The number is 72. I have no solutions because I don't know how to solve it systematically. I have done it using trial and error.
1 x 1 x 1 = 1
2 x 2 x 2 = 8
3 x 3 x 3 = 27
4 x 4 x 4 = 64
5 x 5 x 5 =125
6 x 6 x 6 = 216
7 x 7 x 7 =343
8 x 8 x 8 = 512
9 x 9 x 9 =729
1 - 144 = negative number and has no cube root, therefore, 1 is not the number.
8 - 144 = negative number and has no cube root, therefore, 8 is not the number.
27 - 144 = (the same)
64 - 144 = (the same)
125 - 144 = (the same)
216 - 144 = 72, the answer is positive, therefore, 72 could be the number.
343 - 144 = 199, the answer is positive, therefore, 199 could be the number.
512 - 144 = 368, the answer is positive, therefore, 368 could be the number.
729 - 144 = 585, the answer is positive, therefore, 585 could be the number.
The numbers left that we can try are: 199, 368, 585, 72
Let us try:
199 x 3 = 597, the cube root of 597 is an irrational number, it is not the answer.
368 x 3 =1,104, the cube root of 1,104 is complicated, it is not the answer.
585 x 3 = 1, 755, the cube root of 1, 755 is complicated, it is not the answer.
72 x 3 = 216, the cube root of 216 is 6, 72 could be the answer.
The number left, survived, and passed the previous screening is 72.
Let us try it:
[tex] \sqrt[3]{( (3 x 72)} [/tex] = [tex] \sqrt[3]{(72 + 144)} [/tex]
[tex] \sqrt[3]{216} = \sqrt[3]{(72 + 144)} [/tex]
[tex] \sqrt[3]{216} = \sqrt[3]{216} [/tex]
6 = 6 , it is equal, CORRECT!!!
Therefore, the number is 72.
