Question

Find the fifth roots of -1

Answer 1

ur answer is -1





I think so

Answer 2

Consider the Following quintic equation:

${x}^{5} = - 1$

${x}^{5} + 1 = 0$

This can be factorised as:

$(x + 1)( {x}^{4} - {x}^{3} + {x}^{2} - x + 1) = 0$

So, the roots of the above factorised linear and quartic equation are:

$x_{1} = - 1 \: : \: x_{2} = {e}^{i \frac{\pi}{5} } : x _{3} = - { e ^{i \frac{2\pi}{5} } } \\ x_{4} = {e}^{i \frac{3\pi}{5} } : x_{5} = { {e}^{i \frac{4\pi}{5} } }$

OR their appropriate forms can be written as:

x₁ = −1 (Principal Root)

x₂,₅ ≈ 0.80902 ± 0.58779i

x₃,₄ = −0.30902 ± 0.95106i

where, i = √(−1)