Find all the cube roots of 64i in standard form

2sqrt(3) + 2i, -2sqrt(3) + 2i, -4i

Find all the square roots of 25i in standard form

5sqrt(2)2 + 5sqrt(2)2i, -5sqrt(2)2 - 5sqrt(2)2i

Find all the cube roots of -64 in standard form

2 + 2sqrt(3)i, -4, 2 - 2sqrt(3)i

Find all the square roots of 1/2 - sqrt(3)2i in standard form

-sqrt(3)2 + 1/2i, sqrt(3)2 - 1/2i

Find all the cube roots of 27 in standard form

3, -3/2 + 3sqrt(3)2i, -3/2 - 3sqrt(3)2i

Find all the cube roots of -125i in standard form

5i, -5sqrt(3)2 - 5/2i, 5sqrt(3)2 - 5/2i

Find all the square roots of -9i in standard form

-3sqrt(2)2 + 3sqrt(2)2i, 3sqrt(2)2 - 3sqrt(2)2i

Find all the square roots of 49/2 + 49sqrt(3)2i in standard form

7sqrt(3)2 + 7/2i, -7sqrt(3)2 - 7/2i

Find all the cube roots of -i in standard form

i, -sqrt(3)2 - 1/2i, sqrt(3)2 - 1/2i

Find the cube roots of 8(cos 180° + isin 180°)

2(cos 60° + isin 60°), 2(cos 180° + isin 180°), 2(cos 300° + isin 300°)

Worksheets · Precalculus

De Moivre's theorem worksheet

Raising a complex number to a power is easy in polar form: the modulus is raised to the power and the angle is multiplied by it. Doing the same thing by repeated multiplication in standard form is where the afternoon goes. Run backwards, the same rule gives roots: take the nth root of the modulus and divide the angle by n, then add a full turn over n for each of the others.

Select your difficulty

Pick more than one for a sheet that mixes them.

The preview can look cramped on a phone. The PDF and printed sheet come out normal.



Sigma Prep · sigmaprep.io/worksheets Name De Moivre's Theorem Date Period Simplify, or find the roots, using De Moivre's theorem. Use De Moivre's theorem Use De Moivre’s theorem to find ${(\cos 225^{\circ} + i \sin 225^{\circ})}^{2}$ . Give it in trigonometric form, or in standard form if it is real or imaginary. Find ${[3 ⁣ (\cos 120^{\circ} + i \sin 120^{\circ})]}^{4}$ . Give it in trigonometric form, or in standard form if it is real or imaginary. Let $z = (\cos 45^{\circ} + i \sin 45^{\circ})$ . Find $z^{3}$ . Give it in trigonometric form, or in standard form if it is real or imaginary. Let $z = 2 ⁣ (\cos 30^{\circ} + i \sin 30^{\circ})$ . Find $z^{3}$ . Give it in trigonometric form, or in standard form if it is real or imaginary. Use De Moivre’s theorem to find ${[2 ⁣ (\cos 210^{\circ} + i \sin 210^{\circ})]}^{2}$ . Give it in trigonometric form, or in standard form if it is real or imaginary. Find ${[3 ⁣ (\cos 315^{\circ} + i \sin 315^{\circ})]}^{4}$ . Give it in trigonometric form, or in standard form if it is real or imaginary. Let $z = 2 ⁣ (\cos 300^{\circ} + i \sin 300^{\circ})$ . Find $z^{3}$ . Give it in trigonometric form, or in standard form if it is real or imaginary. Find ${(\cos 240^{\circ} + i \sin 240^{\circ})}^{3}$ . Give it in trigonometric form, or in standard form if it is real or imaginary. Let $z = 2 ⁣ (\cos 60^{\circ} + i \sin 60^{\circ})$ . Find $z^{2}$ . Give it in trigonometric form, or in standard form if it is real or imaginary. Use De Moivre’s theorem to find ${[3 ⁣ (\cos 330^{\circ} + i \sin 330^{\circ})]}^{2}$ . Give it in trigonometric form, or in standard form if it is real or imaginary. Find roots in standard form Find all the cube roots of $64 i$ in standard form Find all the square roots of $25 i$ in standard form Find all the cube roots of $- 64$ in standard form Find all the square roots of $\frac{1}{2} - \frac{\sqrt{3}}{2} i$ in standard form Find all the cube roots of $27$ in standard form Find all the cube roots of $- 125 i$ in standard form Find all the square roots of $- 1$ in standard form Find all the square roots of $- 9 i$ in standard form Find all the square roots of $\frac{49}{2} + \frac{49 \sqrt{3}}{2} i$ in standard form Find all the cube roots of $- i$ in standard form Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

