Cách 1:
\(+\left(a^2+b^2+c^2\right)^2=a^4+b^4+c^4+2\left(a^2b^2+b^2c^2+c^2a^2\right)\)
\(+0=\left(a+b+c\right)^2=a^2+b^2+c^2+2\left(ab+bc+ca\right)=14+2\left(ab+bc+ca\right)\)
\(\Rightarrow ab+bc+ca=-7\)
\(+\left(-7\right)^2=\left(ab+bc+ca\right)^2=a^2b^2+b^2c^2+c^2a^2+2\left(ab.bc+bc.ca+ca.ab\right)\)
\(=a^2b^2+b^2c^2+c^2a^2+2abc\left(a+b+c\right)=a^2b^2+b^2c^2+c^2a^2+2abc.0\)
\(\Rightarrow a^2b^2+b^2c^2+c^2a^2=49\)
Từ các điều trên suy ra:
\(14^2=a^4+b^4+c^4+2.49\)
\(\Rightarrow a^4+b^4+c^4=14^2-2.49=98\)
Cách 2:
\(+a+b+c=0\Rightarrow a+b=-c\)
\(+14=a^2+b^2+c^2=a^2+b^2+\left(-a-b\right)^2=a^2+b^2+a^2+b^2+2ab=2\left(a^2+b^2+ab\right)\)
\(\Rightarrow a^2+b^2+ab=7\)
\(+a^4+b^4+c^4=a^4+b^4+\left[-\left(a+b\right)\right]^4=\left(a^2+b^2\right)^2-2a^2b^2+\left(a^2+b^2+2ab\right)^2\)
\(=\left(a^2+b^2\right)^2-2a^2b^2+\left(a^2+b^2\right)^2+4\left(a^2+b^2\right).ab+4a^2b^2\)
\(=2\left(a^2+b^2\right)^2+4\left(a^2+b^2\right).ab+2a^2b^2\)
\(=2\left(a^2+b^2+ab\right)^2\)
\(=2.7^2=98\)
