To find the value of [tex]\( p \)[/tex] in the equation [tex]\( 2^p = \frac{1}{8^4} \)[/tex], let's go through the steps in detail:
1. Rewrite the expression with a common base:
Notice that 8 can be written as a power of 2. Specifically, [tex]\( 8 = 2^3 \)[/tex].
2. Express [tex]\( 8^4 \)[/tex] using the common base:
[tex]\[
8^4 = (2^3)^4
\][/tex]
Using the power of a power rule [tex]\((a^m)^n = a^{mn}\)[/tex], we get:
[tex]\[
(2^3)^4 = 2^{3 \times 4} = 2^{12}
\][/tex]
3. Substitute back into the original equation:
[tex]\[
2^p = \frac{1}{8^4} = \frac{1}{2^{12}}
\][/tex]
4. Rewrite the fraction with a negative exponent:
Using the negative exponent rule [tex]\(\frac{1}{a^n} = a^{-n}\)[/tex], we convert the right-hand side:
[tex]\[
2^p = 2^{-12}
\][/tex]
5. Set the exponents equal to each other:
Since the bases are the same, we can set the exponents equal to solve for [tex]\( p \)[/tex]:
[tex]\[
p = -12
\][/tex]
Therefore, the value of [tex]\( p \)[/tex] is [tex]\( \boxed{-12} \)[/tex].
