(c) Use de Moivre's formula to write down a formula for all o words, give a formula for the complex numbers w, such k = 0, 1, 2,...,n- 1. Use the formula to calculate ll 2in 2= 16exp
(c) Use de Moivre's formula to write down a formula for all o words, give a formula for the complex numbers w, such k = 0, 1, 2,...,n- 1. Use the formula to calculate ll 2in 2= 16exp
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter4: Polynomial And Rational Functions
Section4.3: Zeros Of Polynomials
Problem 35E
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please send handwritten solution step by step Q6 part c
![Q6
In what follows, let i = v-1 denote the imaginary unit.
(a) It z = 4 +5i and w = 2+4i, find
32 + 4w
3z - 4w
Show the steps. Express your answer using integers or rationals for the real and imaginary parts.
(b) Given that x = 1- 4i is one root of the quartic equation
2*- 82' + 54z – 152z + 425 = 0,
tind the other three roots. Show your working in detail.
(c) Use de Moivre's formula to write down a formula for all of the n-th roots of z = rexp(ie). In other
words, give a formula for the complex numbers w, such that (w)" =z, where
k = 0, 1, 2,...,n– 1. Use the formula to caiculate alli of the 4-th roots of the complex number
2= 16exp().
You must show all of your working and must express each root in the representation:
w = R(cos(4,7) + i sin(,7)],
where Rand o, should be expressed in simplified form (e.g., using rationals and/or roots), not as
decimals. If the roots, w, were plotted on an Argand diagram, what type of geometrical figure would
be formed? Draw this figure, labelling the roots.
(d) Given any complex number, z, let ztn denote the repeated power:
if n =0
(2 1 (n - 1)) ifn >1.'
so that n counts the number of exponentiations. For example,
310 = 3, 311= (3f 0)* = 3* = 27, 312= (31 1) = (3') = 27* = 19683.
Calculate i ↑ 0, i f 1, i ↑ 2, i 1 3, and i T 4. Hence deduce a general formula for i ↑n for n >0
(you may find it helptul to write n = 4m + k tfor m > 0 and k = 0, 1,2, 3). Show all working.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F25bfdcbf-4a1c-4959-bc5c-2044369cdf92%2F733079d2-67e5-4199-9da1-41ac4c4f2859%2Fn9o3pg_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Q6
In what follows, let i = v-1 denote the imaginary unit.
(a) It z = 4 +5i and w = 2+4i, find
32 + 4w
3z - 4w
Show the steps. Express your answer using integers or rationals for the real and imaginary parts.
(b) Given that x = 1- 4i is one root of the quartic equation
2*- 82' + 54z – 152z + 425 = 0,
tind the other three roots. Show your working in detail.
(c) Use de Moivre's formula to write down a formula for all of the n-th roots of z = rexp(ie). In other
words, give a formula for the complex numbers w, such that (w)" =z, where
k = 0, 1, 2,...,n– 1. Use the formula to caiculate alli of the 4-th roots of the complex number
2= 16exp().
You must show all of your working and must express each root in the representation:
w = R(cos(4,7) + i sin(,7)],
where Rand o, should be expressed in simplified form (e.g., using rationals and/or roots), not as
decimals. If the roots, w, were plotted on an Argand diagram, what type of geometrical figure would
be formed? Draw this figure, labelling the roots.
(d) Given any complex number, z, let ztn denote the repeated power:
if n =0
(2 1 (n - 1)) ifn >1.'
so that n counts the number of exponentiations. For example,
310 = 3, 311= (3f 0)* = 3* = 27, 312= (31 1) = (3') = 27* = 19683.
Calculate i ↑ 0, i f 1, i ↑ 2, i 1 3, and i T 4. Hence deduce a general formula for i ↑n for n >0
(you may find it helptul to write n = 4m + k tfor m > 0 and k = 0, 1,2, 3). Show all working.
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