How to find the value of this expression?

Question

I just saw this question in one exam. Please help me solve it. I am not able to find any clue on where to begin.

(ignore that tick it might be wrong)

Answer 1

As it seems to be a multiple choice question, we can find the answer without much computing: all factors have the form $$(2n)^4+324=4(4n^4+81).$$ Since there are as many factors in the numerator as in the denominator, the $4$s cancel out, which results in a fraction with odd numerator and denominator, hence this fraction must simplify to an odd number. There's only one odd number in the proposed answers.

Answer 2

Hint You may try with the Sophie Germain identity. As $(10^4+324)=(10^4+4\times3^4)$

Answer 3

Using the following identity:

$$a^4 + 4\cdot 3^4 = (a^2 + 2 \cdot 3^2 - 2\cdot 3\cdot a)(a^2 + 2 \cdot 3^2 + 2\cdot 3\cdot a) = (a(a-6) + 18)(a(a+6)+18)$$

Most of the terms cancel out and you are left with:

$$\frac{58(64)+18}{4(-2)+18} = \frac{3730}{10} = 373$$

As KprimeX mentioned, this flows from the Sophie Germain Identity.