To simplify the expression \( \frac{3639}{\sqrt{913}} \), we can first rationalize the denominator.
The expression can be rewritten as:
\[ \frac{3639}{\sqrt{913}} \cdot \frac{\sqrt{913}}{\sqrt{913}} = \frac{3639 \sqrt{913}}{913} \]
Now, we simplify \( \frac{3639}{913} \).
Let's perform the division:
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We note that 913 can be factored to see if there are any common factors with 3639. We can perform the division:
Performing the division, \[ 3639 \div 913 \approx 4 \] We can calculate: \[ 913 \times 4 = 3652 \] Since \( 3652 > 3639 \), let's try \( 3 \): \[ 913 \times 3 = 2739 \] Now, after some testing, we find \( 3639 - 2739 = 900 \).
Thus, we should check candidates as \( 3 + \frac{900}{913} \).
Calculating the rational number: \[ 3 = \frac{2739}{913} \]
So, \[ \frac{3639}{913} = 3 + \frac{900}{913} \]
Putting it back to the result produces an approximate: \[ \frac{3639 \sqrt{913}}{913} \text{ can be thought as } 3.99 \frac{\sqrt{913}}{1}. \]
Accordingly, we could summarize the final simplified form as approximately: \( 4\sqrt{913} - \frac{900 \sqrt{913}}{913} \).
Thus, combining both: \[ \frac{3639 \sqrt{913}}{913} = 4 \sqrt{913} - \frac{3639 \sqrt{913}}{913} \]
To have whole units, \[ \frac{3639\sqrt{913}}{913} = 3 \sqrt{913} + \frac{900\sqrt{913}}{913}, \] which is roughly \( 4 \sqrt{913} \), in exact form for higher check \( \frac{3639 \sqrt{913}}{913} \approx \underline{4 \sqrt{913}} \).
The complete rationalized result is: \[ \frac{3639 \sqrt{913}}{913} \]
Thus, the exact answer is: \[ \frac{3639\sqrt{913}}{913} \]
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