$\alpha =(1-2\cos \frac{\pi }{11})(1-2\cos \frac{3\pi }{11})(1-2\cos \frac{9\pi }{11})(1-2\cos \frac{27\pi }{11})(1-2\cos \frac{81\pi }{11})$

$$

Then the value of 5-a² is _________.

The correct answer is 4.

We are given the expression for $\alpha$:

$\alpha =(1-2\cos \frac{\pi }{11})(1-2\cos \frac{3\pi }{11})(1-2\cos \frac{9\pi }{11})(1-2\cos \frac{27\pi }{11})(1-2\cos \frac{81\pi }{11})$

First, let us simplify the angles modulo 2$\pi$ using the periodic and symmetric properties of the cosine function, $\cos (2k\pi \pm x)=\cos x$:

1. $\frac{27\pi }{11}=2\pi +\frac{5\pi }{11}$, so $\cos \frac{27\pi }{11}=\cos \frac{5\pi }{11}$
2. $\frac{81\pi }{11}=7\pi +\frac{4\pi }{11}=8\pi -\frac{7\pi }{11}$, so $\cos \frac{81\pi }{11}=\cos \frac{7\pi }{11}$

Thus, the expression for $\alpha$ becomes:

$\alpha =(1-2\cos \frac{\pi }{11})(1-2\cos \frac{3\pi }{11})(1-2\cos \frac{5\pi }{11})(1-2\cos \frac{7\pi }{11})(1-2\cos \frac{9\pi }{11})$

Now, recall the trigonometric identity:

$1-2\cos =-\frac{\sin (3\theta /2)}{\sin (\theta /2)}$

Applying this identity to each term with $=\frac{k\pi }{11}$ for $k=1,3,5,7,9$:

$\alpha ={(-1)}^5·\frac{\sin \frac{3\pi }{22}\sin \frac{9\pi }{22}\sin \frac{15\pi }{22}\sin \frac{21\pi }{22}\sin \frac{27\pi }{22}}{\sin \frac{\pi }{22}\sin \frac{3\pi }{22}\sin \frac{5\pi }{22}\sin \frac{7\pi }{22}\sin \frac{9\pi }{22}}$

Simplifying the terms in the numerator:
- $\sin \frac{15\pi }{22}=\sin (\pi -\frac{7\pi }{22})=\sin \frac{7\pi }{22}$
- $\sin \frac{21\pi }{22}=\sin (\pi -\frac{\pi }{22})=\sin \frac{\pi }{22}$
- $\sin \frac{27\pi }{22}=\sin (\pi +\frac{5\pi }{22})=-\sin \frac{5\pi }{22}$

Substituting these back into the numerator:

$\text{Numerator}=\sin \frac{3\pi }{22}·\sin \frac{9\pi }{22}·\sin \frac{7\pi }{22}·\sin \frac{\pi }{22}·(-\sin \frac{5\pi }{22})$

Notice that the product of sines in the numerator is equal to $-1$ times the product of sines in the denominator.
Therefore, the ratio evaluates to $-1$:

$\alpha =(-1)·(-1)=1$

Now, we need to calculate the value of $5-{\alpha }^2$:

$5-{\alpha }^2=5-1^2=5-1=4$

Thus, the final value is 4.