Từ \(a+b+c=0\Leftrightarrow\left(a+b+c\right)^2=0\)
\(\Leftrightarrow a^2+b^2+c^2+2\left(ab+bc+ca\right)=0\)
\(\Leftrightarrow ab+bc+ca=\dfrac{-\left(a^2+b^2+c^2\right)}{2}=\dfrac{-1}{2}\)
\(\Leftrightarrow\left(ab+bc+ca\right)^2=\dfrac{1}{4}\)
\(\Leftrightarrow a^2b^2+b^2c^2+c^2a^2+2abc\left(a+b+c\right)=\dfrac{1}{4}\)
\(\Leftrightarrow a^2b^2+b^2c^2+c^2a^2=\dfrac{1}{4}\) (vì a+b+c=0)
Từ \(a^2+b^2+c^2=1\Leftrightarrow\left(a^2+b^2+c^2\right)^2=1\)
\(\Leftrightarrow a^4+b^4+c^4+2\left(a^2b^2+b^2c^2+c^2a^2\right)=1\)
\(\Leftrightarrow a^4+b^4+c^4+\dfrac{1}{2}=1\)
\(\Leftrightarrow a^4+b^4+c^4=\dfrac{1}{2}\)
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