Okay, let's tackle this problem step by step. The question is asking to find the value of k such that (3^k)^6 equals 3^6. Hmm, exponents can sometimes be tricky, but maybe if I remember the exponent rules correctly, I can figure this out. Let me start by recalling some basic exponent properties.

First, I know that when you have an exponent raised to another exponent, you can multiply the exponents. So (a^m)^n equals a^(m*n). Let me apply that here. If I take (3^k)^6, according to the rule, that should be 3^(k*6). So the left side of the equation becomes 3^(6k). 

The right side of the equation is 3^6. Since the bases are the same (both are base 3), the exponents must be equal for the equation to hold true. That means 6k must equal 6. So, setting up the equation 6k = 6. To solve for k, I can divide both sides by 6. That gives k = 6/6, which simplifies to k = 1. 

Wait, let me check that again. If k is 1, then (3^1)^6 is 3^6, which is indeed equal to the right side. So that seems to work. Let me verify if there's another possible solution. For example, could there be a negative exponent that also works? Let's see. If k were -1, then (3^-1)^6 would be (1/3)^6, which is 1/(3^6), which is not equal to 3^6. So negative exponents don't work here. What about fractions? If k were 2/3, then (3^(2/3))^6 would be 3^( (2/3)*6 ) = 3^4, which is 81, but the right side is 3^6 = 729. So that's not equal either. Therefore, the only solution is k = 1.

Alternatively, maybe I can use logarithms to solve for k. Taking the logarithm base 3 of both sides: log_3( (3^k)^6 ) = log_3(3^6). Using the power rule for logarithms, that becomes 6 * log_3(3^k) = 6 * log_3(3). Then 6 * k * log_3(3) = 6 * 1. Since log_3(3) is 1, that simplifies to 6k = 6, leading again to k = 1. So that confirms the answer again.

Another way to think about it: if (3^k)^6 = 3^6, then the exponents must be equal because the bases are the same. Therefore, 6k = 6, so k = 1. Yep, that's straightforward. I don't see any other possible values for k that would satisfy this equation. 

Just to make sure I didn't make a mistake in applying exponent rules, let me recall that (a^m)^n = a^(m*n) and that a^m * a^n = a^(m+n). These are fundamental exponent properties. Here, we're dealing with a single exponent raised to another exponent, so the first property is definitely applicable. 

If I plug k = 1 back into the original equation: (3^1)^6 = 3^6. The left side is 3^6, which equals the right side. So it checks out. Therefore, the solution is correct.
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