ĐK: y>0,y khác 1

a, \(M=\frac{y\sqrt{y}}{\sqrt{y}\left(\sqrt{y}-1\right)}-\frac{2y-\sqrt{y}}{\sqrt{y}\left(\sqrt{y}-1\right)}=\frac{y\sqrt{y}-2y+\sqrt{y}}{\sqrt{y}\left(\sqrt{y}-1\right)}\)

\(=\frac{\sqrt{y}\left(y-2\sqrt{y}+1\right)}{\sqrt{y}\left(\sqrt{y}-1\right)}=\frac{\sqrt{y}\left(\sqrt{y}-1\right)^2}{\sqrt{y}\left(\sqrt{y}-1\right)}=\sqrt{y}-1\)

b, Thay y vào M ta dc: \(M=\sqrt{3+\sqrt{8}}-1=\sqrt{2+2\sqrt{2}.1+1}-1\)

\(=\sqrt{\left(\sqrt{2}+1\right)^2}-1=\left|\sqrt{2}+1\right|-1=\sqrt{2}+1-1=\sqrt{2}\)

\(\hept{\begin{cases}ax+by=17\\3bx+ay=-29\end{cases}}\)

Thay x=1; y=-4 vào hệ phương trình ta có:

\(\hept{\begin{cases}x-4y=17\\-12x+y=-29\end{cases}}\)

\(\Leftrightarrow\hept{\begin{cases}x=17+4y\\-12x+3y=-29\end{cases}}\)

\(\Leftrightarrow\hept{\begin{cases}x=17+4y\\-12\left(17+4y\right)+3y=-29\end{cases}}\)

\(\Leftrightarrow\hept{\begin{cases}x=17+4y\\-204-45y=-29\end{cases}}\)

\(\Leftrightarrow\hept{\begin{cases}x=17+4y\\y=-\frac{35}{8}\end{cases}}\)

\(\Leftrightarrow\hept{\begin{cases}x=\frac{13}{9}\\y=-\frac{35}{8}\end{cases}}\)

Vậy hpt ...

Cầm máy tính ra giải là xong

???????????????????????????????

\(A=\frac{1}{a^3+b^3}+\frac{1}{a^2b}+\frac{1}{ab^2}\ge\frac{1}{\left(a+b\right)\left(a^2-ab+b^2\right)}+\frac{4}{ab\left(a+b\right)}\)

\(\ge\left(\frac{1}{a^2-ab+b^2}+\frac{1}{ab}+\frac{1}{ab}+\frac{1}{ab}\right)+\frac{1}{ab}\)

\(\ge\frac{\left(1+1+1+1\right)^2}{\left(a+b\right)^2}+\frac{1}{ab}\ge\frac{16}{\left(a+b\right)^2}+\frac{1}{\frac{\left(a+b\right)^2}{4}}\ge16+4=20\)

Đẳng thức xảy ra khi \(a=b=\frac{1}{2}\)