algebra

Sum of cubes factorization, which can be generalized to all odd powers of n:

 

a^3 + b^3 = (a+b)(a^2-ab+b^2)

replace a and b with c and 1/c respectively in this case to get:

 

c^3 + (1/c)^3 = (c+1/c)(c^2-1+1/c^2)

since we are already told c^2 + 1/c^2 = 14, we can substitute to get:

 

c^3 + (1/c)^3 = (c+1/c)(14-1) = (c+1/c)(13)

now all we need to find is the value of c+1/c

 

realize that:

 

(c+1/c)^2 = c^2 + 1/c^2 + 2 = 14 + 2 = 16 

this is because we are given that c^2 + 1/c^2 = 14, so just substitute

(c+1/c) = sqrt(16) = +-4

Since we want the largest value possible of the expression, we can assume it's positive, because if it's negative, then -4(13) would likewise be negative. Our answer is then:

 

4(13) = 52