The batting average for $30\text{ innings}$ of a cricket player is $40\text{ runs}$. His highest score exceeds his lowest score by $100\text{ runs}$. If these two innings are not included, the average of the remaining $28\text{ innings}$ is $38\text{ runs}$. Then the lowest score of the player is:

Let the highest score be $H$ and the lowest score be $L$.
From the given information:

$$\text{Total runs in }30\text{ innings}=30\times 40=1,200$$

$$\text{Total runs in }28\text{ innings}=28\times 38=1,064$$

The sum of the highest and lowest scores is the difference between these totals:

$$H+L=1200-1064=136\text{— (Equation 1)}$$

We are also given that the highest score exceeds the lowest by $100\text{ runs}$:

$$H-L=100\text{— (Equation 2)}$$

Subtracting Equation 2 from Equation 1:

$$(H+L)-(H-L)=136-100$$

$$2L=36$$

$$L=18$$

The lowest score of the player is $18$.