Determine the polar form of the product of z and w: $\begin{array}{l}z=4\left(\cos \frac{\pi}{6}+i \sin \frac{\pi}{6}\right) \\ w=4\left(\cos \frac{\pi}{7}+i \sin \frac{\pi}{7}\right)\end{array}$

Step 1 ：Given two complex numbers in polar form: \(z=4\left(\cos \frac{\pi}{6}+i \sin \frac{\pi}{6}\right)\) and \(w=4\left(\cos \frac{\pi}{7}+i \sin \frac{\pi}{7}\right)\)

Step 2 ：The product of two complex numbers in polar form is given by multiplying their magnitudes and adding their angles.

Step 3 ：First, calculate the product of the magnitudes: \(4 \times 4 = 16\)

Step 4 ：Next, add the angles: \(\frac{\pi}{6} + \frac{\pi}{7} = \frac{13\pi}{42}\)

Step 5 ：Therefore, the product of \(z\) and \(w\) in polar form is \(16\left(\cos \frac{13\pi}{42} + i \sin \frac{13\pi}{42}\right)\)

Step 6 ：Final Answer: The polar form of the product of \(z\) and \(w\) is \(\boxed{16\left(\cos \frac{13\pi}{42} + i \sin \frac{13\pi}{42}\right)}\)