Let z=2e10πi21 and w=eπi7. Then if [(z−w)6=reiθ] where r≥0 and 0≤θ≤2π what is the ordered pair (r,θ)?

(z−w)6=reiθ |(z−w)6|=|reiθ| |(z−w)|6=r z−w=eπi7(2eπi3−1) |z−w|=|eπi7(2eπi3−1)|=|eπi7|⋅|(2eπi3−1)| |z−w|=1⋅|(2eπi3−1)|=3 r=|(z−w)|6=27 Arg(z−w)6=6⋅Arg(z−w) z−w=eπi7(2eπi3−1) arg(z−w)=9π14

Check the solution to "Let z=2e10πi21 and w=eπi7. Then if..."

treetopssan

Answered question

2022-01-30

Let

z = 2 e^{10 π \frac{i}{21}} and w = e^{π \frac{i}{7}}

. Then if

[{(z - w)}^{6} = r e^{i θ}]

where

r \geq 0

and

0 \leq θ \leq 2 π

what is the ordered pair

(r, θ)

Answer & Explanation

Madelyn Townsend

Beginner2022-01-31Added 13 answers

{(z - w)}^{6} = r e^{i θ}

| {(z - w)}^{6} | = | r e^{i θ} |

{| (z - w) |}^{6} = r

z - w = e^{π \frac{i}{7}} (2 e^{π \frac{i}{3}} - 1)

| z - w | = | e^{π \frac{i}{7}} (2 e^{π \frac{i}{3}} - 1) | = | e^{π \frac{i}{7}} | \cdot | (2 e^{π \frac{i}{3}} - 1) |

| z - w | = 1 \cdot | (2 e^{π \frac{i}{3}} - 1) | = \sqrt{3}

r = {| (z - w) |}^{6} = 27

A r {g (z - w)}^{6} = 6 \cdot A r g (z - w)

z - w = e^{π \frac{i}{7}} (2 e^{π \frac{i}{3}} - 1)

a r g (z - w) = \frac{9 π}{14}

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