द्विपद प्रमेय का प्रयोग करते हुए निम्न का मान ज्ञात कीजिये -
` (99) ^(5 ) `

Explanation

(i) ` (99) ^ 5 = ( 100 - 1 ) ^ 5 `
` = ""^ ( 5 ) C _ 0 xx ( 100 ) ^ 5 - ""^ 5 C _ 1 xx (100) ^ 4 + ""^ 5 C _ 2 xx ( 100 ) ^ 3 - ""^ 5 C _ 3 xx (100 ) ^ 2 + ""^ 5 C _ 4 xx (100) ^ 1 - ""^ 5 C _ 5 xx (100) ^(0 ) `
` = (100) ^( 5 ) - 5 xx (100) ^ 4 + 10 xx (100) ^ 3 - 10 xx (100 ) ^ 2 + 5 xx 100 - 1 `
` = 10 ^ 10 - 5 xx 10^ 8 + 10 ^ 7 - 10^ 5 + 5 xx 10 ^2 -1 `
` = ( 10^(10 ) + 10 ^7 + 5 xx 10 ^ 2 ) - ( 5 xx 10 ^8 + 10 ^ 5 + 1 ) `
` = 10010000500 -500100001 = 9509900499 `
(ii) ` (102 ) ^6 = (100 + 2 ) ^ 6 `
` = ""^ 6 C _ 0 xx (100) ^6 + ""^ 6 C _ 1 xx (100) ^ 5 xx 2 + ""^ 6 C _ 2 xx (100) ^ 4 xx 2 ^ 2 + ""^ 6 C _ 3 xx (100)^ 3 xx 2 ^ 3 + ""^6 C _ 4 xx (100) ^ 2 xx 2 ^ 4 + ""^ 6 C _ 5 xx (100) ^ 1 xx 2 ^ 5 + ""^ 6 C _ 6 (100 ) ^ 0 xx 2 ^ 6 `
` = 10^ (12 ) + 12 xx 10 ^(10 ) + 6 xx 10 ^(9) + 16 + 10^7 + 24xx 10^5 + 192 xx 10^ 2 + 64 `
= 1126162419264
(iii) ` (10.1)^ 5 = (10 + 0.1)^ 5 `
` = ""^ 5 C _ 0(10 ) ^ 5 (0.1) ^ 5 + ""^ 5 C_ 1 xx (100)^ 4 xx (0.1 ) + ""^ 5C _ 2 xx (10) ^ 3 xx (0.1 ) ^ 2 + ""^ 5 C _ 3 xx (10)^ 2 xx (0.1 ) ^ 3 + ""^5 C _ 4 xx (10)^ 1 xx (0.1)^ 4 + ""^ 5 C _ 5 xx (10)^0 xx (0.1) ^ 5 `
` = (10)^ 5 + 5 xx 10^ 4 xx 0.1 + 10 xx 10 ^ 3 xx (0.1)^ 2 + 10 xx (10)^ 2 xx (0.1 ) ^ 3 + 5 xx 10 xx (0.1 ) ^ 4 + (0.1 ) ^ 5 `
` = 10 ^ 5 + 5 xx 10^ 2 + 1 + 5 xx 0.001 + 0.0000 1 `
` = 100000 + 5000 100 + 1 + 0.005 + 0.0000 1 = 105101.00501 `