If a1 - 3-b1 - 3-c1 - 3-0- thena a-b-c-0b a-b-c3-27 a b cc a-b-c-3 a b cd a3-b3-c3-0

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Identity (x+y)(x-y)

If a1 / 3+b1 ...

Question

If a¹^/3 + b¹^/3 + c¹^/3 = 0, then

(a) a + b + c = 0

(b) (a + b + c)³ =27abc

(c) a + b + c = 3abc

(d) a³ + b³ + c³ = 0

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Solution

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a + b + c = 0

a^{3} + b^{3} + c^{3}

⎡ ⎢ ⎣ \begin{matrix} a_{1} & a_{2} & a_{3} b_{1} & b_{2} & b_{3} c_{1} & c_{2} & c_{3} \end{matrix} ⎤ ⎥ ⎦

¯ ¯ ¯ a = (a_{1}, a_{2}, a_{3}); ¯ ¯ b = (b_{1}, b_{2}, b_{3})

¯ ¯ c = (c_{1}, c_{2}, c_{3})

> 0

\frac{a^{3}}{b^{3}} + \frac{b^{3}}{c^{3}} + \frac{c^{3}}{a^{3}} \geq 3

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

{(a + b + c)}^{3}

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