W15 - Examples

Complex algebra

Complex multiplication

Compute the products:

(a) (b)

Solution

(a)

Expand:

Simplify :

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(b)

Expand:

Simplify :

Complex division

Compute the following divisions of complex numbers:

(a) (b) (c) (d)

Solution

(a)

Conjugate is :

Simplify:

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(b)

Conjugate is :

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(c)

Factor out the :

Use :

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(d)

Denominator conjugate is :

Simplify:

Complex product, quotient, power using Euler

Start with two complex numbers:

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Product :

Quotient :

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Power :

Notice:

Simplify:

Thus: .

Complex power from Cartesian

Compute .

Solution

First convert to exponential form:

Compute the power:

Finding all 4th roots of 16

Compute all the roots of .

Solution

Write .

Evaluate roots formula:

Simplify:

Finding 2nd roots of 2i

Find both roots of .

Solution

Write . Evaluate roots formula:

Compute the options: :

Convert to rectangular:

Some roots of unity

Find the and and and and and roots of the number .

Solution

Write . Evaluate roots formula. There is no possible :

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Write . Evaluate roots formula in terms of :

Compute the two options, :

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Evaluate roots formula in terms of :

Compute the options: :

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Evaluate roots formula:

Compute the options: :

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Evaluate roots formula:

Compute the options: :

Don’t simplify, it’s not feasible.

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Evaluate roots formula:

Compute the options: :

Simplify: