Find $z w$ and $\frac{z}{w}$. Write each answer in polar form and in exponential form.
\[
\begin{array}{l}
\mathrm{z}=9\left(\cos \frac{\pi}{8}+i \sin \frac{\pi}{8}\right) \\
\mathrm{w}=9\left(\cos \frac{\pi}{9}+i \sin \frac{\pi}{9}\right)
\end{array}
\]

Step 1 ：Given two complex numbers in polar form, $z = 9(\cos(\frac{\pi}{8}) + i\sin(\frac{\pi}{8}))$ and $w = 9(\cos(\frac{\pi}{9}) + i\sin(\frac{\pi}{9}))$

Step 2 ：The product of two complex numbers in polar form is obtained by multiplying their magnitudes and adding their angles, so $zw = r_1r_2(\cos(\theta_1 + \theta_2) + i\sin(\theta_1 + \theta_2))$

Step 3 ：Substitute $r_1 = 9$, $r_2 = 9$, $\theta_1 = \frac{\pi}{8}$, and $\theta_2 = \frac{\pi}{9}$ into the formula, we get $zw = 9*9(\cos(\frac{\pi}{8} + \frac{\pi}{9}) + i\sin(\frac{\pi}{8} + \frac{\pi}{9}))$

Step 4 ：Simplify to get $zw = 81(\cos(\frac{17\pi}{72}) + i\sin(\frac{17\pi}{72}))$

Step 5 ：The exponential form of a complex number is $re^{i\theta}$, so the exponential form of $zw$ is $zw = 81e^{i\frac{17\pi}{72}}$

Step 6 ：The quotient of two complex numbers in polar form is obtained by dividing their magnitudes and subtracting their angles, so $\frac{z}{w} = \frac{r_1}{r_2}(\cos(\theta_1 - \theta_2) + i\sin(\theta_1 - \theta_2))$

Step 7 ：Substitute $r_1 = 9$, $r_2 = 9$, $\theta_1 = \frac{\pi}{8}$, and $\theta_2 = \frac{\pi}{9}$ into the formula, we get $\frac{z}{w} = \frac{9}{9}(\cos(\frac{\pi}{8} - \frac{\pi}{9}) + i\sin(\frac{\pi}{8} - \frac{\pi}{9}))$

Step 8 ：Simplify to get $\frac{z}{w} = (\cos(\frac{\pi}{72}) + i\sin(\frac{\pi}{72}))$

Step 9 ：The exponential form of a complex number is $re^{i\theta}$, so the exponential form of $\frac{z}{w}$ is $\frac{z}{w} = e^{i\frac{\pi}{72}}$

Step 10 ：So, the product $zw$ in polar form is $81(\cos(\frac{17\pi}{72}) + i\sin(\frac{17\pi}{72}))$ and in exponential form is $\boxed{81e^{i\frac{17\pi}{72}}}$

Step 11 ：The quotient $\frac{z}{w}$ in polar form is $(\cos(\frac{\pi}{72}) + i\sin(\frac{\pi}{72}))$ and in exponential form is $\boxed{e^{i\frac{\pi}{72}}}$