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Assume\( a = 3^n -9\) and\( b = 9^n -3.\)
We have the equation\( a^3 + b^3 = (a +b)^3\) or\( 3ab(a + b) =0,\) since\( (a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3 = a^3 + b^3 + 3ab(a +b).\)
Then we have\( a = 0, b =0\) or\( a + b =0.\)
Case 1:\( a =0\)
Since we have\( 3^n – 9 =0,\) we have\( n =2.\)
Case 2:\( b =0\)
Since we have\( 9^n – 3 =0\) , we have\( n =½,\) which is not a solution, since\(n\) is a positive integer.
Case 3:\( a + b =0\)
\( 3^n + 9^n – 12 =0\)
\( => 9^n + 3^n – 12 =0\)
[m] => 3^{2n} + 3^n –
...
Assume\( a = 3^n -9\) and\( b = 9^n -3.\)
We have the equation\( a^3 + b^3 = (a +b)^3\) or\( 3ab(a + b) =0,\) since\( (a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3 = a^3 + b^3 + 3ab(a +b).\)
Then we have\( a = 0, b =0\) or\( a + b =0.\)
Case 1:\( a =0\)
Since we have\( 3^n – 9 =0,\) we have\( n =2.\)
Case 2:\( b =0\)
Since we have\( 9^n – 3 =0\) , we have\( n =½,\) which is not a solution, since\(n\) is a positive integer.
Case 3:\( a + b =0\)
\( 3^n + 9^n – 12 =0\)
\( => 9^n + 3^n – 12 =0\)
[m] => 3^{2n} + 3^n –
...
