1. \(\left(x-1\right)^{x+2}=\left(x-1\right)^{x+4}\)
\(\Rightarrow3^x+2.3^x.3+2.3^x=729\)
\(\Rightarrow3^x.\left(1+2.3+2\right)=729\)
\(\Rightarrow3^x+9=729\)
\(\Rightarrow3^x.3^2=3^6\)
\(\Rightarrow3^x=3^4\)
\(\Rightarrow x=4\)
2. Ta có: \(2\left(x+y\right)=5\left(y+z\right)=3\left(z+x\right)\)
\(\Rightarrow\dfrac{2\left(x+y\right)}{30}=\dfrac{5\left(y+z\right)}{30}=\dfrac{3\left(z+x\right)}{30}\)
\(\Rightarrow\dfrac{x+y}{15}=\dfrac{y+z}{16}=\dfrac{z+x}{10}\)
Áp dụng t/c dãy tỉ số bằng nhau ta có:
\(\dfrac{x+y}{15}=\dfrac{z+x}{10}=\dfrac{x+y-z-x}{15-10}=\dfrac{y-z}{5}\left(1\right)\)
\(\dfrac{z+x}{10}=\dfrac{y+z}{6}=\dfrac{z+x-y-z}{10-6}=\dfrac{x-y}{4}\left(2\right)\)
Từ (1), (2) => \(\dfrac{x-y}{4}=\dfrac{y-z}{5}\left(=\dfrac{z+x}{10}\right)\)
