(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 calculate all 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() + i sin(d.)], where R and , 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.
(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 calculate all 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() + i sin(d.)], where R and , 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.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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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%2F386c0047-98b4-4cc6-892b-be9a6eecab06%2Fo8dvbu_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.
Transcribed Image Text:ALPHABET
In all questions, you should work with an alphabet of n = 37 symbols, as indicated below.
For individual symbols, you should use integer codes in the range 0 to 36 inclusive (therefore, you will be
mainly using arithmetic modulo n = 37), as indicated by the following table:
A
в
DEFG
0 1
JK
2
3
4
5
7
8
LM
O P
14 15
Q|R
S
10 11
12
13
16
17
18
19
TU
20 21
V.
W
Y
?
22
23
24 25 26 27
28
29
@
30 31
32 33
34 35 36
So, for example, the symbol C should be encoded as the integer 3.
Note: the space character (used to separate words) should be encoded by the integer 0.
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