^3 - 3abc = 0$

$a^3 + b^3 + c^3 = 3abc$

$\boxed{\red{\bold{Putting \: values \:of \:a , b, c}}}$

${(-12)}^3 + 7^3 + 5^3 = 3 × -12 × 7 × 5$

${(-12)}^3 + 7^3 + 5^3 = -1260$

$\blue{\bold{\underline{Hence\: the \: answer\: is \:-1260}}}$

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