Question

If a^1/3+b^1/3+c^1/3=0,then prove that a+b+c=3a^1/3b^1/3c^1/3

Answer 1

Given :

$a^{1/3}+b^{1/3}+c^{1/3}=0$

$\implies a^{1/3}+b^{1/3}=-c^{1/3}$ .......................(1)

Cubing both sides :

$\implies (a^{1/3}+b^{1/3})^3=(-c^{1/3})^3$

Use formula : $(a+b)^3=a^3+b^3+3ab(a+b)$

$\implies a+b+3a^{1/3}b^{1/3}(a^{1/3}+b^{1/3})=-c$

$\implies a+b+3a^{1/3}b^{1/3}(-c^{1/3})=-c$   [ From (1) ]

$\implies a+b+c-3a^{1/3}b^{1/3}c^{1/3}=0$

$\implies a+b+c=3a^{1/3}b^{1/3}c^{1/3}$ [P.R.O.V.E.D]

Hope it helps

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