Let's solve the equation [tex]\( 8^{2n} = \frac{1}{64} \)[/tex] step by step.
1. Rewrite [tex]\(\frac{1}{64}\)[/tex] in terms of base 8:
We start by expressing 64 as a power of 8. Notice that:
[tex]\[
64 = 8^2
\][/tex]
Therefore:
[tex]\[
\frac{1}{64} = \frac{1}{8^2}
\][/tex]
Using the property of exponents [tex]\( a^{-m} = \frac{1}{a^m} \)[/tex], we can rewrite [tex]\(\frac{1}{8^2}\)[/tex] as:
[tex]\[
\frac{1}{8^2} = 8^{-2}
\][/tex]
2. Set up the equation with the same base:
Substitute [tex]\( \frac{1}{64} \)[/tex] with [tex]\( 8^{-2} \)[/tex]:
[tex]\[
8^{2n} = 8^{-2}
\][/tex]
3. Equate the exponents:
Since the bases are the same (both are base 8), we can equate the exponents:
[tex]\[
2n = -2
\][/tex]
4. Solve for [tex]\(n\)[/tex]:
To solve for [tex]\( n \)[/tex], we divide both sides of the equation [tex]\( 2n = -2 \)[/tex] by 2:
[tex]\[
n = \frac{-2}{2}
\][/tex]
Simplify the right-hand side:
[tex]\[
n = -1
\][/tex]
Therefore, the value of [tex]\( n \)[/tex] is [tex]\( \boxed{-1} \)[/tex].
