If a + 2b + 3c = 0, prove that a 3 + 8b 3 + 27c 3 = 18abc.

Given, a + 2b + 3c = 0 a + 2b = -3c. Cubing both sides, (a + 2b) 3 = (-3c) 3 a 3 + (2b) 3 + 3(a)(2b)(a + 2b) = -27c 3 a 3 + 8b 3 + 6ab(-3c) = -27c 3 a 3 + 8b 3 - 18abc = -27c 3 a 3 + 8b 3 + 27c 3 = 18abc. Hence, proved that a 3 + 8b 3 + 27c 3 = 18abc.

Mathematics

If a + 2b + 3c = 0, prove that a³ + 8b³ + 27c³ = 18abc.

Expansions

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Answer

Given,

a + 2b + 3c = 0
a + 2b = -3c.

Cubing both sides,

(a + 2b)³ = (-3c)³

a³ + (2b)³ + 3(a)(2b)(a + 2b) = -27c³

a³ + 8b³ + 6ab(-3c) = -27c³

a³ + 8b³ - 18abc = -27c³

a³ + 8b³ + 27c³ = 18abc.

Hence, proved that a³ + 8b³ + 27c³ = 18abc.

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If a + 2b + 3c = 0, prove that a³ + 8b³ + 27c³ = 18abc.

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