Think of baking soda ($NaHCO_3$) as the raw flour dough and washing soda ($Na_2CO_3 \cdot 10H_2O$) as the baked bread; baking soda is heated and processed to transform it into the final chemical product used for cleaning.

Solvay makes Soda from Salt and Baking Soda: S-S-B (Solvay - Sodium chloride - Bicarbonate).

The Solvay process involves the reaction of sodium chloride, ammonia, and carbon dioxide in water to precipitate sodium bicarbonate: $\text{Na Cl} + NH_3 + CO_2 + H_2O \rightarrow NaHCO_3 \downarrow + NH_4Cl$. The separated sodium bicarbonate is then thermally decomposed (calcined) at high temperatures to form anhydrous sodium carbonate (soda ash): $2NaHCO_3 \xrightarrow{\Delta} Na_2CO_3 + H_2O + CO_2$. Finally, dissolving soda ash in water and recrystallizing yields washing soda containing ten molecules of water of crystallization: $Na_2CO_3 + 10H_2O \rightarrow Na_2CO_3 \cdot 10H_2O$.