29. \( \left(7 d^{\frac{3}{2}} \cdot 2 g^{\frac{5}{6}}\right)\left(2 g^{\frac{3}{2}} \cdot 7 d^{\frac{5}{6}}\right) \)

To simplify the expression \( \left(7 d^{\frac{3}{2}} \cdot 2 g^{\frac{5}{6}}\right)\left(2 g^{\frac{3}{2}} \cdot 7 d^{\frac{5}{6}}\right) \), we first rearrange and combine like terms. 1. Start with the numerical coefficients: \[ 7 \cdot 2 \cdot 2 \cdot 7 = 196 \] 2. Now, for the \(d\) terms: \[ d^{\frac{3}{2}} \cdot d^{\frac{5}{6}} = d^{\frac{3}{2} + \frac{5}{6}} \] Convert \(\frac{3}{2}\) to sixths: \[ \frac{3}{2} = \frac{9}{6} \quad \text{so} \quad d^{\frac{9}{6}} \cdot d^{\frac{5}{6}} = d^{\frac{9 + 5}{6}} = d^{\frac{14}{6}} = d^{\frac{7}{3}} \] 3. Now for the \(g\) terms: \[ g^{\frac{5}{6}} \cdot g^{\frac{3}{2}} = g^{\frac{5}{6} + \frac{3}{2}} \] Convert \(\frac{3}{2}\) to sixths: \[ \frac{3}{2} = \frac{9}{6} \quad \text{so} \quad g^{\frac{5}{6}} \cdot g^{\frac{9}{6}} = g^{\frac{5 + 9}{6}} = g^{\frac{14}{6}} = g^{\frac{7}{3}} \] Putting it all together, we get: \[ \left(7 d^{\frac{3}{2}} \cdot 2 g^{\frac{5}{6}}\right)\left(2 g^{\frac{3}{2}} \cdot 7 d^{\frac{5}{6}}\right) = 196 d^{\frac{7}{3}} g^{\frac{7}{3}} \] Thus, the final simplified expression is: \[ \boxed{196 d^{\frac{7}{3}} g^{\frac{7}{3}}} \]