` - 7 - 24 i ` का वर्गमूल निकालिये |

Explanation

माना कि ` sqrt ( - 7 - 24i ) = x + iy `, जहाँ x और y वास्तविक संख्याएँ है |
वर्ग करने पर, ` - 7 - 24i = ( x + iy ) ^ 2 = x ^ 2 + i ^ 2 y ^ 2 + 2 x y i `
या ` - 7 - 24 i = x ^ 2 - y^ 2 + i 2 xy `
Real और Imaginary part को बराबर करने पर,
` x ^ 2 - y ^ 2 = - 7 " " `...(1)
` 2xy = -24 ` या ` xy = -12" " `...(2)
अब ` ( x ^ 2 + y^ 2 ) ^ 2 = ( x ^ 2 - y ^ 2 ) ^ 2 + 4 x ^ 2 y ^ 2 = ( -7 ) ^ 2 + 4 ( -12 ) ^ 2 = 49 + 576 = 625`
` therefore x ^ 2 + y ^ 2 = pm 25 ` लेकिन ` x^ 2 + y ^ 2 cancel (lt ) 0 `
` therefore x ^ 2 + y^ 2 = 25 " " `...(3)
(1) और (3 ) को जोड़ने पर ` 2x ^ 2 = 18 rArr x ^ 2 = 9 rArr x = pm 3 " " `...(4)
अब (2 ) से, जब ` x = 3, y = - 4 `
तथा जब ` x = -3, y = 4 `
अतः अभीष्ट वर्गमूल = ` x + iy = 3 - 4i , - 3 + 4 i = - ( 3 - 4i ) `
अतः ` sqrt ( - 7 - 24 i ) = pm ( 3 - 4 i ) `
`{:("Second Method": ),( - 7 - 24i ) ,( = 3^ 3 + ( 4i ) ^ 2 - 2*3* 4i ) ,( = (3 - 4i ) ^ 2 ) , ( therefore sqrt(-7 - 24i ) = pm ( 3- 4i)):} :| {:("Rough"),( (24) /(2) = 12 = 3 xx 4" या" 2xx 6 "या " 1 xx 12 ) , ( 3"," 4 to 3"," 4i to 3 ^ 2 + 16 i ^ 2 = -7 ) , ( 2"," 6 to 2"," 12i ) , ("real part"= -7 (-ve)),(therefore "बड़े गुणनखण्ड के साथ" i "ले"):}`