We will make use of the expansion of the cube of a binomial
#(a+b)^3 = a^3+3a^2b+3ab^2+b^3#
as well as the quadratic formula
#ax^2+bx+c=0 => x = (-b+-sqrt(b^2-4ac))/(2a)#
Proceeding,
#root(3)(m+9) = 3 + root(3)(m-9)#
#=> m+9 = (3+root(3)(m-9))^3#
#=> m+9 = 27 + 27root(3)(m-9) + 9(root(3)(m-9))^2+m-9#
#=> 9(root(3)(m-9))^2 + 27root(3)(m-9) + 9 = 0#
#=> (root(3)(m-9))^2 + 3root(3)(m-9) + 1 = 0#
#=> root(3)(m-9) = (-3+-sqrt((-3)^2-4(1)(1)))/(2(1))#
#=> root(3)(m-9) = (-3+-sqrt(5))/2#
#=> 2root(3)(m-9) = -3 +-sqrt(5)#
#=> 8(m-9) = (-3 +- sqrt(5))^3#
#=> 8m - 72 = -27 +- 27sqrt(5) - 45 +- 15sqrt(5)#
#=> 8m - 72 = -72 +- 32sqrt(5)#
#=> 8m = +-32sqrt(5)#
#=> m = +-4sqrt(5)#
#:. |m| = 4sqrt(5)#
