1) \( 36^{\log _{6} 5}+10^{1-\log _{10} 2}-8^{\log _{2} 3} \)

To simplify \( 36^{\log_{6} 5} + 10^{1-\log_{10} 2} - 8^{\log_{2} 3} \), we can tackle each term one at a time. 1. **Calculating \( 36^{\log_{6} 5} \)**: \[ 36 = 6^2 \implies 36^{\log_{6} 5} = (6^2)^{\log_{6} 5} = 6^{2 \log_{6} 5} = 6^{\log_{6} (5^2)} = 5^2 = 25 \] 2. **Calculating \( 10^{1-\log_{10} 2} \)**: We can rewrite \( 1 - \log_{10} 2 \) as \( \log_{10} (10) - \log_{10} (2) = \log_{10} \left( \frac{10}{2} \right) = \log_{10} 5 \). \[ 10^{1-\log_{10} 2} = 10^{\log_{10} 5} = 5 \] 3. **Calculating \( 8^{\log_{2} 3} \)**: \[ 8 = 2^3 \implies 8^{\log_{2} 3} = (2^3)^{\log_{2} 3} = 2^{3 \log_{2} 3} = 2^{\log_{2} (3^3)} = 3^3 = 27 \] Now, compiling all of our results together: \[ 36^{\log_{6} 5} + 10^{1-\log_{10} 2} - 8^{\log_{2} 3} = 25 + 5 - 27 = 30 - 27 = 3 \] Thus, the final answer is \[ \boxed{3} \]