\( \frac { ( \frac { 105 } { 99 } + \frac { 2 } { 3 } + 1 , \sqrt { 27 } ) ^ { - 2 } } { \sqrt[ 3 ] { 2 - \frac { 55 } { 27 } } } = \)

To solve the expression \[ \frac { ( \frac { 105 } { 99 } + \frac { 2 } { 3 } + 1 , \sqrt { 27 } ) ^ { - 2 } } { \sqrt[ 3 ] { 2 - \frac { 55 } { 27 } } }, \] we will break it down step by step. ### Step 1: Simplify \( \frac{105}{99} \) First, we simplify \( \frac{105}{99} \): \[ \frac{105}{99} = \frac{105 \div 3}{99 \div 3} = \frac{35}{33}. \] ### Step 2: Simplify \( \frac{2}{3} + 1 \) Next, we simplify \( \frac{2}{3} + 1 \): \[ 1 = \frac{3}{3} \quad \Rightarrow \quad \frac{2}{3} + 1 = \frac{2}{3} + \frac{3}{3} = \frac{5}{3}. \] ### Step 3: Combine the fractions Now we combine \( \frac{35}{33} + \frac{5}{3} \): To add these fractions, we need a common denominator. The least common multiple of 33 and 3 is 33. \[ \frac{5}{3} = \frac{5 \times 11}{3 \times 11} = \frac{55}{33}. \] Now we can add: \[ \frac{35}{33} + \frac{55}{33} = \frac{35 + 55}{33} = \frac{90}{33}. \] We can simplify \( \frac{90}{33} \): \[ \frac{90 \div 3}{33 \div 3} = \frac{30}{11}. \] ### Step 4: Calculate \( \sqrt{27} \) Next, we calculate \( \sqrt{27} \): \[ \sqrt{27} = \sqrt{9 \times 3} = \sqrt{9} \cdot \sqrt{3} = 3\sqrt{3}. \] ### Step 5: Form the tuple and raise to the power of -2 Now we have: \[ \left( \frac{30}{11}, 3\sqrt{3} \right). \] We need to find the magnitude of this tuple: \[ \sqrt{\left( \frac{30}{11} \right)^2 + (3\sqrt{3})^2}. \] Calculating each part: \[ \left( \frac{30}{11} \right)^2 = \frac{900}{121}, \] \[ (3\sqrt{3})^2 = 9 \cdot 3 = 27. \] Now we convert 27 to a fraction with a denominator of 121: \[ 27 = \frac{27 \cdot 121}{121} = \frac{3267}{121}. \] Now we can add: \[ \sqrt{\frac{900}{121} + \frac{3267}{121}} = \sqrt{\frac{900 + 3267}{121}} = \sqrt{\frac{4167}{121}} = \frac{\sqrt{4167}}{11}. \] ### Step 6: Raise to the power of -2 Now we raise this to the power of -2: \[ \left( \frac{\sqrt{4167}}{11} \right)^{-2} = \frac{11^2}{4167} = \frac{121}{4167}. \] ### Step 7: Simplify the denominator \( \sqrt[3]{2 - \frac{55}{27}} \) Now we simplify the denominator: \[ 2 - \frac{55}{27} = \frac{2 \cdot 27}{27} - \frac{55}{27} = \frac{54 - 55}{27} = \frac{-1}{27}. \] Now we find the cube root: \[ \sqrt[3]{\frac{-1}{27}} = \frac{\sqrt[3]{-1}}{\sqrt[3]{27}} = \frac{-1}{3}. \] ### Step 8: Final calculation Now we can substitute back into the expression: \[ \frac{\frac{121}{4167}}{\frac{-1}{3}} = \frac{121}{4167} \cdot \left(-3\right) = \frac{-363}{4167}. \] Thus, the