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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.

Find the exact numerical value of the expression 2sin(π/8)cos(π/8).

This problem demonstrates how to find the exact numerical value of 2sin(π/8)cos(π/8) by applying the double angle identity for sine. Learn the step-by-step process of using the identity 2sin(A)cos(A) = sin(2A) to simplify trigonometric expressions. Ideal for students in Grades 10-12.

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