Đề ??

\(bx^2=ay^{2^{ }}=\dfrac{x^2}{\dfrac{1}{b}}=\dfrac{y^2}{\dfrac{1}{a}}=\dfrac{x^2+y^2}{\dfrac{a+b}{ab}}=\dfrac{ab}{a+b}.\)

\(\Leftrightarrow\dfrac{x^2}{a}=\dfrac{1}{a+b}=\dfrac{y^2}{b}.\)

\(\dfrac{x^{2016}}{a^{1008}}+\dfrac{y^{2016}}{b^{1008}}=\left(\dfrac{x^2}{a}\right)^{1008}+\left(\dfrac{y^2}{b}\right)^{1008}=2.\left(\dfrac{1}{a+b}\right)^{1008}=\dfrac{2}{\left(a +b\right)^{1008}}\left(dpcm\right)\)

Theo bài ra ta có:

\(bx^2=ay^2\) \(\Rightarrow\dfrac{x^2}{a}=\dfrac{y^2}{b}\)

\(x^2+y^2=1\)

Áp dụng tính chất dãy tỉ số bằng nhau ta được:

\(\dfrac{x^2}{a}=\dfrac{y^2}{b}=\dfrac{x^2+y^2}{a+b}=\dfrac{1}{a+b}\)

\(\dfrac{x^2}{a}=\dfrac{y^2}{b}=\dfrac{1}{a+b}\) \(\left(1\right)\)

Từ \(\left(1\right)\) suy ra :

\(\dfrac{x^{2016}}{a^{1008}}+\dfrac{y^{2016}}{b^{1008}}\) \(=\dfrac{\left(x^2\right)^{1008}}{a^{1008}}+\dfrac{\left(y^2\right)^{1008}}{b^{1008}}\)

\(=\left(\dfrac{x^2}{a}\right)^{1008}+\left(\dfrac{y^2}{b}\right)^{1008}\)

\(=\left(\dfrac{1}{a+b}\right)^{1008}+\left(\dfrac{1}{a+b}\right)^{1008}\)

\(=2\cdot\left(\dfrac{1}{a+b}\right)^{1008}\)

\(=2\cdot\dfrac{1^{1008}}{\left(a+b\right)^{1008}}\)

\(=2\cdot\dfrac{1}{\left(a+b\right)^{1008}}\)

\(=\dfrac{2}{a+b}^{1008}\)

Vậy \(\dfrac{x^{2016}}{a^{1008}}+\dfrac{y^{2016}}{b^{1008}}=\dfrac{2}{a+b}^{1008}\)

Ta có: \(\hept{\begin{cases}x^2+y^2=1\\\frac{x^4}{a}+\frac{y^4}{b}=\frac{\left(x^2+y^2\right)^2}{a+b}\end{cases}}\)

\(\Leftrightarrow b\left(a+b\right)x^4+a\left(a+b\right)y^4=ab\left(x^4+2x^2y^2+y^4\right)\)

\(\Leftrightarrow b^2x^4+a^2y^4-2abx^2y^2=0\)

\(\Leftrightarrow\left(bx^2-ay^2\right)^2=0\)

\(\frac{x^2}{a}=\frac{y^2}{b}=\frac{x^2+y^2}{a+b}=\frac{1}{a+b}\)

\(\Rightarrow\frac{x^{2016}}{a^{1008}}=\frac{y^{2016}}{b^{1008}}=\frac{1}{\left(a+b\right)^{1008}}\)

\(\Rightarrow\frac{x^{2016}}{a^{1008}}+\frac{y^{2016}}{b^{21008}}=\frac{2}{\left(a+b\right)^{1008}}\)

Em vào câu hỏi tương tự tham khảo: 

Ta có: \(x^2+y^2=1\Leftrightarrow x^4+2x^2y^2+y^4=1\)

Khi đó: \(\frac{x^4}{a}+\frac{y^4}{b}=\frac{x^4+2x^2y^2+y^4}{a+b}\)

<=> \(\left(a+b\right)\left(\frac{x^4}{a}+\frac{y^4}{b}\right)=x^4+2x^2y^2+y^4\)

