Let's start by simplifying the given expression step-by-step.
Given the expression:
[tex]\[
\left(4^2\right)^4
\][/tex]
We can simplify this using the power of a power rule, which states:
[tex]\[
(a^m)^n = a^{m \cdot n}
\][/tex]
In this case, our base [tex]\(a\)[/tex] is [tex]\(4\)[/tex], our inner exponent [tex]\(m\)[/tex] is [tex]\(2\)[/tex], and our outer exponent [tex]\(n\)[/tex] is [tex]\(4\)[/tex]. Applying the power of a power rule:
[tex]\[
\left(4^2\right)^4 = 4^{2 \cdot 4}
\][/tex]
Now, we just need to perform the multiplication in the exponent:
[tex]\[
2 \cdot 4 = 8
\][/tex]
Therefore, we can rewrite the expression as:
[tex]\[
4^8
\][/tex]
So, the simplified form of the given expression [tex]\(\left(4^2\right)^4\)[/tex] is [tex]\(\boxed{4^8}\)[/tex].
