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2014-12-20T12:00:45+08:00

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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:
    
 \sqrt[3]{( (3 x 72)}  \sqrt[3]{(72 + 144)}
 \sqrt[3]{216} =  \sqrt[3]{(72 + 144)}
 \sqrt[3]{216} =  \sqrt[3]{216}
          6             =          6 , it is equal, CORRECT!!!

Therefore, the number is 72.

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