The problem requires simplifying the given expression $\frac{4^8}{4^3}$ and rewriting it in the form $4^n$.

Solution:

Using the laws of exponents, specifically the rule: $\frac{a^m}{a^n} = a^{m-n},$ we subtract the exponents.

$\frac{4^8}{4^3} = 4^{8-3}.$

Simplify the exponent: $8 - 3 = 5.$

So, the final answer is: $4^5.$

Let me know if you want further details or have any questions!

Related Questions:

1. How would the rule of exponents change if the base numbers were different?
2. What is the numerical value of $4^5$?
3. How do we simplify an expression like $\frac{a^m b^n}{a^p b^q}$?
4. Can this approach be applied to negative exponents? Why or why not?
5. What are some real-world examples where exponent simplification is useful?

Tip:

Always check that the bases are the same when using the rules of exponents!