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x^1/3 + y^1/3 + z^1/3 = 0 find (x + y + z)^3

x^1/3 + y^1/3 + z^1/3 = 0
find (x + y + z)^3
 

Grade:10

1 Answers

Arun Kumar IIT Delhi
askIITians Faculty 256 Points
6 years ago
Hello Student
\\$Let $x^{1/3} = a, y^{1/3} = b, z^{1/3} = c \\$The question then becomes, if $(a+b+c) = 0 \\$ then what is the value of $(a^3+b^3+c^3)^3. \\$We have a+b $= -c. \\=> (a+b)^3 = (-c)^3 = -c^3. \\=> a^3+b^3+3*a*(b^2)+3*(a^2)*b = -c^3 \\=> a^3+b^3+c^3 = -{ 3*a*(b^2)+3*(a^2)*b } = -3*a*b*(a+b) = -3*a*b*(-c) = 3abc. \\=> (a^3+b^3+c^3)^3 = (3abc)^3 = 27(a^3)(b^3)(c^3) = 27xyz.
Thanks & Regards
Arun Kumar
Btech, IIT Delhi
Askiitians Faculty

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