The problem in the image appears to ask for simplification of the following expression:

$4^8 : 2^4$

This is a division of powers with the same base. To simplify:

Step-by-step solution:

1. Write the expression clearly: $\frac{4^8}{2^4}$

2. Express 4 as a power of 2: $4 = 2^2$

   So, the expression becomes: $\frac{(2^2)^8}{2^4}$

3. Apply the power rule $(a^m)^n = a^{m \cdot n}$: $\frac{2^{16}}{2^4}$

4. Subtract the exponents (since the bases are the same): $2^{16 - 4} = 2^{12}$

Thus, the simplified expression is:

$2^{12}$

This equals $4096$.

Would you like more details on any step? Here are a few related questions you might find useful:

1. How do you handle divisions of powers with different bases?
2. What is the general rule for simplifying exponents when dividing?
3. Can you provide an example of multiplying expressions with exponents?
4. What are the rules for adding or subtracting exponents when the base is the same?
5. How do you convert complex numbers into simpler forms using exponent rules?

Tip: When simplifying powers with division, always check if the bases are the same before applying exponent rules.