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 $\boxed{{\displaystyle \frac{16}{3}}}$.