How do you simplify root8(-1066)?

1 Answer
George C.
Aug 13, 2016

root(8)(-1066) is not simplifiable as such, but is expressible in the form a+bi:

root(8)(-1066) =1/2 root(8)(1066)sqrt(2+sqrt(2))+(1/2 root(8)(1066)sqrt(2-sqrt(2))) i

Explanation:

First note that 1066 = 2*13*41 has no square factors, let alone any higher powers, so this radical does not simplify as such, but we can get it into the form a+bi.

I will use:

cos(pi/8) = 1/2sqrt(2+sqrt(2))

sin(pi/8) = 1/2sqrt(2-sqrt(2))

(cos theta + i sin theta)^n = (cos n theta + i sin n theta) " " (de Moivre)

So we find:

root(8)(-1066) = (1066(cos pi + i sin pi))^(1/8)

=root(8)(1066)(cos pi + i sin pi)^(1/8)

=root(8)(1066)(cos (pi/8) + i sin (pi/8))

=root(8)(1066)cos(pi/8)+(root(8)(1066)sin(pi/8)) i

=1/2 root(8)(1066)sqrt(2+sqrt(2))+(1/2 root(8)(1066)sqrt(2-sqrt(2))) i

Note that this is the principal 8th root, there are 7 others which may be found by repeatedly multiplying by:

cos (pi/4) + i sin (pi/4) = sqrt(2)/2 + (sqrt(2)/2)i

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