Here's how to write the balanced equation for the saponification of glyceryl tristearate with aqueous sodium hydroxide:

1. Identify the Reactants:

* Glyceryl tristearate: This is a triglyceride, a fat or oil, with the formula C₅₇H₁₁₀O₆.

* Sodium hydroxide: This is a strong base, NaOH (aq).

2. Identify the Products:

* Glycerol: A triol with the formula C₃H₈O₃.

* Sodium stearate: A soap, with the formula C₁₇H₃₅CO₂Na.

3. Write the Unbalanced Equation:

C₅₇H₁₁₀O₆ (s) + NaOH (aq) → C₃H₈O₃ (aq) + C₁₇H₃₅CO₂Na (aq)

4. Balance the Equation:

To balance the equation, we need to adjust the coefficients in front of each molecule:

C₅₇H₁₁₀O₆ (s) + 3 NaOH (aq) → C₃H₈O₃ (aq) + 3 C₁₇H₃₅CO₂Na (aq)

Explanation of Balancing:

* Carbon (C): There are 57 carbons on the left and 57 on the right.

* Hydrogen (H): There are 110 hydrogens on the left and 110 on the right.

* Oxygen (O): There are 6 oxygens on the left and 6 on the right.

* Sodium (Na): There are 3 sodiums on the left and 3 on the right.

Therefore, the balanced equation for the saponification of glyceryl tristearate with aqueous sodium hydroxide is:

C₅₇H₁₁₀O₆ (s) + 3 NaOH (aq) → C₃H₈O₃ (aq) + 3 C₁₇H₃₅CO₂Na (aq)