What is the use of knowing elasticity of average product?
If you know Elasticity of average product you can identify how your output per worker or machine will change as $L$ or $K$ increases.

Recall the definition of elasticity is:

$$\mathcal{E}=\frac{\Delta y/y}{\Delta x/x}$$

average product of labor and capital:

$$AP_L=\frac{Q}{L}$$
$$AP_K=\frac{Q}{K}$$

therefore the elasticity of average product of labor and capital is:

$$\Gamma_L=\frac{\Delta AP/AP}{\Delta L/L}$$
$$\Gamma_K=\frac{\Delta AP/AP}{\Delta K/K}$$

This is useful for evaluating how existing worker or capital productivity will be impacted as a result of an increase in capital or labor.