` a^n = [ ( a - b ) + b ] ^ n `
` = ""^ nC _ 0 ( a - b )^n + ""^ n C _ 1 ( a - b ) ^ ( n - 1 ) + b + … + ""^n C _ ( n - 1 ) ( a- b ) b ^ ( n - 1 ) + ""^ nC _ n b ^n `
` = ( a - b ) ^n + ""^n C _ 1 ( a - b ) ^ (n - 1 ) * b + ... + ""^n C _ ( n -1 ) ( a - b ) b ^( n - 1 ) + b^n + b ^n [ because ""^ n C _ 0 = ""^ n C _ n = 1 ] `
` rArr a ^ n - b ^n = ( a - b ) ^n + ""^ n C _ 1 ( a- b ) ^ (n - 1 ) * b + ... + ""^n C _ ( n - 1 ) ( a - b ) b ^ ( n -1 ) `
` = ( a - b) [ ( a - b ) ^(n - 1 ) + ""^ n C _ 1 ( a - b ) ^(n - 2 ) * b + ... + ""^ n C _ ( n - 1 ) b ^ (n - 1 ) ] `
स्पष्टतः दायाँ पक्ष (a -b ) से विभाजित होगा |
` rArr ` बायाँ पक्ष = ` ( a ^n - b ^n ) ` भी ` ( a- b ) ` से विभाजित होगा |