Given that $\cos \alpha=\frac{3}{5}$ and $0< \alpha< \frac{\pi}{2}$, determine the exact value of $\cos \frac{\alpha}{2}$.
\[
\cos \frac{\alpha}{2}=
\]
Use for any numbers in the expression. Rationalize all denominators

Step 1 ：We are given that \(\cos \alpha=\frac{3}{5}\) 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 \(\cos \frac{\alpha}{2} = \sqrt{\frac{1+\cos \alpha}{2}}\).

Step 3 ：Substitute the given value of \(\cos \alpha\) into this formula to find the value of \(\cos \frac{\alpha}{2}\).

Step 4 ：\(\cos \frac{\alpha}{2} = \sqrt{\frac{1+\frac{3}{5}}{2}}\)

Step 5 ：\(\cos \frac{\alpha}{2} = \sqrt{\frac{8}{10}}\)

Step 6 ：\(\cos \frac{\alpha}{2} = \sqrt{0.8}\)

Step 7 ：\(\cos \frac{\alpha}{2} = 0.8944271909999159\)

Step 8 ：Final Answer: The exact value of \(\cos \frac{\alpha}{2}\) is \(\boxed{0.8944271909999159}\)