De Moivre's Theorem

Date Period

Simplify, or find the roots, using De Moivre's theorem.

Use De Moivre's theorem
Use De Moivre’s theorem to find ${(\cos 225^{\circ} + i \sin 225^{\circ})}^{2}$ . Give it in trigonometric form, or in standard form if it is real or imaginary.
Find ${[3 ⁣ (\cos 120^{\circ} + i \sin 120^{\circ})]}^{4}$ . Give it in trigonometric form, or in standard form if it is real or imaginary.
Let $z = (\cos 45^{\circ} + i \sin 45^{\circ})$ . Find $z^{3}$ . Give it in trigonometric form, or in standard form if it is real or imaginary.
Let $z = 2 ⁣ (\cos 30^{\circ} + i \sin 30^{\circ})$ . Find $z^{3}$ . Give it in trigonometric form, or in standard form if it is real or imaginary.
Use De Moivre’s theorem to find ${[2 ⁣ (\cos 210^{\circ} + i \sin 210^{\circ})]}^{2}$ . Give it in trigonometric form, or in standard form if it is real or imaginary.
Find ${[3 ⁣ (\cos 315^{\circ} + i \sin 315^{\circ})]}^{4}$ . Give it in trigonometric form, or in standard form if it is real or imaginary.
Let $z = 2 ⁣ (\cos 300^{\circ} + i \sin 300^{\circ})$ . Find $z^{3}$ . Give it in trigonometric form, or in standard form if it is real or imaginary.
Find ${(\cos 240^{\circ} + i \sin 240^{\circ})}^{3}$ . Give it in trigonometric form, or in standard form if it is real or imaginary.
Let $z = 2 ⁣ (\cos 60^{\circ} + i \sin 60^{\circ})$ . Find $z^{2}$ . Give it in trigonometric form, or in standard form if it is real or imaginary.
Use De Moivre’s theorem to find ${[3 ⁣ (\cos 330^{\circ} + i \sin 330^{\circ})]}^{2}$ . Give it in trigonometric form, or in standard form if it is real or imaginary.

Find roots in standard form
Find all the cube roots of $64 i$ in standard form
Find all the square roots of $25 i$ in standard form
Find all the cube roots of $- 64$ in standard form
Find all the square roots of $\frac{1}{2} - \frac{\sqrt{3}}{2} i$ in standard form
Find all the cube roots of $27$ in standard form
Find all the cube roots of $- 125 i$ in standard form
Find all the square roots of $- 1$ in standard form
Find all the square roots of $- 9 i$ in standard form
Find all the square roots of $\frac{49}{2} + \frac{49 \sqrt{3}}{2} i$ in standard form
Find all the cube roots of $- i$ in standard form



