1. **State the problem:** Simplify the expression $$-2 + \left( \frac{2 - \left(-\frac{1}{2}\right)}{-3 + (-1)^{-5}} \right)^2 \times \left( \frac{\frac{5}{2^2} - \frac{3}{4} + 1}{(-2)^2 + \left(\frac{1}{2}\right)^2} \right)^2 + \left( \frac{\frac{2}{3} - 1}{1 - \frac{2}{3}} - \frac{3}{2} - 2 \right) $$ 2. **Recall important rules:** - Powers: $a^{-n} = \frac{1}{a^n}$ - Order of operations: parentheses, exponents, multiplication/division, addition/subtraction - Simplify step-by-step inside parentheses before applying powers 3. **Simplify each part step-by-step:** - Numerator of first big fraction inside parentheses: $$2 - \left(-\frac{1}{2}\right) = 2 + \frac{1}{2} = \frac{4}{2} + \frac{1}{2} = \frac{5}{2}$$ - Denominator of first big fraction: $$-3 + (-1)^{-5}$$ Since $(-1)^{-5} = \frac{1}{(-1)^5} = \frac{1}{-1} = -1$, so denominator is $-3 + (-1) = -4$ - First big fraction: $$\frac{5/2}{-4} = \frac{5}{2} \times \frac{1}{-4} = -\frac{5}{8}$$ - Square it: $$\left(-\frac{5}{8}\right)^2 = \frac{25}{64}$$ - Numerator of second big fraction: $$\frac{5}{2^2} - \frac{3}{4} + 1 = \frac{5}{4} - \frac{3}{4} + 1 = \frac{2}{4} + 1 = \frac{1}{2} + 1 = \frac{3}{2}$$ - Denominator of second big fraction: $$(-2)^2 + \left(\frac{1}{2}\right)^2 = 4 + \frac{1}{4} = \frac{16}{4} + \frac{1}{4} = \frac{17}{4}$$ - Second big fraction: $$\frac{3/2}{17/4} = \frac{3}{2} \times \frac{4}{17} = \frac{12}{34} = \frac{6}{17}$$ - Square it: $$\left(\frac{6}{17}\right)^2 = \frac{36}{289}$$ - Multiply the two squared fractions: $$\frac{25}{64} \times \frac{36}{289} = \frac{900}{18596}$$ Simplify numerator and denominator by dividing numerator and denominator by 4: $$\frac{225}{4649}$$ 4. **Simplify the last big parentheses:** - Numerator: $$\frac{2}{3} - 1 = \frac{2}{3} - \frac{3}{3} = -\frac{1}{3}$$ - Denominator: $$1 - \frac{2}{3} = \frac{3}{3} - \frac{2}{3} = \frac{1}{3}$$ - Fraction: $$\frac{-\frac{1}{3}}{\frac{1}{3}} = -1$$ - Then subtract $\frac{3}{2}$ and 2: $$-1 - \frac{3}{2} - 2 = -1 - 1.5 - 2 = -4.5 = -\frac{9}{2}$$ 5. **Combine all parts:** $$-2 + \frac{225}{4649} - \frac{9}{2}$$ - Convert all to a common denominator $4649$: $$-2 = -\frac{9298}{4649}, \quad -\frac{9}{2} = -\frac{20920.5}{4649}$$ - Sum: $$-\frac{9298}{4649} + \frac{225}{4649} - \frac{20920.5}{4649} = \frac{-9298 + 225 - 20920.5}{4649} = \frac{-29993.5}{4649}$$ - This is approximately $-6.45$, which contradicts the given answer 36. **Re-examine the problem:** The user states the answer is 36, so let's check the last parentheses carefully: $$\left( \frac{\frac{2}{3} - 1}{1 - \frac{2}{3}} - \frac{3}{2} - 2 \right)$$ Calculate stepwise: - $$\frac{2}{3} - 1 = -\frac{1}{3}$$ - $$1 - \frac{2}{3} = \frac{1}{3}$$ - Fraction: $$\frac{-\frac{1}{3}}{\frac{1}{3}} = -1$$ - Then $$-1 - \frac{3}{2} - 2 = -1 - 1.5 - 2 = -4.5$$ So this part is $-\frac{9}{2}$. Now, reconsider the multiplication of the squared fractions: $$\left( -\frac{5}{8} \right)^2 = \frac{25}{64}$$ $$\left( \frac{6}{17} \right)^2 = \frac{36}{289}$$ Multiply: $$\frac{25}{64} \times \frac{36}{289} = \frac{900}{18596}$$ Simplify numerator and denominator by dividing numerator and denominator by 4: $$\frac{225}{4649}$$ Now sum all: $$-2 + \frac{225}{4649} - \frac{9}{2}$$ Convert to decimals: -2 = -2 225/4649 ≈ 0.0484 -9/2 = -4.5 Sum: $$-2 + 0.0484 - 4.5 = -6.4516$$ This does not match 36. **Check if the last parentheses are grouped differently:** If the last term is: $$\left( \frac{\frac{2}{3} - 1}{1 - \frac{2}{3}} \right) - \frac{3}{2} - 2$$ Calculate: - $$\frac{2}{3} - 1 = -\frac{1}{3}$$ - $$1 - \frac{2}{3} = \frac{1}{3}$$ - Fraction: $$-1$$ Then: $$-1 - \frac{3}{2} - 2 = -1 - 1.5 - 2 = -4.5$$ No change. **Try to re-check the first big fraction denominator:** $$-3 + (-1)^{-5}$$ Since $(-1)^5 = -1$, so $(-1)^{-5} = \frac{1}{-1} = -1$ So denominator is $-3 + (-1) = -4$ No error here. **Try to re-check numerator of second big fraction:** $$\frac{5}{2^2} - \frac{3}{4} + 1 = \frac{5}{4} - \frac{3}{4} + 1 = \frac{2}{4} + 1 = \frac{1}{2} + 1 = \frac{3}{2}$$ Denominator: $$(-2)^2 + \left(\frac{1}{2}\right)^2 = 4 + \frac{1}{4} = \frac{17}{4}$$ Fraction: $$\frac{3/2}{17/4} = \frac{3}{2} \times \frac{4}{17} = \frac{12}{34} = \frac{6}{17}$$ Square: $$\left(\frac{6}{17}\right)^2 = \frac{36}{289}$$ No error. **Try to calculate the entire expression numerically:** - First big fraction squared: $\left(-\frac{5}{8}\right)^2 = 0.390625$ - Second big fraction squared: $\left(\frac{6}{17}\right)^2 \approx 0.12456$ - Multiply: $0.390625 \times 0.12456 \approx 0.0487$ - Add $-2$: $-2 + 0.0487 = -1.9513$ - Add last parentheses $-4.5$: $-1.9513 - 4.5 = -6.4513$ This confirms previous result. **Conclusion:** The expression evaluates approximately to $-6.45$, not 36. **If the user states the answer is 36, possibly the expression or grouping is different.** **Final answer based on given expression:** $$\boxed{-6.4513 \approx -\frac{29993.5}{4649}}$$