Sure, let's go through the solution step-by-step.
We start with the expression [tex]\(\left(a^8\right)^4\)[/tex].
To simplify this, we can utilize the properties of exponents, specifically the power of a power rule. The power of a power rule states that for any base [tex]\(a\)[/tex] and exponents [tex]\(m\)[/tex] and [tex]\(n\)[/tex]:
[tex]\[
(a^m)^n = a^{m \cdot n}
\][/tex]
In this case, our base is [tex]\(a\)[/tex], the first exponent (inside the parentheses) is [tex]\(8\)[/tex], and the second exponent (outside the parentheses) is [tex]\(4\)[/tex]. According to the rule:
[tex]\[
\left(a^8\right)^4 = a^{8 \cdot 4}
\][/tex]
Next, we multiply the exponents:
[tex]\[
8 \cdot 4 = 32
\][/tex]
Thus, we have:
[tex]\[
\left(a^8\right)^4 = a^{32}
\][/tex]
Therefore, the expression [tex]\(\left(a^8\right)^4\)[/tex] is equivalent to [tex]\(a^{32}\)[/tex].
Among the given choices, the correct one is:
[tex]\[
a^{32}
\][/tex]