Sigma Prep · sigmaprep.io/worksheets Name De Moivre's TheoremAnswer keyVersion 1 Date Period Simplify, or find the roots, using De Moivre's theorem. Use De Moivre's theorem Use De Moivre’s theorem to find ${(\cos 225^{\circ} + i \sin 225^{\circ})}^{2}$ . Give it in trigonometric form, or in standard form if it is real or imaginary. $i$ Find ${[3 ⁣ (\cos 120^{\circ} + i \sin 120^{\circ})]}^{4}$ . Give it in trigonometric form, or in standard form if it is real or imaginary. $81 ⁣ (\cos 120^{\circ} + i \sin 120^{\circ})$ Let $z = (\cos 45^{\circ} + i \sin 45^{\circ})$ . Find $z^{3}$ . Give it in trigonometric form, or in standard form if it is real or imaginary. $\cos 135^{\circ} + i \sin 135^{\circ}$ Let $z = 2 ⁣ (\cos 30^{\circ} + i \sin 30^{\circ})$ . Find $z^{3}$ . Give it in trigonometric form, or in standard form if it is real or imaginary. $8 i$ Use De Moivre’s theorem to find ${[2 ⁣ (\cos 210^{\circ} + i \sin 210^{\circ})]}^{2}$ . Give it in trigonometric form, or in standard form if it is real or imaginary. $4 ⁣ (\cos 60^{\circ} + i \sin 60^{\circ})$ Find ${[3 ⁣ (\cos 315^{\circ} + i \sin 315^{\circ})]}^{4}$ . Give it in trigonometric form, or in standard form if it is real or imaginary. $- 81$ Let $z = 2 ⁣ (\cos 300^{\circ} + i \sin 300^{\circ})$ . Find $z^{3}$ . Give it in trigonometric form, or in standard form if it is real or imaginary. $- 8$ Find ${(\cos 240^{\circ} + i \sin 240^{\circ})}^{3}$ . Give it in trigonometric form, or in standard form if it is real or imaginary. $1$ Let $z = 2 ⁣ (\cos 60^{\circ} + i \sin 60^{\circ})$ . Find $z^{2}$ . Give it in trigonometric form, or in standard form if it is real or imaginary. $4 ⁣ (\cos 120^{\circ} + i \sin 120^{\circ})$ Use De Moivre’s theorem to find ${[3 ⁣ (\cos 330^{\circ} + i \sin 330^{\circ})]}^{2}$ . Give it in trigonometric form, or in standard form if it is real or imaginary. $9 ⁣ (\cos 300^{\circ} + i \sin 300^{\circ})$ Find roots in standard form Find all the cube roots of $64 i$ in standard form $2 \sqrt{3} + 2 i$ , $- 2 \sqrt{3} + 2 i$ , $- 4 i$ Find all the square roots of $25 i$ in standard form $\frac{5 \sqrt{2}}{2} + \frac{5 \sqrt{2}}{2} i$ , $- \frac{5 \sqrt{2}}{2} - \frac{5 \sqrt{2}}{2} i$ Find all the cube roots of $- 64$ in standard form $2 + 2 \sqrt{3} i$ , $- 4$ , $2 - 2 \sqrt{3} i$ Find all the square roots of $\frac{1}{2} - \frac{\sqrt{3}}{2} i$ in standard form $- \frac{\sqrt{3}}{2} + \frac{1}{2} i$ , $\frac{\sqrt{3}}{2} - \frac{1}{2} i$ Find all the cube roots of $27$ in standard form $3$ , $- \frac{3}{2} + \frac{3 \sqrt{3}}{2} i$ , $- \frac{3}{2} - \frac{3 \sqrt{3}}{2} i$ Find all the cube roots of $- 125 i$ in standard form $5 i$ , $- \frac{5 \sqrt{3}}{2} - \frac{5}{2} i$ , $\frac{5 \sqrt{3}}{2} - \frac{5}{2} i$ Find all the square roots of $- 1$ in standard form $i$ , $- i$ Find all the square roots of $- 9 i$ in standard form $- \frac{3 \sqrt{2}}{2} + \frac{3 \sqrt{2}}{2} i$ , $\frac{3 \sqrt{2}}{2} - \frac{3 \sqrt{2}}{2} i$ Find all the square roots of $\frac{49}{2} + \frac{49 \sqrt{3}}{2} i$ in standard form $\frac{7 \sqrt{3}}{2} + \frac{7}{2} i$ , $- \frac{7 \sqrt{3}}{2} - \frac{7}{2} i$ Find all the cube roots of $- i$ in standard form $i$ , $- \frac{\sqrt{3}}{2} - \frac{1}{2} i$ , $\frac{\sqrt{3}}{2} - \frac{1}{2} i$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

De Moivre's TheoremAnswer keyVersion 1

Date Period

Simplify, or find the roots, using De Moivre's theorem.

