1. **State the problem:** Simplify the expression $$2 \cdot \frac{7}{2} \cdot \frac{3}{2} + 2 \cdot \frac{3}{2} \cdot \frac{9}{2} + 2 \cdot \frac{7}{2} \cdot \frac{9}{2}$$. 2. **Recall multiplication rules:** When multiplying fractions, multiply numerators together and denominators together. 3. **Calculate each term separately:** - First term: $$2 \cdot \frac{7}{2} \cdot \frac{3}{2} = \frac{2}{1} \cdot \frac{7}{2} \cdot \frac{3}{2} = \frac{2 \times 7 \times 3}{1 \times 2 \times 2} = \frac{42}{4} = \frac{21}{2}$$. - Second term: $$2 \cdot \frac{3}{2} \cdot \frac{9}{2} = \frac{2}{1} \cdot \frac{3}{2} \cdot \frac{9}{2} = \frac{2 \times 3 \times 9}{1 \times 2 \times 2} = \frac{54}{4} = \frac{27}{2}$$. - Third term: $$2 \cdot \frac{7}{2} \cdot \frac{9}{2} = \frac{2}{1} \cdot \frac{7}{2} \cdot \frac{9}{2} = \frac{2 \times 7 \times 9}{1 \times 2 \times 2} = \frac{126}{4} = \frac{63}{2}$$. 4. **Add the results:** $$\frac{21}{2} + \frac{27}{2} + \frac{63}{2} = \frac{21 + 27 + 63}{2} = \frac{111}{2}$$. 5. **Final answer:** $$\boxed{\frac{111}{2}}$$ or as a decimal, $$55.5$$. This completes the simplification.