a: \( \frac{\left.\left(\frac{3}{4}\right)^{-3}\left(\frac{3}{4}\right)^{6}\left(\frac{3}{4}\right)^{9}+\frac{256}{81}\right)^{-2}}{\left(\frac{4}{3}\right)^{-5}\left(\frac{3}{4}\right)^{1}\left[\left(\frac{3}{4}\right)^{5}\right]^{2}\left(\frac{27}{64}\right)^{3}}= \)

To simplify the expression \[ a = \frac{\left.\left(\frac{3}{4}\right)^{-3}\left(\frac{3}{4}\right)^{6}\left(\frac{3}{4}\right)^{9}+\frac{256}{81}\right)^{-2}}{\left(\frac{4}{3}\right)^{-5}\left(\frac{3}{4}\right)^{1}\left[\left(\frac{3}{4}\right)^{5}\right]^{2}\left(\frac{27}{64}\right)^{3}} \] we will simplify both the numerator and the denominator step by step. ### Step 1: Simplify the Numerator The numerator is \[ \left(\frac{3}{4}\right)^{-3}\left(\frac{3}{4}\right)^{6}\left(\frac{3}{4}\right)^{9} + \frac{256}{81} \] Using the property of exponents, we can combine the terms: \[ \left(\frac{3}{4}\right)^{-3 + 6 + 9} = \left(\frac{3}{4}\right)^{12} \] Thus, the numerator becomes: \[ \left(\frac{3}{4}\right)^{12} + \frac{256}{81} \] Next, we need to express \(\frac{256}{81}\) in terms of \(\left(\frac{3}{4}\right)^{12}\). We can rewrite \(\frac{256}{81}\) as: \[ \frac{256}{81} = \frac{4^4}{3^4} = \left(\frac{4}{3}\right)^{4} \] Now we have: \[ \left(\frac{3}{4}\right)^{12} + \left(\frac{4}{3}\right)^{4} \] To combine these, we can express \(\left(\frac{4}{3}\right)^{4}\) in terms of \(\left(\frac{3}{4}\right)^{12}\): \[ \left(\frac{4}{3}\right)^{4} = \left(\frac{3}{4}\right)^{-4} \] Thus, the numerator becomes: \[ \left(\frac{3}{4}\right)^{12} + \left(\frac{3}{4}\right)^{-4} \] ### Step 2: Simplify the Denominator The denominator is \[ \left(\frac{4}{3}\right)^{-5}\left(\frac{3}{4}\right)^{1}\left[\left(\frac{3}{4}\right)^{5}\right]^{2}\left(\frac{27}{64}\right)^{3} \] First, simplify \(\left[\left(\frac{3}{4}\right)^{5}\right]^{2}\): \[ \left(\frac{3}{4}\right)^{10} \] Now, the denominator becomes: \[ \left(\frac{4}{3}\right)^{-5} \left(\frac{3}{4}\right)^{1} \left(\frac{3}{4}\right)^{10} \left(\frac{27}{64}\right)^{3} \] Combining the powers of \(\left(\frac{3}{4}\right)\): \[ \left(\frac{4}{3}\right)^{-5} \left(\frac{3}{4}\right)^{11} \left(\frac{27}{64}\right)^{3} \] Next, simplify \(\left(\frac{27}{64}\right)^{3}\): \[ \left(\frac{27}{64}\right)^{3} = \frac{27^3}{64^3} = \frac{19683}{262144} \] Now, we can express \(\left(\frac{4}{3}\right)^{-5}\) as: \[ \left(\frac{3}{4}\right)^{5} \] Thus, the denominator becomes: \[ \left(\frac{3}{4}\right)^{5} \left(\frac{3}{4}\right)^{11} \cdot \frac{19683}{262144} = \left(\frac{3}{4}\right)^{16} \cdot \frac{19683}{262144} \] ### Step 3: Combine the Results Now we can rewrite \(a\): \[ a = \frac{\left(\left(\frac{3}{4}\right)^{12} + \