Use De Moivre's theorem
Use De Moivre’s theorem to find ${(\cos 225^{\circ} + i \sin 225^{\circ})}^{2}$ . Give it in trigonometric form, or in standard form if it is real or imaginary. $i$
Find ${[3 ⁣ (\cos 120^{\circ} + i \sin 120^{\circ})]}^{4}$ . Give it in trigonometric form, or in standard form if it is real or imaginary. $81 ⁣ (\cos 120^{\circ} + i \sin 120^{\circ})$
Let $z = (\cos 45^{\circ} + i \sin 45^{\circ})$ . Find $z^{3}$ . Give it in trigonometric form, or in standard form if it is real or imaginary. $\cos 135^{\circ} + i \sin 135^{\circ}$
Let $z = 2 ⁣ (\cos 30^{\circ} + i \sin 30^{\circ})$ . Find $z^{3}$ . Give it in trigonometric form, or in standard form if it is real or imaginary. $8 i$
Use De Moivre’s theorem to find ${[2 ⁣ (\cos 210^{\circ} + i \sin 210^{\circ})]}^{2}$ . Give it in trigonometric form, or in standard form if it is real or imaginary. $4 ⁣ (\cos 60^{\circ} + i \sin 60^{\circ})$
Find ${[3 ⁣ (\cos 315^{\circ} + i \sin 315^{\circ})]}^{4}$ . Give it in trigonometric form, or in standard form if it is real or imaginary. $- 81$
Let $z = 2 ⁣ (\cos 300^{\circ} + i \sin 300^{\circ})$ . Find $z^{3}$ . Give it in trigonometric form, or in standard form if it is real or imaginary. $- 8$
Find ${(\cos 240^{\circ} + i \sin 240^{\circ})}^{3}$ . Give it in trigonometric form, or in standard form if it is real or imaginary. $1$
Let $z = 2 ⁣ (\cos 60^{\circ} + i \sin 60^{\circ})$ . Find $z^{2}$ . Give it in trigonometric form, or in standard form if it is real or imaginary. $4 ⁣ (\cos 120^{\circ} + i \sin 120^{\circ})$
Use De Moivre’s theorem to find ${[3 ⁣ (\cos 330^{\circ} + i \sin 330^{\circ})]}^{2}$ . Give it in trigonometric form, or in standard form if it is real or imaginary. $9 ⁣ (\cos 300^{\circ} + i \sin 300^{\circ})$

Find roots in standard form
Find all the cube roots of $64 i$ in standard form $2 \sqrt{3} + 2 i$ , $- 2 \sqrt{3} + 2 i$ , $- 4 i$
Find all the square roots of $25 i$ in standard form $\frac{5 \sqrt{2}}{2} + \frac{5 \sqrt{2}}{2} i$ , $- \frac{5 \sqrt{2}}{2} - \frac{5 \sqrt{2}}{2} i$
Find all the cube roots of $- 64$ in standard form $2 + 2 \sqrt{3} i$ , $- 4$ , $2 - 2 \sqrt{3} i$
Find all the square roots of $\frac{1}{2} - \frac{\sqrt{3}}{2} i$ in standard form $- \frac{\sqrt{3}}{2} + \frac{1}{2} i$ , $\frac{\sqrt{3}}{2} - \frac{1}{2} i$
Find all the cube roots of $27$ in standard form $3$ , $- \frac{3}{2} + \frac{3 \sqrt{3}}{2} i$ , $- \frac{3}{2} - \frac{3 \sqrt{3}}{2} i$
Find all the cube roots of $- 125 i$ in standard form $5 i$ , $- \frac{5 \sqrt{3}}{2} - \frac{5}{2} i$ , $\frac{5 \sqrt{3}}{2} - \frac{5}{2} i$
Find all the square roots of $- 1$ in standard form $i$ , $- i$
Find all the square roots of $- 9 i$ in standard form $- \frac{3 \sqrt{2}}{2} + \frac{3 \sqrt{2}}{2} i$ , $\frac{3 \sqrt{2}}{2} - \frac{3 \sqrt{2}}{2} i$
Find all the square roots of $\frac{49}{2} + \frac{49 \sqrt{3}}{2} i$ in standard form $\frac{7 \sqrt{3}}{2} + \frac{7}{2} i$ , $- \frac{7 \sqrt{3}}{2} - \frac{7}{2} i$
Find all the cube roots of $- i$ in standard form $i$ , $- \frac{\sqrt{3}}{2} - \frac{1}{2} i$ , $\frac{\sqrt{3}}{2} - \frac{1}{2} i$

How to do these

${(2 ⁣ (\cos 30^{\circ} + i \sin 30^{\circ}))}^{3}$

The modulus 2 is cubed, giving 8.
The angle 30 is multiplied by 3, giving 90.

