(v+6)^3+72=0

Question

is a real number. Write your answer in simplified radical form. (If there is more than one solution, separate them with commas.)

Answer 1

A³+B³=(A+B)(A²-AB+B²)=A³-A²B+AB²+A²B-AB²+B³).

So when A=v+6 and B=∛72:

(v+6)³+72=(v+6+∛72)((v+6)²-(v+6)∛72+∛72²).

∛72=∛(8×9)=2∛9, so v+6+2∛9=0, v=-6-2∛9, which can also be written v=-6-2(3^⅔).

There are two more zeroes but they are complex conjugates. There's only one real solution for v.