To find the value of [tex]\( k \)[/tex] when [tex]\((3^6)^k = 3^2\)[/tex], follow these steps:
1. Understand the Problem:
We are given the equation [tex]\((3^6)^k = 3^2\)[/tex] and need to solve for [tex]\( k \)[/tex].
2. Apply the Exponentiation Rule:
The problem involves exponents, so we use the property of exponents [tex]\((a^m)^n = a^{m \cdot n}\)[/tex]. Applying this rule to our equation:
[tex]\[
(3^6)^k = 3^{6k}
\][/tex]
Therefore, the original equation [tex]\((3^6)^k = 3^2\)[/tex] can be rewritten as:
[tex]\[
3^{6k} = 3^2
\][/tex]
3. Equate the Exponents:
Since the bases on both sides of the equation are the same (both are 3), we can set the exponents equal to each other:
[tex]\[
6k = 2
\][/tex]
4. Solve for [tex]\( k \)[/tex]:
To find the value of [tex]\( k \)[/tex], solve the equation [tex]\( 6k = 2 \)[/tex] by isolating [tex]\( k \)[/tex]:
[tex]\[
k = \frac{2}{6}
\][/tex]
Simplify the fraction:
[tex]\[
k = \frac{1}{3}
\][/tex]
So, the value of [tex]\( k \)[/tex] is [tex]\(\boxed{0.3333333333333333}\)[/tex].
