Farewell 2016

Algebra Level 3

If a 3 + b 3 + c 3 = 0 \sqrt[3]{a} + \sqrt[3]{b} + \sqrt[3]{c} = 0 , then ( a + b + c ) 3 (a+b+c)^3 must simplify to which of the following expressions?

0 0 − 1 -1 81 a b c 81 abc 27 a b c 27abc 1 1 3 a b c 3abc 9 a b c 9abc

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Krishna Shankar by Krishna Shankar , 23, India

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2 solutions

From the identity

( a 3 ) 3 + ( b 3 ) 3 + ( c 3 ) 3 = ( a 3 + b 3 + c 3 ) [ ( a 3 ) 2 + ( b 3 ) 2 + ( c 3 ) 2 − a 3 b 3 − b 3 c 3 − c 3 a 3 ] + 3 a 3 b 3 c 3 a + b + c = ( 0 ) [ ( a 3 ) 2 + ( b 3 ) 2 + ( c 3 ) 2 − a 3 b 3 − b 3 c 3 − c 3 a 3 ] + 3 a b c 3 = 3 a b c 3 ⟹ ( a + b + c ) 3 = 27 a b c \begin{aligned} (\sqrt [3] a)^3 + (\sqrt [3] b)^3 + (\sqrt [3] c)^3 & = {\color{#D61F06}(\sqrt [3] a + \sqrt [3] b + \sqrt [3] c)} [(\sqrt [3] a)^2 + (\sqrt [3] b)^2+ (\sqrt [3] c)^2 - \sqrt [3] a \sqrt [3] b - \sqrt [3] b \sqrt [3] c - \sqrt [3] c \sqrt [3] a] + 3 \sqrt [3] a \sqrt [3] b \sqrt [3] c \\ a + b + c & = {\color{#D61F06}(0)} [(\sqrt [3] a)^2 + (\sqrt [3] b)^2+ (\sqrt [3] c)^2 - \sqrt [3] a \sqrt [3] b - \sqrt [3] b \sqrt [3] c - \sqrt [3] c \sqrt [3] a] + 3 \sqrt [3] {abc} \\ & = 3\sqrt [3] {abc} \\ \implies (a+b+c)^3 & = \boxed{27abc} \end{aligned}

2 Helpful 0 Interesting 0 Brilliant 1 Confused

If x + y + z = 0 , x 3 + y 3 + z 3 = 3 x y z x+y+z=0, x^3+y^3+z^3=3xyz

Putting x = a 3 , y = b 3 , z = c 3 , x=\sqrt[3]{a}, y=\sqrt[3]{b}, z=\sqrt[3]{c},

a + b + c = 3 a 3 b 3 c 3 a+b+c=3\sqrt[3]{a}\sqrt[3]{b}\sqrt[3]{c}

⟹ ( a + b + c ) 3 = 27 a b c \implies (a+b+c)^3=\boxed{27abc}

0 Helpful 0 Interesting 0 Brilliant 0 Confused

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