2016/090) Find the value of $\frac{18^3+7^3+3.18 *7 *25}{3^6+6*243 *2 + 15 *81 *4+20 *27 *8 + 15 *9 *16 + 6*3*32 + 64}$

We have numerator = $18^3+7^3+3.18*7*25 = 18^3+7^3 + 3 *18 *7 (18+7) = (18+7)^3 = 25^3 = 5^6$
denominator = $3^6+6*243*2 + 15*81*4+20*27*8 + 15*9*16 + 6*3*32 + 64$*
$= 3^6 + 6 * 3^5 * 2 + 15 * 3^4 * 2^2  + 20 * 3^3 *2^3 + 15 * 3^2 * 2^4 + 6 * 3 * 2^5 + 2^6$
$= {6 \choose 0}3^6 + {6 \choose 1} * 3^5 * 2 + {6 \choose 2} * 3^4 * 2^2  + {6 \choose 3} * 3^3 *2^3 + {6 \choose 4} * 3^2 * 2^4 + {6 \choose 5} * 3 * 2^5 + {6 \choose 6}2^6$
$= ( 3+2)^6 = 5^6$
so given expression is 1