Question
If $\text{a}^{\frac{1}{3}}+\text{b}^{\frac{1}{3}}+\text{c}^{\frac{1}{3}}=0,$ than.
✓
Answer
$\text{a}^{\frac{1}{3}}+\text{b}^{\frac{1}{3}}+\text{c}^{\frac{1}{3}}=0$
$\Rightarrow\text{a}^{\frac{1}{3}}+\text{b}^{\frac{1}{3}}=-\text{c}^{\frac{1}{3}}$
$\Rightarrow\Big[(\text{a}^{\frac{1}{3}})(\text{b}^{\frac{1}{3}})\Big]^3=\Big(-\text{c}^{\frac{1}{3}}\Big)^3$
$\Rightarrow\text{a}+\text{b}+\Big[3\times\text{a}^{\frac{1}{3}}\times\text{b}^{\frac{1}{3}}\Big(\text{a}^{\frac{1}{3}}+\text{b}^{\frac{1}{3}}\Big)\Big]=-\text{c}$
$\Rightarrow\text{a}+\text{b}+3\times\text{a}^{\frac{1}{3}}\times\text{b}^{\frac{1}{3}}\Big(-\text{c}^{\frac{1}{3}}\Big)=-\text{c}$
$\Rightarrow\text{a}+\text{b}+\text{c}=3\times\text{a}^{\frac{1}{3}}\times\text{b}^{\frac{1}{3}}\times\text{c}^{\frac{1}{3}}$
$\Rightarrow(\text{a}+\text{b}+\text{c})^3=\Big(3\times\text{a}^{\frac{1}{3}}\times\text{b}^{\frac{1}{3}}\times\text{c}^{\frac{1}{3}}\Big)$
$\Rightarrow(\text{a}+\text{b}+\text{c})^3=27$ abc
$\Rightarrow\text{a}^{\frac{1}{3}}+\text{b}^{\frac{1}{3}}=-\text{c}^{\frac{1}{3}}$
$\Rightarrow\Big[(\text{a}^{\frac{1}{3}})(\text{b}^{\frac{1}{3}})\Big]^3=\Big(-\text{c}^{\frac{1}{3}}\Big)^3$
$\Rightarrow\text{a}+\text{b}+\Big[3\times\text{a}^{\frac{1}{3}}\times\text{b}^{\frac{1}{3}}\Big(\text{a}^{\frac{1}{3}}+\text{b}^{\frac{1}{3}}\Big)\Big]=-\text{c}$
$\Rightarrow\text{a}+\text{b}+3\times\text{a}^{\frac{1}{3}}\times\text{b}^{\frac{1}{3}}\Big(-\text{c}^{\frac{1}{3}}\Big)=-\text{c}$
$\Rightarrow\text{a}+\text{b}+\text{c}=3\times\text{a}^{\frac{1}{3}}\times\text{b}^{\frac{1}{3}}\times\text{c}^{\frac{1}{3}}$
$\Rightarrow(\text{a}+\text{b}+\text{c})^3=\Big(3\times\text{a}^{\frac{1}{3}}\times\text{b}^{\frac{1}{3}}\times\text{c}^{\frac{1}{3}}\Big)$
$\Rightarrow(\text{a}+\text{b}+\text{c})^3=27$ abc
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