<=> \(\frac{b}{a}x^4+\frac{a}{b}y^4=2x^2y^2\)

<=> \(\frac{x^4}{a^2}+\frac{y^4}{b^2}-\frac{2x^2y^2}{ab}=0\)

<=> \(\left(\frac{x^2}{a}-\frac{y^2}{b}\right)^2=0\)

<=> \(\frac{x^2}{a}=\frac{y^2}{b}=\frac{x^2+y^2}{a+b}=\frac{1}{a+b}\)( dãy tỉ số bằng nhau)

Khi đó: \(\frac{x^{2016}}{a^{1008}}+\frac{y^{2016}}{b^{1008}}=2\frac{x^{2016}}{a^{1008}}=\frac{2}{\left(a+b\right)^{1008}}\)

2. Đặt c + d = x

Ta có: \(a+b+c+d=0\Rightarrow a+b+x=0\Rightarrow a^3+b^3+c^3+d^3=3abx\)

\(\Rightarrow a^3+b^3+c^3+d^3+3cd\left(c+d\right)=3ab\left(c+d\right)\)

\(\Rightarrow a^3+b^3+c^3+d^3=3ab\left(c+d\right)-3cd\left(c+d\right)=3\left(ab-cd\right)\left(c+d\right)\)

Câu 4:

      \(a^{2016}+b^{2016}+c^{2016}=a^{1008}b^{1008}+b^{1008}c^{1008}+c^{1008}+a^{1008}\)

\(\Rightarrow2a^{2016}+2b^{2016}+2c^{2016}-2a^{1008}b^{1008}-2b^{1008}c^{1008}-2c^{1008}a^{1008}=0\)

\(\Rightarrow\left(a^{1008}-b^{1008}\right)^2+\left(b^{1008}-c^{1008}\right)^2+\left(c^{1008}-a^{1008}\right)^2=0\)

\(\Rightarrow a^{1008}=b^{1008},b^{1008}=c^{1008},c^{1008}=a^{1008}\)

\(\Rightarrow a=b,b=c,c=a\) (vì a,b,c > 0 nên \(a\ne-b,b\ne-c,c\ne-a\) )

\(\Rightarrow a-b=0,b-c=0,a-c=0\)

Thay vào A ta tính được A = 0

\(a^{2016}+b^{2016}+c^{2016}=a^{1008}b^{1008}+b^{1008}c^{1008}+c^{1008}a^{1008}\)

\(\Rightarrow2a^{2016}+2b^{2016}+2c^{2016}=2a^{1008}b^{1008}+2b^{1008}c^{1008}+2c^{1008}a^{1008}\)

\(\Rightarrow\left(a^{2016}-2a^{1008}b^{1008}+b^{1008}\right)+\left(b^{2016}-2b^{1008}c^{1008}+c^{1008}\right)\)\(+\left(c^{2016}-2c^{1008}a^{1008}+a^{2016}\right)=0\)

\(\Rightarrow\left(a^{1008}-b^{1008}\right)^2+\left(b^{1008}-c^{1008}\right)^2+\left(c^{1008}-a^{1008}\right)=0\)

Vì \(\hept{\begin{cases}\left(a^{1008}-b^{1008}\right)^2\ge0\\\left(b^{1008}-c^{1008}\right)^2\ge0\\\left(c^{1008}-a^{1008}\right)^2\ge0\end{cases}}\)

\(\Rightarrow\left(a^{1008}-b^{1008}\right)^2+\left(b^{1008}-c^{1008}\right)^2+\left(c^{1008}-a^{1008}\right)^2\ge0\)

Dấu " = " xảy ra: \(\Leftrightarrow\hept{\begin{cases}a^{1008}-b^{1008}=0\\b^{1008}-c^{1008}=0\\c^{1008}-a^{1008}=0\end{cases}\Leftrightarrow}\hept{\begin{cases}a^{1008}=b^{1008}\\b^{1008}=c^{1008}\\c^{1008}=a^{1008}\end{cases}\Leftrightarrow}a=b=c\)

Thay a=b=c vào A ta có: \(A=\left(a-a\right)^{15}+\left(a-a\right)^{2015}+\left(a-a\right)^{2016}=0\)