# How to find the third root of z^3=1?

In summary, the student attempted to solve for z in C using z^3-1=0, but was unable to find an answer because he mistakenly applied a different formulae than what was given to him. DeMoivre's formula was another way of finding the roots which were spaced 120 degrees apart with the same modulus of 1.

## Homework Statement

in a given activity: solve for z in C the equation: z^3=1

## Homework Equations

prove that the roots are 1, i, and i^2

## The Attempt at a Solution

using z^3-1=0 <=> Z^3-1^3 == a^3-b^3=(a-b)(a^2+2ab+b^2)
it's clear the solution are 1 and i^2=-1 but i didn't find "i" as a solution by using this method

Are you sure the equation is right or the roots are right?

1 is a root but i and i^2 aren't

When you plug them into z^3 = 1 you should get 1 each time but i gives -i = 1 and i^2 gives -1 = 1 right?

## Homework Statement

in a given activity: solve for z in C the equation: z^3=1

## Homework Equations

prove that the roots are 1, i, and i^2

## The Attempt at a Solution

using z^3-1=0 <=> Z^3-1^3 == a^3-b^3=(a-b)(a^2+2ab+b^2)
it's clear the solution are 1 and i^2=-1 but i didn't find "i" as a solution by using this method

You will see the following request in many threads here at PF, especially in those started by relative new-comers.

Please state the entire problem as it was given to you.

jedishrfu said:
Are you sure the equation is right or the roots are right?

1 is a root but i and i^2 aren't

When you plug them into z^3 = 1 you should get 1 each time but i gives -i = 1 and i^2 gives -1 = 1 right?

the mistake was in applying the formulae a^3-b^3 which equals to :

• a3 – b3 = (ab)(a2 + ab + b2)
i wrote 2ab instead of ab only without myltiplying it by 2
so, again the roots are: -1/2+sqrt(3)/2*i and -1/2-sqrt(3)/2

by using the D=b^2-4ac for the quadratic one and a=b for the first term which is between brackets

## 1. What is the third root of z^3=1?

The third root of z^3=1 is 1, because 1 multiplied by itself three times equals 1.

## 2. How do you find the third root of z^3=1?

To find the third root of z^3=1, you can take the cube root of both sides of the equation. This means finding the number that, when multiplied by itself three times, gives you 1. In this case, the third root is 1.

## 3. Is 1 the only solution to z^3=1?

No, there are two other solutions to z^3=1. These are -0.5 + 0.866i and -0.5 - 0.866i, where i is the imaginary number equal to the square root of -1.

## 4. How do you express the other solutions to z^3=1?

The other two solutions to z^3=1 can be expressed as complex numbers, where the real part is -0.5 and the imaginary part is either 0.866 or -0.866.

## 5. Can you use a calculator to find the third root of z^3=1?

Yes, most scientific calculators have a cube root function that can be used to find the third root of z^3=1. You can also use the calculator to raise 1 to the power of 1/3, which will also give you the answer of 1.