find the three cubic roots of 8 write the roots in standard

find the three cubic roots of -8. write the roots in standard form

Solution

x³ = -8

==> x³ + 8 = 0

==> x³ + 2³ = 0   ; we have a³ + b³ = (a + b)(a² - ab + b²)

==> (x + 2)(x² - x(2) + 2²) = 0

==> (x + 2)(x² - 2x + 4) = 0

==> x + 2 = 0 , x² - 2x + 4 = 0

==> x = -2 , x² - 2x + 4 = 0

consider x² - 2x + 4 = 0

==> x² - 2x + 1 + 3 = 0

==> (x - 1)² = -3    ; since a² - 2ab + b² = (a - b)²

==> (x - 1) = (-3) , -(-3)

==> x = 1 + (-3) , 1 - (-3)

==> x = 1 + i3 , 1 - i3 ; since i = (-1)

Hence the three cubic roots of -8 are -2 , 1 - i3 , 1 + i3