What is always one solution of \(z^n=1\) for positive integer \(n\)?

How many distinct complex roots does \(z^7=1\) have?

List the square roots of unity.

Write the fourth roots of unity in rectangular form.

Write the cube roots of unity in exponential form.

Find the argument spacing between adjacent sixth roots of unity.

List the sixth roots of unity in exponential form.

Show that \(e^{4\pi i/5}\) is a fifth root of unity.

Write the fifth roots of unity using \(k=0,1,2,3,4\).

Convert the primitive cube roots of unity to rectangular form.

Verify that the four fourth roots of unity all have modulus \(1\).

Explain why using \(k=0,1,\ldots,n-1\) gives all independent roots of \(z^n=1\).

Find the product of all fourth roots of unity.

Find the sum of all cube roots of unity.

Which sixth roots of unity are also cube roots of unity?

For which integers \(k\) with \(0\le k\le7\) is \(e^{2\pi ik/8}\) a primitive eighth root of unity?

Find all fourth roots of unity that satisfy \(\operatorname{Im}(z)\ge0\).

A learner lists \(e^{2\pi ik/5}\) for every integer \(k\) as infinitely many fifth roots of unity. Explain the mistake.

Prove that every \(n\)-th root of unity has modulus \(1\).

Show that the product of two \(n\)-th roots of unity is again an \(n\)-th root of unity.