Question

Cho a, b, c thỏa mãn : a² - 20b + 81 = 0; b² + 18c + 9 = 0; c² + 6a + 100 = 0. Tính giá trị biểu thức M = ( a + 2 )²⁰¹⁷ + ( b - 9 )²⁰¹⁸ +

( c + 9 )²⁰¹⁸

Answer 1

Cộng vế với vế, ta có: 

       \(a^2-20b+81+b^2+18c+9+c^2+6a+100=0\)

\(\Rightarrow\left(a^2+6a+9\right)+\left(b^2-20b+100\right)+\left(c^2+18c+81\right)=0\)

\(\Rightarrow\left(a^2+2.a.3+3^2\right)+\left(b^2-2.b.10+10^2\right)+\left(c^2+2.9.c+9^2\right)=0\)

\(\Rightarrow\left(a+3\right)^2+\left(b-10\right)^2+\left(c+9\right)^2=0\)

\(\Rightarrow\hept{\begin{cases}a+3=0\\b-10=0\\c+9=0\end{cases}\Rightarrow}\hept{\begin{cases}a=-3\\b=10\\c=-9\end{cases}}\)

Khi đó: \(M=\left(a+2\right)^{2017}+\left(b-9\right)^{2018}+\left(c+9\right)^{2018}\)

               \(=\left(-3+2\right)^{2017}+\left(10-9\right)^{2018}+\left(-9+9\right)^{2018}\)

               \(=-1+1+0=0\)