b) \( \frac{\frac{2}{9}-\left(\frac{3}{2}\right)^{-1}}{5^{0}-5: 4^{-2}} \)

To solve the expression \( \frac{\frac{2}{9}-\left(\frac{3}{2}\right)^{-1}}{5^{0}-5: 4^{-2}} \), let's break it down step by step. 1. **Calculate \(\left(\frac{3}{2}\right)^{-1}\)**: - The reciprocal of \(\frac{3}{2}\) is \(\frac{2}{3}\). 2. **Calculate \(\frac{2}{9} - \left(\frac{3}{2}\right)^{-1}\)**: - This becomes \(\frac{2}{9} - \frac{2}{3}\). - To subtract these fractions, convert \(\frac{2}{3}\) to a fraction with a denominator of 9: \(\frac{2}{3} = \frac{6}{9}\). - Now, \(\frac{2}{9} - \frac{6}{9} = \frac{-4}{9}\). 3. **Calculate \(5^{0}\)**: - Any non-zero number raised to the power of 0 is 1, so \(5^0 = 1\). 4. **Calculate \(5 : 4^{-2}\)**: - First, calculate \(4^{-2} = \frac{1}{4^2} = \frac{1}{16}\). - Then \(5 : \frac{1}{16} = 5 \times 16 = 80\). 5. **Combine these results**: - Now we have \(1 - 80 = -79\). 6. **Final calculation**: - Substitute back into the original expression: \[\frac{\frac{-4}{9}}{-79} = \frac{4}{9 \times 79} = \frac{4}{711}\]. The final value of the expression is \(\frac{4}{711}\).