Lời giải:
Chia cả hai vế cho \(5^x\):
\(\left(\frac{2}{5}\right)^x+\left(\frac{3}{5}\right)^x=1\)
Nếu \(x>1\)
Do \(\frac{2}{5}<1;\frac{3}{5}<1 \Rightarrow \left(\frac{2}{5}\right)^x<\frac{2}{5}; \left(\frac{3}{5}\right)^x< \frac{3}{5}\)
\(\Rightarrow \left(\frac{2}{5}\right)^x+\left(\frac{3}{5}\right)^x< \frac{2}{5}+\frac{3}{5}=1\) (vô lý)
Nếu \(x<1 \)
Do \(\frac{2}{5}; \frac{3}{5}<1 \Rightarrow \left(\frac{2}{5}\right)^x>\frac{2}{5}; \left(\frac{3}{5}\right)^x>\frac{3}{5}\)
\(\Rightarrow \left(\frac{2}{5}\right)^x+\left(\frac{3}{5}\right)^x>\frac{2}{5}+\frac{3}{5}=1\) (vô lý)
Từ 2 TH trên suy ra \(x=1\)