Let's simplify the equation step by step.
We have:
\[ 5^{-5} \cdot (3^{\sqrt{5}})^{12} \]
First, simplify \((3^{\sqrt{5}})^{12}\):
\[ (3^{\sqrt{5}})^{12} = 3^{12\sqrt{5}} \]
Thus, the equation becomes:
\[ 5^{-5} \cdot 3^{12\sqrt{5}} \]
Now, let's compare this expression to 15:
We know that:
\[ 15 = 3^1 \cdot 5^1 \]
For the two sides to be equal, the powers of 3 and 5 would need to match exactly.
The left-hand side has the term \(3^{12\sqrt{5}}\) and the term \(5^{-5}\), while the right-hand side has \(3^1\) and \(5^1\).
Since \(12\sqrt{5}\) is not equal to 1, and \(-5\) is also not equal to 1, we can conclude that the original equation is false.
Thus, the answer is:
\[ \text{2 for false} \]
So, the final answer is 2.
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