The following formula I <(Orson Pratt, Sen.)> discovered April 25th 1876, in the 19th Ward, Salt Lake City, Utah.

Find an 8th root of    √-1, or <of>  x8 = √-1 .

x = [½ + ½ (√(2+1)/2√2 )½ ]½ + [½ – ½ (√(2+1)/2√2 )½ ]½   .

Find a 16th root of =√-1  ; or of x16 = √-1 .

x = {½ + ½ [½ + ½ (√(2+1)/2√2 )½ ]½}½ + {½-½ [½ + ½ (√(2+1)/2√2 )½ ]½}½  √-1 .

Find an <expression for the 2n> roots of x ²ⁿ = -1; as a root of √-1)²ⁿ .

x = ±{½ ± ½ [½ ± ½ (….§ ½ ± ½ · 1/√2 §½….)½]½}½

± {½ ± [ ½ ± ½ ( …. § ½±½ · 1/√2 ….)½]½}½  .

In each term of this rest <expression> there must be n-1 factors, under the sign of the square root.

The second sign in each of the two terms must be opposite; that is, if one is positive the other must be negative.  The other signs may have every possible change, and every change will give a (2ⁿ)th root, or one of the values of x.

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Find the sixteen imaginary roots of x16 + 1 = 0.
x16 + 1= (x8 – √-1  ) (x8 + √-1 ) = 0.

x = ±[½+½(½+½ · 1/√2 ) ½]½ ±[½-½(½+½. 1/√2 )½]½ (√-1) ,   x²=+(½+½ · 1/√2 )½  (√-1) ; x4= + 1/√2 + 1/√2 (√-1)   ; x16 = -1

x=±[½-½(½+½ . 1/√2 )½] ½ +[½+½(½+½ · 1/√2 )½]½ (√-1),  x² = -(½+½ · 1/√2 )½ (√-1) ;x4=+ (1/√2) (√-1)   ; x8= (√-1) 😡16=-1

x = ±[½+½(1/√2  -½ · 1/√2)½]½  ±[½- ½ (½-½· 1/√2 )½]½ (√-1) ; x²=+(½-½ (1/√2) )½ – (½+½ · (1/√2) )½  (√-1) ; x4 = – 1/√2 – 1/√2  (√-1)  ; x8 =  (√-1); x16 = – 1

x = ±[½-½(½-½ · (1/√2) )½]½±[½+ ½(½-½ · (1/√2) )½]½  (√-1) ; x²=-(½-½ ·(1/√2) )½+(½+½·(1/√2) )½  (√-1) ; x4 = -(1/√2) – (1/√2)  (√-1) ; x8 =  (√-1) ; x16 = -1

x = ±[½+½(½-½· (1/√2) )½]½±[½- ½.(½-½· (1/√2) )½]½ (√-1) ; x²=+(½-½· (1/√2 )½+(½+½· (1/√2) )½ (√-1) ; x4 = – (1/√2) + (1/√2) (√-1)  ; x8 – (√-1) ;  x16 = – 1

X = ±[½-½(½-½· (1/√2) )½]½ ±[½+ ½.(½-½· (1/√2) )½]½ (√-1) ; x²=-(½-½· (1/√2) )½-(½+½· (1/√2) )½ (√-1) ; x4 = – (1/√2) + (1/√2) (√-1) ; x8 = (√-1); x16 = – 1

x = ±[½+½(½+½.· (1/√2) )½]½  ±[½- ½.(½+½· (1/√2) )½]½ (√-1) ; x²=+(½+½· (1/√2)  )½-(½-½·  (1/√2) )½ (√-1) ; x4 = -(1/√2)  – (1/√2) (√-1)  ; x8 = (√-1) ;  x16 = – 1

x = ±[½-½(½+½· (1/√2) )½]½±[½+ ½.(½+½· (1/√2) )½]½ (√-1) ; x²-(½+½· (1/√2) )½+(½-½· (1/√2) )½ (√-1) ; x4 = – (1/√2) – (1/√2) (√-1) ; x8 = (√-1);  x16 = – 1

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[Transcribed by Nora Fowers, Shannon Devenport, Dick Grigg; Feb. 2011]

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