To find the largest integer whose square root is an irrational number between 3 and 4, we start by identifying the interval for the squares of 3 and 4.
Calculating these squares:
- The square of 3 is \(3^2 = 9\).
- The square of 4 is \(4^2 = 16\).
This means we are looking for integers \(n\) such that:
\[ 9 < n < 16 \]
The integers in this range are 10, 11, 12, 13, 14, and 15.
Next, we want to find the largest integer from this list that has an irrational square root. The square roots of these integers are:
- \(\sqrt{10}\), which is irrational.
- \(\sqrt{11}\), which is irrational.
- \(\sqrt{12}\), which is irrational.
- \(\sqrt{13}\), which is irrational.
- \(\sqrt{14}\), which is irrational.
- \(\sqrt{15}\), which is irrational.
Since all these values have irrational square roots and fall within the specified range, the largest integer in our considered range is 15.
Therefore, the largest integer whose square root is an irrational number between 3 and 4 is:
\[ \boxed{15} \]
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