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

Find an 8^th 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 16^th 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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