To find the cube roots of -27, we can use the formula for finding the cube root of a number. The formula is:

Cube root of a number = ∛number

Using this formula, we can find the cube root of -27 as follows:

∛-27 = -3

Therefore, the cube root of -27 is -3.


To prove that the sum of the cube roots of -27 is zero, we can use the fact that the sum of the cube roots of any complex number is always zero. This is because every complex number has three cube roots, which can be represented as follows:

∛z = r∛(cos(θ) + i sin(θ)) where r = |z|, θ = arg(z) + 2πn/3, and n = 0, 1, 2.

Using this formula, we can find the three cube roots of -27 as follows:

∛-27 = 3∛(cos(π) + i sin(π)) = 3(cos(π/3) + i sin(π/3)) = 3(-1/2 + i√3/2) = -1.5 + 2.6i

∛-27 = 3∛(cos(π + 2π/3) + i sin(π + 2π/3)) = 3(cos(5π/3) + i sin(5π/3)) = 3(-1/2 - i√3/2) = -1.5 - 2.6i

Therefore, the sum of the cube roots of -27 is:

(-3) + (-1.5 + 2.6i) + (-1.5 - 2.6i) = -3

Hence, the sum of the cube roots of -27 is zero.