Reid Ellis
02/02/2023 · Middle School

c) \( \sqrt{(0,5-0, \hat{5}): 0,8 \hat{3}+\frac{23}{45}}-\left(-\frac{3}{2}\right)^{-2}= \)

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To solve the expression \[ \sqrt{(0.5 - 0. \hat{5}): 0.8 \hat{3} + \frac{23}{45}} - \left(-\frac{3}{2}\right)^{-2}, \] we need to clarify the notation used, particularly the parts with the hats (e.g., \(0. \hat{5}\) and \(0.8 \hat{3}\)). Assuming \(0. \hat{5}\) represents the repeating decimal \(0.555...\) (which is equal to \(\frac{5}{9}\)), and \(0.8 \hat{3}\) represents the repeating decimal \(0.833...\) (which is equal to \(\frac{5}{6}\)), we can rewrite the expression as follows: 1. **Convert the repeating decimals:** - \(0. \hat{5} = \frac{5}{9}\) - \(0.8 \hat{3} = \frac{5}{6}\) 2. **Substituting these values into the expression:** \[ \sqrt{(0.5 - \frac{5}{9}): \frac{5}{6} + \frac{23}{45}} - \left(-\frac{3}{2}\right)^{-2} \] 3. **Calculate \(0.5 - \frac{5}{9}\):** \[ 0.5 = \frac{1}{2} = \frac{9}{18}, \quad \frac{5}{9} = \frac{10}{18} \] \[ 0.5 - \frac{5}{9} = \frac{9}{18} - \frac{10}{18} = -\frac{1}{18} \] 4. **Now calculate \(-\frac{1}{18} : \frac{5}{6}\):** \[ -\frac{1}{18} \div \frac{5}{6} = -\frac{1}{18} \cdot \frac{6}{5} = -\frac{6}{90} = -\frac{1}{15} \] 5. **Next, calculate \(-\frac{1}{15} + \frac{23}{45}\):** - Convert \(-\frac{1}{15}\) to a fraction with a denominator of 45: \[ -\frac{1}{15} = -\frac{3}{45} \] \[ -\frac{3}{45} + \frac{23}{45} = \frac{20}{45} = \frac{4}{9} \] 6. **Now we have:** \[ \sqrt{\frac{4}{9}} = \frac{2}{3} \] 7. **Calculate \(\left(-\frac{3}{2}\right)^{-2}\):** \[ \left(-\frac{3}{2}\right)^{-2} = \left(-\frac{2}{3}\right)^{2} = \frac{4}{9} \] 8. **Finally, substitute back into the expression:** \[ \frac{2}{3} - \frac{4}{9} \] - Convert \(\frac{2}{3}\) to a fraction with a denominator of 9: \[ \frac{2}{3} = \frac{6}{9} \] \[ \frac{6}{9} - \frac{4}{9} = \frac{2}{9} \] Thus, the final result is \[ \boxed{\frac{2}{9}}. \]

Quick Answer

The final result is \( \frac{2}{9} \).
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