Simplify the following without the use of a calculator: \( \begin{array}{lll}7.27^{-\frac{1}{3}} \cdot \underbrace{81^{-\frac{1}{4}}} \cdot 9^{2} & \text { 8. }\left(\frac{36}{64}\right)^{-\frac{3}{2}} & \text { 9. }(2,25)^{-\frac{1}{2}}\end{array} \)

Let's simplify each expression step by step. **Expression 8: \(7.27^{-\frac{1}{3}} \cdot 81^{-\frac{1}{4}} \cdot 9^{2}\)** 1. Rewrite \(7.27\) as \((7.27)^{-\frac{1}{3}}\): - First, \(7.27\) is not a perfect cube but we can approximate or reduce it. However, let's convert it to a fraction for more precision: \[ 7.27 = \frac{727}{100} \] - Therefore, \[ 7.27^{-\frac{1}{3}} = \left(\frac{727}{100}\right)^{-\frac{1}{3}} = \frac{100^{\frac{1}{3}}}{727^{\frac{1}{3}}} \] 2. Simplify \(81^{-\frac{1}{4}}\): - \(81 = 3^4\), so: \[ 81^{-\frac{1}{4}} = (3^4)^{-\frac{1}{4}} = 3^{-1} = \frac{1}{3} \] 3. Simplify \(9^2\): - \(9 = 3^2\), thus: \[ 9^2 = (3^2)^2 = 3^{4} = 81 \] Now we put it all together: \[ \frac{100^{\frac{1}{3}}}{727^{\frac{1}{3}}} \cdot \frac{1}{3} \cdot 81 = \frac{100^{\frac{1}{3}} \cdot 81}{3 \cdot 727^{\frac{1}{3}}} \] Therefore this is simplified but requires \(100^{1/3}\) and \(727^{1/3}\) calculations to evaluate further depending on context. For now, we leave it in that form. **Expression 8 Result: \(\frac{100^{\frac{1}{3}} \cdot 81}{3 \cdot 727^{\frac{1}{3}}}\)** --- **Expression 9: \(\left(\frac{36}{64}\right)^{-\frac{3}{2}}\)** 1. Simplify \(\frac{36}{64}\): - Both can be simplified by their greatest common divisor of \(4\): \[ \frac{36}{64} = \frac{9}{16} \] 2. Now compute \(\left(\frac{9}{16}\right)^{-\frac{3}{2}}\): - The negative exponent means we flip the fraction: \[ = \left(\frac{16}{9}\right)^{\frac{3}{2}} = \frac{16^{\frac{3}{2}}}{9^{\frac{3}{2}}} \] 3. Next, calculate \(16^{\frac{3}{2}}\) and \(9^{\frac{3}{2}}\): - \(16^{\frac{3}{2}} = (4^2)^{\frac{3}{2}} = 4^3 = 64\) - \(9^{\frac{3}{2}} = (3^2)^{\frac{3}{2}} = 3^3 = 27\) Therefore, the final result is: \[ \frac{64}{27} \] **Expression 9 Result: \(\frac{64}{27}\)** --- **Expression 10: \((2.25)^{-\frac{1}{2}}\)** 1. Rewrite \(2.25\) as \(\left(\frac{9}{4}\right)\): \[ 2.25 = \frac{9}{4} \] 2. Now compute \((\frac{9}{4})^{-\frac{1}{2}}\): - This means: \[ = \left(\frac{4}{9}\right)^{\frac{1}{2}} = \frac{\sqrt{4}}{\sqrt{9}} = \frac{2}{3} \] **Expression 10 Result: \(\frac{2}{3}\)** --- To summarize: - For \(7.27^{-\frac{1}{3}} \cdot 81^{-\frac{1}{4}} \cdot 9^2\), we have \(\frac{100^{\frac{1}{3}} \cdot 81}{3 \cdot