Powers of Complex Numbers

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De Moivre's Theorem for powers and roots of complex numbers.

Want a deeper conceptual understanding? Try our interactive lesson! (Plus Only)

Powers of complex numbers

AHL AA 1.14

Now that we know how to represent complex numbers in the form reiθ, we can understand De Moivre's Theorem, which gives us a shortcut to finding powers of complex numbers:

z=reiθ⇒zn=(reiθ)n=rneinθ

And since reiθ=rcisθ:

[r(cosθ+isinθ)]n=(rcisθ)n=rneinθ=rncis(nθ)📖

Roots of complex numbers

AHL AA 1.14

De Moivre's Theorem can also be used to find the nth roots of complex numbers:

n√reiθ=(reiθ)1/n=n√r⋅eiθ/n🚫

or equivalently

n√rcisθ=n√rcis(nθ)🚫

However, since cisθ=cis(θ+2kπ) for any k∈Z, then we actually have

n√rcisθ=n√rcis(nθ+2kπ),k=0,1…n−1🚫

Note that k stops at n−1 since when k=n we have

cis(nθ+2nπ)=cis(θ+2π)=cisθ

Nice work completing Powers of Complex Numbers, here's a quick recap of what we covered:

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Mixed Practice

Complex Numbers

Powers of Complex Numbers

0 of 0 exercises completed

De Moivre's Theorem for powers and roots of complex numbers.

Want a deeper conceptual understanding? Try our interactive lesson! (Plus Only)

Powers of complex numbers

AHL AA 1.14

Now that we know how to represent complex numbers in the form reiθ, we can understand De Moivre's Theorem, which gives us a shortcut to finding powers of complex numbers:

z=reiθ⇒zn=(reiθ)n=rneinθ

And since reiθ=rcisθ:

[r(cosθ+isinθ)]n=(rcisθ)n=rneinθ=rncis(nθ)📖

Roots of complex numbers

AHL AA 1.14

De Moivre's Theorem can also be used to find the nth roots of complex numbers:

n√reiθ=(reiθ)1/n=n√r⋅eiθ/n🚫

or equivalently

n√rcisθ=n√rcis(nθ)🚫

However, since cisθ=cis(θ+2kπ) for any k∈Z, then we actually have

n√rcisθ=n√rcis(nθ+2kπ),k=0,1…n−1🚫

Note that k stops at n−1 since when k=n we have

cis(nθ+2nπ)=cis(θ+2π)=cisθ

Nice work completing Powers of Complex Numbers, here's a quick recap of what we covered:

Skills covered

Mixed Practice

Exercises checked off

I'm Plex, here to help you understand this concept!

Mixed Practice

Complex Numbers

Powers of Complex Numbers

0 of 0 exercises completed

De Moivre's Theorem for powers and roots of complex numbers.

Want a deeper conceptual understanding? Try our interactive lesson! (Plus Only)

Powers of complex numbers

AHL AA 1.14

Now that we know how to represent complex numbers in the form reiθ, we can understand De Moivre's Theorem, which gives us a shortcut to finding powers of complex numbers:

z=reiθ⇒zn=(reiθ)n=rneinθ

And since reiθ=rcisθ:

[r(cosθ+isinθ)]n=(rcisθ)n=rneinθ=rncis(nθ)📖

Roots of complex numbers

AHL AA 1.14

De Moivre's Theorem can also be used to find the nth roots of complex numbers:

n√reiθ=(reiθ)1/n=n√r⋅eiθ/n🚫

or equivalently

n√rcisθ=n√rcis(nθ)🚫

However, since cisθ=cis(θ+2kπ) for any k∈Z, then we actually have

n√rcisθ=n√rcis(nθ+2kπ),k=0,1…n−1🚫

Note that k stops at n−1 since when k=n we have

cis(nθ+2nπ)=cis(θ+2π)=cisθ

Nice work completing Powers of Complex Numbers, here's a quick recap of what we covered:

Skills covered

Mixed Practice

Exercises checked off

I'm Plex, here to help you understand this concept!

Generating starter questions...

Mixed Practice

Complex Numbers

Powers of Complex Numbers

0 of 0 exercises completed

De Moivre's Theorem for powers and roots of complex numbers.

Want a deeper conceptual understanding? Try our interactive lesson! (Plus Only)

Powers of complex numbers

AHL AA 1.14

Now that we know how to represent complex numbers in the form reiθ, we can understand De Moivre's Theorem, which gives us a shortcut to finding powers of complex numbers:

z=reiθ⇒zn=(reiθ)n=rneinθ

And since reiθ=rcisθ:

[r(cosθ+isinθ)]n=(rcisθ)n=rneinθ=rncis(nθ)📖

Roots of complex numbers

AHL AA 1.14

De Moivre's Theorem can also be used to find the nth roots of complex numbers:

n√reiθ=(reiθ)1/n=n√r⋅eiθ/n🚫

or equivalently

n√rcisθ=n√rcis(nθ)🚫

However, since cisθ=cis(θ+2kπ) for any k∈Z, then we actually have

n√rcisθ=n√rcis(nθ+2kπ),k=0,1…n−1🚫

Note that k stops at n−1 since when k=n we have

cis(nθ+2nπ)=cis(θ+2π)=cisθ

Nice work completing Powers of Complex Numbers, here's a quick recap of what we covered:

Skills covered

Mixed Practice

Exercises checked off

I'm Plex, here to help you understand this concept!

Generating starter questions...