If `a ,b ,c` are non-zero real numbers, then find the minimum value of the expression `(((a^4+ 3a^2+1)(b^4+5b^2+1)(c^4+7c^2+1))/(a^2b^2c^2))` which is not divisible by prime number.

Answer

315

Explanation

We have `((a^(4)+3a^(2)+1)/a^(2))((b^(2)+5b^(2)+1)/(b^(2)))((c^(4)+7c^(2)+1)/(c^(2)))`
`=(a^(2)+(1)/(a^(2))+3)(b^(2)+(1)/(b^(2))+5)(c^(2)+(1)/(c^(2))+7)`
`=((a-(1)/(a))^(2)+5)((b-(1)/(b))^(2)+7)((c-(1)/(c))^(2)+9)`
So, minimum value is `5xx7xx9=315`.