1. Angular momentum of a planet in circular orbit

For a planet of mass $m$ orbiting a star of mass $M$ in a circular orbit of radius $r$,

$L = mvr$

where $v$ is the orbital speed.

Using gravitation as centripetal force,

$\frac{GMm}{r^2} = \frac{mv^2}{r}$

So,

$v = \sqrt{\frac{GM}{r}}$

Hence,

$L = mr\sqrt{\frac{GM}{r}} = m\sqrt{GMr}$

Thus,

$L \propto m\sqrt{r}$

for planets orbiting the same star.

2. Apply to planets $A$ and $B$

Given:

$R_B = 2R_A$

and mass of planet $B$ is $4\sqrt{2}$ times mass of planet $A$:

$m_B = 4\sqrt{2}\, m_A$

Therefore,

$\frac{L_B}{L_A} = \frac{m_B\sqrt{R_B}}{m_A\sqrt{R_A}}$

Substitute the given ratios:

$\frac{L_B}{L_A} = \left(4\sqrt{2}\right)\sqrt{\frac{R_B}{R_A}}$

Since

$\frac{R_B}{R_A} = 2,$

we get

$\frac{L_B}{L_A} = 4\sqrt{2}\cdot \sqrt{2} = 4\times 2 = 8$

3. Final integer answer

$\boxed{8}$