To find the smallest positive integer \( n \) such that \( \sqrt[4]{675 + n} \) is an integer, we set \( \sqrt[4]{675 + n} = k \) where \( k \) is a positive integer. This leads us to the equation:
\[ 675 + n = k^4 \]
From this, we can isolate \( n \):
\[ n = k^4 - 675 \]
Since we want \( n \) to be a positive integer, we require:
\[ k^4 > 675 \]
To find the smallest \( k \), we first need to calculate \( \sqrt[4]{675} \). We start by estimating \( \sqrt[4]{675} \):
\[ \sqrt[4]{675} = \sqrt{\sqrt{675}} \]
Calculating \( \sqrt{675} \), we have:
\[ 675 = 25 \times 27 = 25 \times 9 \times 3 = 5^2 \times 3^3 \]
Thus,
\[ \sqrt{675} = \sqrt{25 \times 27} = 5\sqrt{27} = 5 \cdot 3\sqrt{3} = 15\sqrt{3} \]
Now, estimating \( \sqrt{3} \approx 1.732 \):
\[ \sqrt{675} \approx 15 \times 1.732 \approx 25.98 \]
Now calculating \( \sqrt[4]{675} \):
\[ \sqrt[4]{675} \approx \sqrt{25.98} \approx 5.10 \]
Thus, the smallest integer \( k \) must be at least \( 6 \) because \( 5^4 = 625 \) is less than \( 675 \).
Now we calculate \( n \) for \( k = 6 \):
\[ n = 6^4 - 675 = 1296 - 675 = 621 \]
Next, we check if \( k = 5 \) also provides an integer but we already determined that \( n < 0 \) since \( 5^4 = 625\) which is less than \( 675 \).
Therefore, the minimum \( k \geq 6 \).
To confirm:
\[ k = 6 \Rightarrow n = 621 \Rightarrow 675 + 621 = 1296 \Rightarrow \sqrt[4]{1296} = 6 \]
This confirms that indeed, \( \sqrt[4]{675 + n} \) is an integer.
Thus, the smallest positive integer \( n \) such that \( \sqrt[4]{675 + n} \) is an integer is
\[ \boxed{621} \]