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Misc 3 - Reduce (1/1 + 4i − 2/1 + ๐‘–) (3 − 4i/5 + i) - Miscellaneous

  1. Chapter 5 Class 11 Complex Numbers
  2. Serial order wise
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Misc 3 Reduce (1/(1 + 4๐‘–)โˆ’2/(1 + ๐‘–)) ((3 โˆ’ 4๐‘–)/(5 + ๐‘–)) to the standard form. We have to solve and then make it in Standard form a + ib (1/(1 + 4๐‘–)โˆ’2/(1 + ๐‘–)) ((3 โˆ’ 4๐‘–)/(5 + ๐‘–)) = (((1 + ๐‘–) โˆ’ 2(1 โˆ’ 4๐‘–))/(1 โˆ’ 4๐‘–)(1 + ๐‘–) ) ((3 โˆ’ 4๐‘–)/(5 + ๐‘–)) = ((1 + ๐‘– โˆ’ 2 + 8๐‘–)/(1 + ๐‘– โˆ’4๐‘– โˆ’4๐‘–2)) ((3 โˆ’ 4๐‘–)/(5 + ๐‘–)) = ((1 + ๐‘– โˆ’ 2 + 8๐‘–)/(1 + ๐‘– โˆ’4๐‘– โˆ’ 4 ร—(โˆ’1))) ((3 โˆ’ 4๐‘–)/(5 + ๐‘–)) = ((1 + ๐‘– โˆ’ 2 + 8๐‘–)/(1 + ๐‘– โˆ’ 4๐‘– + 4)) ((3 โˆ’ 4๐‘–)/(5 + ๐‘–)) = (((1 โˆ’ 2) + ๐‘– + 8๐‘–)/(1 +4 + ๐‘– โˆ’4๐‘– )) ((3 โˆ’ 4๐‘–)/(5 + ๐‘–)) = ((โˆ’1 + 9๐‘–)/(5 โˆ’ 3๐‘–)) ((3 โˆ’ 4๐‘–)/(5 + ๐‘–)) = ((โˆ’1 + 9๐‘–) (3 โˆ’ 4๐‘–))/(5 โˆ’ 3๐‘–)( 5 + ๐‘–) = (โˆ’1(3 โˆ’ 4๐‘–) + 9๐‘–(3 โˆ’ 4๐‘–))/(5 ( 5 + ๐‘–) โˆ’ 3๐‘–( 5 + ๐‘–) ) = (โˆ’3 + 4๐‘– +27๐‘– โˆ’36๐‘–2)/(25 + 5๐‘– โˆ’15๐‘– โˆ’3๐‘–2) = (โˆ’3 + 4๐‘– +27๐‘– โˆ’36 ร— โˆ’1)/(25 + 5๐‘– โˆ’15๐‘– โˆ’3 ร— โˆ’1) = (โˆ’3 + 4๐‘– +27๐‘– + 36)/(25 + 5๐‘– โˆ’15๐‘– + 3) = (โˆ’3 + 36+ 4๐‘– +27๐‘– )/(25 + 3 + 5๐‘– โˆ’15๐‘– ) = (33 +31๐‘– )/(28 โˆ’10๐‘– ) = (33 +31๐‘– )/(2(14 โˆ’5๐‘–) ) = 1/2ร—(33 +31๐‘– )/((14 โˆ’5๐‘–) ) Rationalizing = 1/2ร—(33 + 31๐‘– )/((14 โˆ’ 5๐‘–) ) ร— ((14 + 5i))/((14 + 5i)) = 1/2ร—(33(14 + 5i) + 31๐‘–(14 + 5i))/((14 โˆ’ 5๐‘–)(14 + 5i)) = 1/2ร—(462 + 165๐‘– + 434๐‘– + 155๐‘–2)/(142 โˆ’ (5๐‘–)2) = 1/2ร—(462 + 165๐‘– + 434๐‘– + 155๐‘–2)/(196 โˆ’ 25๐‘–2) " = " 1/2ร—(462 + 165๐‘– + 434๐‘– + 155(โˆ’1))/(196 โˆ’25(โˆ’1)) " = " 1/2ร—(462 + 165๐‘– + 434๐‘– โˆ’ 155)/(196 + 25) " = " 1/2ร—(462 โˆ’ 155 + 165๐‘– + 434๐‘– )/221 " = " (307 + 599๐‘– )/2(221) " = " (307 + 599๐‘– )/442 = 307/442 + 599/442 ๐‘– This is the required standard form.

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Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He provides courses for Mathematics from Class 9 to 12. You can ask questions here.
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