W15 - HW Solutions

03 - Complex forms - exponential to Cartesian

Write each number in the form .

(a) (b)

Solution

(a)

State Euler’s formula.

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Plug given expression into formula with and .

(b)

Write as expression product of exponents.

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Plug computed expression into Euler’s formula with and

05 - Complex roots using polar

Find the three cube () roots of .

Write your answer in the form .

Solution

Write

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Use the roots formula where , , and .

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Write out roots by evaluating at

09 - Complex products and quotients using polar

For each pair of complex numbers and , compute:

(a)

(b)

(Use polar forms with .)

Solution

(a)

Compute .

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Compute . Multiply numerator and denominator by the conjugate and simplify.

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Compute . Multiply numerator and denominator by the conjugate and simplify.

(b)

Compute .

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Compute . Multiply numerator and denominator by the conjugate and simplify.

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Compute . Multiply numerator and denominator by the conjugate and simplify.

10 - Complex powers using polar

Using De Moivre’s Theorem, write each number in the form .

(a) (b)

(First convert to polar/exponential, then compute the power, then convert back.)

Solution

(a)

Convert to polar form.

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Use De Moivre’s Theorem.

(b)

Convert to polar form.

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Use De Moivre’s Theorem.

11 - Complex roots using polar

Find each of the indicated roots. Try to write your answer in form if that is not hard, otherwise leave it in polar form.

(a) The four roots of .

(b) The three cube () roots of .

Try to write your answer in form if that is not hard, otherwise leave it in polar form.

Solution

(a)

Write

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Use the roots formula where and .

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Write out roots by evaluating at .

(b)

Write

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Use the roots formula where , , and .

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Write out roots by evaluating at