The expression given is:

$2 \sin\left(\frac{\pi}{8}\right) \cos\left(\frac{\pi}{8}\right)$

This can be simplified using the double angle identity for sine:

$2 \sin(A) \cos(A) = \sin(2A)$

So, applying this identity to the given expression with $A = \frac{\pi}{8}$:

$2 \sin\left(\frac{\pi}{8}\right) \cos\left(\frac{\pi}{8}\right) = \sin\left(2 \times \frac{\pi}{8}\right) = \sin\left(\frac{\pi}{4}\right)$

The sine of $\frac{\pi}{4}$ is a well-known value:

$\sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}$

Thus, the exact numerical value of the expression is:

$\frac{\sqrt{2}}{2}$

Would you like further details on the steps?

Here are five related questions for practice:

1. How would you simplify $\sin(2A)$ if $A = \frac{\pi}{6}$?
2. What is the value of $\cos\left(\frac{\pi}{4}\right)$?
3. Can you express $\sin\left(\frac{\pi}{3}\right)$ in terms of simpler radicals?
4. How would you use the double angle identity to simplify $2 \sin\left(\frac{\pi}{12}\right) \cos\left(\frac{\pi}{12}\right)$?
5. What are other common trigonometric identities that involve both sine and cosine?

Tip: When simplifying trigonometric expressions, always check for common identities like the double angle or sum-to-product formulas.