1. **State the problem:** Calculate the value of $27 \frac{2}{3} - \left( \frac{1}{8} \right)^{\frac{1}{3}}$. 2. **Convert mixed number to improper fraction:** $$27 \frac{2}{3} = \frac{3 \times 27 + 2}{3} = \frac{81 + 2}{3} = \frac{83}{3}$$ 3. **Evaluate the cube root:** $$\left( \frac{1}{8} \right)^{\frac{1}{3}} = \frac{1^{\frac{1}{3}}}{8^{\frac{1}{3}}} = \frac{1}{2}$$ 4. **Rewrite the expression:** $$\frac{83}{3} - \frac{1}{2}$$ 5. **Find common denominator and subtract:** $$\frac{83}{3} - \frac{1}{2} = \frac{83 \times 2}{3 \times 2} - \frac{1 \times 3}{2 \times 3} = \frac{166}{6} - \frac{3}{6} = \frac{166 - 3}{6} = \frac{163}{6}$$ 6. **Convert to mixed number:** $$\frac{163}{6} = 27 \frac{1}{6}$$ **Final answer:** $27 \frac{1}{6}$