The answer is $8 ⁣ (\cos 90^{\circ} + i \sin 90^{\circ})$ .

Where students go wrong

Multiplying the modulus by the power instead of raising it. A modulus of 2 cubed is 8, not 6. The angle is the one that multiplies.

Also called de moivre or powers of complex numbers.

What you can put on this worksheet

De Moivre's theorem: Find ${[3 ⁣ (\cos 210^{\circ} + i \sin 210^{\circ})]}^{3}$ . Give it in trigonometric form, or in standard form if it is real or imaginary. → $- 27 i$
Powers in standard form: Find ${(- 2)}^{3}$ → $- 8$
The n roots of a complex number: Find the cube roots of $8 ⁣ (\cos 180^{\circ} + i \sin 180^{\circ})$ → $2 ⁣ (\cos 60^{\circ} + i \sin 60^{\circ})$ , $2 ⁣ (\cos 180^{\circ} + i \sin 180^{\circ})$ , $2 ⁣ (\cos 300^{\circ} + i \sin 300^{\circ})$
Roots in standard form: Find all the cube roots of $- 8$ in standard form → $1 + \sqrt{3} i$ , $- 2$ , $1 - \sqrt{3} i$

Questions about these worksheets

Yes. Every de moivre's theorem sheet is free to print and free to download as a PDF. There is no account to make, no email to hand over and no limit on how many you take.

Yes. The answer key prints on a second page, and you can choose whether it shows each question beside its answer or just the answers in a list.

Yes. Print straight from the page, or download the PDF and print that. The sheet is laid out for paper rather than squeezed off a screen, so there is room to work under each question, and the answer key comes out on its own page.

Precalculus is usually taken around 11th or 12th grade, though schools vary and plenty of students meet it earlier or later. Pick the difficulty rather than the grade: the easy level suits a first lesson on it, the hard level suits review before a test.

Yes, as many as you like. The questions are built fresh each time rather than picked from a fixed set of files, so pressing Generate gives a new sheet. That is what you want for a second class, a retake, or two students sitting next to each other.

That is the closest comparison, and Kuta Software is good software. It is also paid software you install on a computer. These worksheets run in a browser tab for free, with no account and nothing to install. Like Kuta, the de moivre's theorem questions are generated when you ask for them rather than pulled from a fixed set of files, so two classes never get the same sheet, and the answer key comes with it.

Yes. There are two levels, and you can tick more than one for a sheet that mixes them, in which case the questions come out easiest first. The harder levels are not just larger numbers: they ask for something the easy ones do not.

You pick from 4: de moivre's theorem, powers in standard form, the n roots of a complex number and roots in standard form. Tick as many as you want and set how many of each, or let it spread them evenly.

Multiplying the modulus by the power instead of raising it. A modulus of 2 cubed is 8, not 6. The angle is the one that multiplies.

De Moivre's theorem worksheet

Select your difficulty

Use De Moivre's theorem

Find roots in standard form

Use De Moivre's theorem

Find roots in standard form

How to do these

Where students go wrong

What you can put on this worksheet

Questions about these worksheets

Are these de moivre's theorem worksheets free?+

Do the de moivre's theorem worksheets come with an answer key?+

Are these de moivre's theorem worksheets printable?+

What grade are these de moivre's theorem worksheets for?+

Can I get a different version of the same de moivre's theorem worksheet?+

Is this a free alternative to Kuta Software for de moivre's theorem?+

Can I choose how hard the de moivre's theorem questions are?+

What kinds of question are on a de moivre's theorem worksheet?+

What do students get wrong with de moivre's theorem?+