Calculate the moles of ethanoic acid and the number of water molecules formed in the reaction between ethanoic acid and potassium hydroxide.

$$\text{Average volume of A} = 14.40 \text{ cm}^3 = 14.40 \times 10^{-3} \text{ dm}^3 = 0.01440 \text{ dm}^3$$

This conversion is correct.

$$\text{Moles of CH}_3\text{COOH} = \text{Concentration} \times \text{Volume}$$ $$\text{Moles of CH}_3\text{COOH} = 0.100 \text{ mol dm}^{-3} \times 0.01440 \text{ dm}^3$$ $$\text{Moles of CH}_3\text{COOH} = 0.00144 \text{ mol}$$ $$\text{Moles of CH}_3\text{COOH} = 1.44 \times 10^{-3} \text{ mol}$$

This calculation is correct.

The reaction is between ethanoic acid ($\text{CH}_3\text{COOH}$) and potassium hydroxide ($\text{KOH}$).

$$\text{CH}_3\text{COOH} + \text{KOH} \rightarrow \text{CH}_3\text{COOK} + \text{H}_2\text{O}$$

From the balanced equation, the mole ratio of $\text{CH}_3\text{COOH} : \text{H}_2\text{O}$ is $1:1$.
This is correct.

Since the mole ratio is $1:1$, the moles of $\text{H}_2\text{O}$ formed are equal to the moles of $\text{CH}_3\text{COOH}$ used.

$$\text{Moles of H}_2\text{O formed} = 1.44 \times 10^{-3} \text{ mol}$$

This is correct.

$$\text{Number of water molecules} = \text{Moles} \times \text{Avogadro's number} (\text{N}_{\text{A}})$$

Using $\text{N}_{\text{A}} = 6.02 \times 10^{23} \text{ mol}^{-1}$:

$$\text{Number of water molecules} = 1.44 \times 10^{-3} \text{ mol} \times 6.02 \times 10^{23} \text{ mol}^{-1}$$ $$\text{Number of water molecules} = (1.44 \times 6.02) \times (10^{-3} \times 10^{23})$$ $$\text{Number of water molecules} = 8.6688 \times 10^{(-3+23)}$$ $$\text{Number of water molecules} = 8.6688 \times 10^{20} \text{ molecules}$$

Your calculation for the numerical part ($1.44 \times 6.02 = 8.6688$) is correct, but there is an error in the exponent. The exponent should be $10^{20}$, not $10^{23}$.

Therefore, the final answer for the number of water molecules is:

$$\boxed{\text{Number of water molecules} = 8.6688 \times 10^{20} \text{ molecules}}$$

The first part of your answer is correct, but the second part has an error in the exponent for the number of water molecules.