The given expression is:

$\frac{2^{5\log \frac{1}{5}} + 9^{\log 3}}{9^{\log 3} \times 3^{\log 3}}$

Let's break this down step-by-step:

Step 1: Simplify $2^{5 \log \frac{1}{5}}$

We can use the property of logarithms $a^{\log_b c} = c^{\log_b a}$.

Here, $2^{5 \log \frac{1}{5}} = \left( \frac{1}{5} \right)^{\log 2^5} = \left( \frac{1}{5} \right)^{\log 32} = \left( \frac{1}{5} \right)^{5 \log 2}$.

So, it becomes $32^{\log 5} = 5^{-5 \log 2}$.

Step 2: Simplify $9^{\log 3}$

Using the property $a^{\log_b c} = c^{\log_b a}$, we get: $9^{\log 3} = (3^2)^{\log 3} = 3^{2 \log 3} = 3^{2}$

Step 3: Simplify $3^{\log 3}$

[ 3^{\log 3} = 3^1 =1 The full solution is:

loglog