To find an expression equivalent to [tex]\((a^8)^4\)[/tex], we can use the power rule for exponents. The power rule states that when raising a power to another power, you multiply the exponents. Mathematically, this is expressed as:
[tex]\[
(a^m)^n = a^{m \cdot n}
\][/tex]
Let's apply this rule to the given expression [tex]\((a^8)^4\)[/tex]:
1. Identify the inner exponent [tex]\( m \)[/tex] and the outer exponent [tex]\( n \)[/tex]. In this case, [tex]\( m = 8 \)[/tex] and [tex]\( n = 4 \)[/tex].
2. Multiply the exponents together:
[tex]\[
m \cdot n = 8 \cdot 4
\][/tex]
3. Calculate the product of the exponents:
[tex]\[
8 \cdot 4 = 32
\][/tex]
4. Substitute this result back into the expression:
[tex]\[
(a^8)^4 = a^{32}
\][/tex]
Therefore, the expression equivalent to [tex]\((a^8)^4\)[/tex] is:
[tex]\[
a^{32}
\][/tex]
So, the correct answer is [tex]\( a^{32} \)[/tex].
