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Let $N = 2^4\times 3^4\times 5^2$ . Notice that $N$ is a perfect square,

Number of positive factors of $N$ will be $(4 + 1)(4 + 1)(2 + 1) = 75$

Now if all positive factors are written in ascending order then middle position is $\dfrac{75 + 1}{2} = 38$ .

Using the fact that, in case of the ascending order of the positive factors of a perfect square, it's middle position is equal to the square root of the number, our required answer becomes $\sqrt N = 2^2\times 3^2\times 5 = 180$