It seems like you're referring to some notation with exponents that includes different bases and possibly repeating decimals. Let's clarify what you mean and simplify accordingly.
Based on what you've written, it looks like you want to simplify the following expression using the product rule for exponents:
\[ 5^{\tan} \cdot 5^5 \]
According to the product rule for exponents, which states that \( a^m \cdot a^n = a^{m+n} \), we can combine the exponents as long as the bases are the same.
Assuming "\(\tan\)" stands for a value we can represent, and combining the \(\tan\) and \(5\) gives us:
\[ 5^{\tan + 5} \]
Now, if we see any specific notation like \( \displaystyle 25 \) meaning \( 5^2 \) or if you meant something else with \( 50 \) and \( 15 \) in the superscript, please clarify.
If we're still assuming to simplify in the context of the bases repeating, and you meant something like:
- \( 5^{25} \) (which is \( 25 \) small on top)
Then, without additional context of what \( 25 \small{50} \), \( 5^{15} \), or other notations represent, their simplification wouldn’t follow a standard exponent property.
Please provide a little more context or clarify your expression so I can help you further!
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