The equation $6^{x}+8^{x}=10^{x}$ has

• A

  no real root
• B

  infinitely many rational roots
• Correct answer
  C

  exactly one real root
• D

  two distinct real roots

Solution

$6^{x}+8^{x}=10^{x}\left(\frac{3}{5}\right)^{x}+\left(\frac{4}{5}\right)^{x}=1 \Rightarrow\left(\frac{3}{5}\right)^{x}=1-\left(\frac{4}{5}\right)^{x}$ $\therefore \quad 6^{x}+8^{x}=10^{x}$ has exactly one real root

Hence, the correct answer is option: 'C'.