Find the exact length of the polar curve $r=\cos ^{4}(\theta / 4)$.
Length $=$

Step 1 ：First, we need to find the length of the polar curve $r=\cos^{4}(\theta / 4)$. The formula for the length of a polar curve is given by $L = \int_{a}^{b} \sqrt{r^{2} + (\frac{dr}{d\theta})^{2}} d\theta$.

Step 2 ：Here, $r=\cos^{4}(\theta / 4)$, so we need to find $\frac{dr}{d\theta}$.

Step 3 ：Taking the derivative of $r$ with respect to $\theta$, we get $\frac{dr}{d\theta} = -\sin(\theta / 4)\cos^{3}(\theta / 4)$.

Step 4 ：Substituting $r$ and $\frac{dr}{d\theta}$ into the formula for $L$, we get $L = \int_{0}^{2\pi} \sqrt{\cos^{8}(\theta / 4) + \sin^{2}(\theta / 4)\cos^{6}(\theta / 4)} d\theta$.

Step 5 ：Since $\sin^{2}(\theta) = 1 - \cos^{2}(\theta)$, we can simplify the integrand to $L = \int_{0}^{2\pi} \sqrt{\cos^{8}(\theta / 4) + (1 - \cos^{2}(\theta / 4))\cos^{6}(\theta / 4)} d\theta$.

Step 6 ：Further simplifying, we get $L = \int_{0}^{2\pi} \cos^{3}(\theta / 4) d\theta$.

Step 7 ：Integrating from $0$ to $2\pi$, we get $L = 8\int_{0}^{\pi/2} \cos^{3}(x) dx$.

Step 8 ：Using the reduction formula for $\int \cos^{n}(x) dx$, where $n$ is odd, we get $L = 8[\frac{1}{3}\cos^{2}(x)\sin(x) + \frac{2}{3}\int \cos(x) dx]$ evaluated from $0$ to $\pi/2$.

Step 9 ：Evaluating the integral and the limits, we get $L = 8[\frac{1}{3}\cos^{2}(x)\sin(x) + \frac{2}{3}\sin(x)]_{0}^{\pi/2}$.

Step 10 ：Finally, evaluating at the limits $0$ and $\pi/2$, we get $L = 8[\frac{1}{3}\cdot 0 + \frac{2}{3}\cdot 1 - 0] = \frac{16}{3}$.

Step 11 ：So, the exact length of the polar curve $r=\cos^{4}(\theta / 4)$ is $\boxed{\frac{16}{3}}$.