Evaluate Expression 72C86E

1. **State the problem:** Evaluate the expression $$- \left(-\frac{2}{3}\right)^2 + \left(\frac{1}{2}\right)^3 \cdot (-1) - \frac{2}{5} \cdot (-2)^3 + \frac{11}{2} - \left(\frac{1}{6}\right)^2 \cdot 3^{-1}$$. 2. **Recall important rules:** - Squaring a negative number inside parentheses affects the sign. - Powers and exponents are evaluated before multiplication. - Negative exponents mean reciprocal: $$a^{-1} = \frac{1}{a}$$. - Multiplication and division are performed from left to right. 3. **Calculate each term:** - $$\left(-\frac{2}{3}\right)^2 = \left(-\frac{2}{3}\right) \times \left(-\frac{2}{3}\right) = \frac{4}{9}$$ - $$\left(\frac{1}{2}\right)^3 = \frac{1}{8}$$ - $$(-2)^3 = -8$$ - $$\left(\frac{1}{6}\right)^2 = \frac{1}{36}$$ - $$3^{-1} = \frac{1}{3}$$ 4. **Substitute back:** $$- \frac{4}{9} + \frac{1}{8} \cdot (-1) - \frac{2}{5} \cdot (-8) + \frac{11}{2} - \frac{1}{36} \cdot \frac{1}{3}$$ 5. **Simplify multiplications:** $$- \frac{4}{9} - \frac{1}{8} + \left(- \frac{2}{5} \times -8 \right) + \frac{11}{2} - \frac{1}{108}$$ 6. **Calculate $$- \frac{2}{5} \times -8$$:** $$- \frac{2}{5} \times -8 = \cancel{-} \frac{2}{5} \times \cancel{-} 8 = \frac{16}{5}$$ 7. **Rewrite expression:** $$- \frac{4}{9} - \frac{1}{8} + \frac{16}{5} + \frac{11}{2} - \frac{1}{108}$$ 8. **Find common denominator for all fractions:** The denominators are 9, 8, 5, 2, and 108. - The least common multiple (LCM) of 9, 8, 5, 2, and 108 is 1080. 9. **Convert each fraction to denominator 1080:** - $$- \frac{4}{9} = - \frac{4 \times 120}{9 \times 120} = - \frac{480}{1080}$$ - $$- \frac{1}{8} = - \frac{1 \times 135}{8 \times 135} = - \frac{135}{1080}$$ - $$\frac{16}{5} = \frac{16 \times 216}{5 \times 216} = \frac{3456}{1080}$$ - $$\frac{11}{2} = \frac{11 \times 540}{2 \times 540} = \frac{5940}{1080}$$ - $$- \frac{1}{108} = - \frac{1 \times 10}{108 \times 10} = - \frac{10}{1080}$$ 10. **Sum all fractions:** $$-480 - 135 + 3456 + 5940 - 10 = ( -480 - 135 - 10 ) + (3456 + 5940) = -625 + 9396 = 8771$$ 11. **Final fraction:** $$\frac{8771}{1080}$$ 12. **Convert to mixed number or decimal:** - $$8771 \div 1080 = 8$$ remainder $$131$$ - So, $$8 \frac{131}{1080}$$ approximately $$8.13$$. **Final answer:** $$8 \frac{131}{1080}$$ or approximately $$8.13$$.