$A=\dfrac{\left(1+\dfrac14\right)\left(3^4+\dfrac14\right).....\left(29^4+\dfrac14\right)}{\left(2^4+\dfrac14\right)\left(4^4+\dfrac14\right).....\left(30^4+\dfrac14\right)}$
$=\dfrac{2^4\left[\left(1+\dfrac14\right)\left(3^4+\dfrac14\right).....\left(29^4+\dfrac14\right)\right]}{2^4\left[\left(2^4+\dfrac14\right)\left(4^4+\dfrac14\right).....\left(30^4+\dfrac14\right)\right]}$
$=\dfrac{(2^4+4)(6^4+4).....(58^4+4)}{(4^4+4)(8^4+4).....(60^4+4)}$ $=\dfrac{(2^4+2.2^2.2+4-2.2^2.2)(6^4+2.6^2.2+4-2.6^2.2).....(58^4+2.58^2.2+4-2.58^2.2)}{(4^4+2.4^2.2+4-2.4^2.2)(8^4+2.8^2.2+4-2.8^2.2).....(60^4+2.60^2.2+4-2.60^2.2)}$
$=\dfrac{[(2^2+2)^2-2^2.2^2][(6^2+2)^2-2^2.6^2].....[(58^2+2)^2-2.58^2.2]}{[(4^2+2)^2-2^2.4^2][(8^2+2)^2-2^2.8^2].....[(60^2+2)^2-2^2.60^2]}$
$=\dfrac{(2^2+2-2.2)(2^2+2+2.2)(6^2+2-2.6)(6^2+2+2.6)....(58^2+2-2.58)(58^2+2+2.58)}{(4^2+2-2.4)(4^2+2+2.4)(8^2+2-2.8)(8^2+2+2.8).....(60^2+2-2.60)(60+2+2.60)}$
$=\dfrac{2.10.26.50.....3250.2482}{10.26.50.82.....3482.3722}$
$=\dfrac2{3722}=\dfrac1{1861}$
