Given that $\cos \alpha=\frac{2}{3}$ and $0<\alpha<\frac{\pi}{2}$, determine the exact value of $\cos \frac{\alpha}{2}$. \[ \cos \frac{\alpha}{2}= \] (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression. Rationalize all denominators.)

Step 1 ：We are given that \(\cos \alpha=\frac{2}{3}\) and \(0<\alpha<\frac{\pi}{2}\). We need to find the value of \(\cos \frac{\alpha}{2}\).

Step 2 ：We can use the half-angle formula for cosine, which is \(\cos \frac{\alpha}{2} = \pm \sqrt{\frac{1+\cos \alpha}{2}}\).

Step 3 ：Since \(0<\alpha<\frac{\pi}{2}\), we know that \(0<\frac{\alpha}{2}<\frac{\pi}{4}\), so \(\cos \frac{\alpha}{2}\) will be positive.

Step 4 ：Therefore, we can ignore the negative root and use the formula \(\cos \frac{\alpha}{2} = \sqrt{\frac{1+\cos \alpha}{2}}\) to find the value of \(\cos \frac{\alpha}{2}\).

Step 5 ：Substituting the given value of \(\cos \alpha\) into the formula, we get \(\cos \frac{\alpha}{2} = \sqrt{\frac{1+\frac{2}{3}}{2}}\).

Step 6 ：Solving this, we find that the exact value of \(\cos \frac{\alpha}{2}\) is \(\boxed{0.9128709291752768}\).