1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
mixer [17]
3 years ago
9

Assume z = x + iy, then find a complex number z satisfying the given equation. d. 2z8 – 2z4 + 1 = 0

Mathematics
1 answer:
kodGreya [7K]3 years ago
3 0

Answer: complex equations has n number of solutions, been n the equation degree. In this case:

Z=\frac{\sqrt[8]{2} }{\sqrt[4]{2}} e^{i11,25°}

Z=\frac{\sqrt[8]{2} }{\sqrt[4]{2}} e^{i101,25°}

Z=\frac{\sqrt[8]{2} }{\sqrt[4]{2}} e^{i191,25°}

Z=\frac{\sqrt[8]{2} }{\sqrt[4]{2}} e^{i281,25°}

Z=\frac{\sqrt[8]{2} }{\sqrt[4]{2}} e^{i78,75°}

Z=\frac{\sqrt[8]{2} }{\sqrt[4]{2}} e^{i168,75°}

Z=\frac{\sqrt[8]{2} }{\sqrt[4]{2}} e^{i258,75°}

Z=\frac{\sqrt[8]{2} }{\sqrt[4]{2}} e^{i348,75°}

Step-by-step explanation:

I start with a variable substitution:

Z^{4} = X

Then:

2X^{2}-2X+1=0

Solving the quadratic equation:

X_{1} =\frac{2+\sqrt{4-4*2*1} }{2*2} \\X_{2} =\frac{2-\sqrt{4-4*2*1} }{2*2}

X=\left \{ {{0,5+0,5i} \atop {0,5-0,5i}} \right.

Replacing for the original variable:

Z=\sqrt[4]{0,5+0,5i}

or Z=\sqrt[4]{0,5-0,5i}

Remembering that complex numbers can be written as:

Z=a+ib=|Z|e^{ic}

Using this:

Z=\left \{ {{{\frac{\sqrt{2}}{2} e^{i45°} } \atop {{\frac{\sqrt{2}}{2} e^{i-45°} }} \right.

Solving for the modulus and the angle:

Z=\left \{ {{\sqrt[4]{\frac{\sqrt{2}}{2} e^{i45}} = \sqrt[4]{\frac{\sqrt{2}}{2} } \sqrt[4]{e^{i45}} } \atop {\sqrt[4]{\frac{\sqrt{2}}{2} e^{i-45}} = \sqrt[4]{\frac{\sqrt{2}}{2} } \sqrt[4]{e^{i-45}} }} \right.

The possible angle respond to:

RAng_{12...n} =\frac{Ang +360*(i-1)}{n}

Been "RAng" the resultant angle, "Ang" the original angle, "n" the degree of the root and "i" a value between 1 and "n"

In this case n=4 with 2 different angles: Ang = 45º and Ang = 315º

Obtaining 8 different angles, therefore 8 different solutions.

You might be interested in
HELP IM TIMED!(giving 50 points)
lutik1710 [3]

Answer:

A and D and E

Step-by-step explanation:

Substitute the variable(in this case) <em>t</em> for 3, then use the order of operations to simplify(PEMDAS), which will give you 33 for both expressions. To see if two expressions are equivalent, you need to substitute the variable for an odd number AND an even number, so next, substitute the variable <em>t </em>for the number 6, which will give you 51 for both expressions. That means the expressions are equivalent

Hope this helps :)

P.S. Mark brainliest pls

6 0
2 years ago
Read 2 more answers
What is the equation of the line that passes through the point
snow_lady [41]

Answer:

3x-2y=2

Step-by-step explanation:

The equation of line having slope \frac{3}{2} and passing through (-4,-7).

The\ equation\ of\ line\ is\ given\ by\ y=mx+c\ where\ m\ is\ the\ slope\ of\ line.\\\\Here\ m=\frac{3}{2}\\\\Equation\ y=\frac{3}{2}x+c\\\\Line\ passes\ through\ (-4,-7),\ it\ will\ satisfy\ the\ equation\ of\ line.\\\\-7=\frac{3}{2}\times (-4)+c\\\\-7=-6+c\\\\c=-7+6\\\\c=-1\\\\y=\frac{3}{2}x-1\\\\2y=3x-2\\\\3x-2y=2

8 0
3 years ago
Check all of the polynomial functions that have 2 as a root.
Leviafan [203]

A,C,D those were the answers i put and they were correct.

5 0
3 years ago
How is simplifying similar to division?
Alex
When you simpilfy a fraction you make it smaller. When you divide a number you make it smaller.

Ex: 5 /5 1
_ = _
10 /5 2
3 0
3 years ago
Read 2 more answers
What's the answer to this??
Evgen [1.6K]
The variable "x" equals 7
3 0
3 years ago
Other questions:
  • How many prime numbers are there between 0 and 25
    7·2 answers
  • Please help me!!!! Fast!!! If your right I'll tell you!​
    8·1 answer
  • 20. Find the chi square value X 2L corresponding to a sample size of 23 and a confidence level of 90%
    11·1 answer
  • find zeros and y intercept and describe the end behavior in proper notation. then sketch the graph. 3x^3+2x^2-12x-8
    15·1 answer
  • Andrew invests $9,000 at the end of each year for 20 years. The rate of interest Andrew gets is 8% annually. Using the tables fo
    7·2 answers
  • Explain why the order of the coordinates does not matter when calculating length in geometry?
    9·2 answers
  • If 25.20 cost for a pack of 9 padlocks how much is it for a per padlocks
    15·2 answers
  • Brooklyn needs to order some new supplies for the restaurant where she works. The
    9·1 answer
  • For 3, 5, Are the triangles similar? If so, state the similarity and the postulate or theorem that justifies your answer.
    13·1 answer
  • NEED HELP ASAP PLEASEE!!!!!!!!!
    15·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!