Explicație pas cu pas:
[tex]A = {63}^{n} + {7}^{n + 1}\cdot {3}^{2n + 1} - {21}^{n}\cdot {3}^{n + 2} = \\ [/tex]
[tex]= {(7 \cdot {3}^{2} )}^{n} + (7\cdot {7}^{n})\cdot (3\cdot {3}^{2n}) - {(3\cdot 7)}^{n}\cdot ({3}^{2}\cdot {3}^{n}) \\ [/tex]
[tex]= {7}^{n}\cdot {3}^{2n} + 21\cdot{7}^{n}\cdot {3}^{2n} - {3}^{n}\cdot {7}^{n} \cdot 9\cdot {3}^{n} \\ [/tex]
[tex]= {7}^{n}\cdot {3}^{2n} + 21\cdot{7}^{n}\cdot {3}^{2n} - 9\cdot {7}^{n} \cdot {3}^{2n} \\ [/tex]
[tex]= {7}^{n} \cdot {3}^{2n}\cdot(1 + 21 - 9) \\ [/tex]
[tex]= {7}^{n} \cdot {3}^{2n}\cdot \bf \red{13}[/tex]
-
[tex]B = {35}^{n} + {7}^{n}\cdot {5}^{n + 2} + 3 \cdot {7}^{n + 1}\cdot {5}^{n} = \\ [/tex]
[tex]= {(7\cdot5)}^{n} + {7}^{n}\cdot ({5}^{2} \cdot {5}^{n}) + 3 \cdot (7\cdot{7}^{n})\cdot {5}^{n} \\ [/tex]
[tex]= {7}^{n} \cdot{5}^{n} + {7}^{n}\cdot 25 \cdot {5}^{n} + 3 \cdot 7\cdot{7}^{n}\cdot {5}^{n} \\ [/tex]
[tex]= {7}^{n} \cdot{5}^{n}\cdot(1 + 25 + 21) \\ [/tex]
[tex]= {7}^{n} \cdot{5}^{n}\cdot \bf \red {47}[/tex]
q.e.d.
