To solve the problem where [tex]\( a^{b^c} = 512 \)[/tex] and we need to find the minimum possible value of [tex]\( abc \)[/tex], let's break it down step-by-step.
1. Prime Factorization of 512:
We start by recognizing that 512 can be expressed as a power of a prime number.
[tex]\[
512 = 2^9
\][/tex]
2. Expressing 512 as [tex]\( a^{b^c} \)[/tex]:
Next, we need to express [tex]\( 512 = a^{b^c} \)[/tex] where [tex]\( a, b, c \)[/tex] are positive integers.
3. Finding values for [tex]\( a, b, c \)[/tex]:
We need to write [tex]\( 2^9 \)[/tex] in the form of [tex]\( a^{b^c} \)[/tex].
One possible way is to recognize that [tex]\( 512 = 2^9 \)[/tex] can be restructured as:
[tex]\[
512 = 2^{3^2}
\][/tex]
Thus, we can let [tex]\( a = 2 \)[/tex], [tex]\( b = 3 \)[/tex], and [tex]\( c = 2 \)[/tex]. This fits the form [tex]\( a^{b^c} \)[/tex].
4. Calculating [tex]\( abc \)[/tex]:
Now, we multiply these values to find [tex]\( abc \)[/tex]:
[tex]\[
abc = 2 \times 3 \times 2 = 12
\][/tex]
So, the minimum possible value of [tex]\( abc \)[/tex] when [tex]\( a^{b^c} = 512 \)[/tex] is [tex]\( 12 \)[/tex].
Thus, the correct answer is:
(b) 12
