using de moivre's theorem to find square roots of (2-2√ 3 i)
Let z = 2 - 2√3 i. In polar form, we have
[tex]z = |z| e^{i\arg(z)}[/tex]
where
[tex]|z| = |2-2\sqrt3| = \sqrt{2^2 + (-2\sqrt3)^2} = 4[/tex]
[tex]\arg(z) = \tan^{-1}\left(-\dfrac{2\sqrt3}2\right) = -\tan^{-1}(\sqrt3) = -\dfrac\pi3[/tex]
Equivalently,
[tex]z = 4\left(\cos\left(-\dfrac\pi3\right) + i\sin\left(-\dfrac\pi3\right)\right)[/tex]
Let w be a complex number such that w ² = z. By DeMoivre's theorem,
[tex]z = 4\left(\cos\left(-\dfrac\pi3\right) + i\sin\left(-\dfrac\pi3\right)\right) \\\\ \implies z^{1/2} = 4^{1/2} \left(\cos\left(\dfrac{-\frac\pi3+2k\pi}{2}\right) + i \sin\left(\dfrac{-\frac\pi3+2k\pi}{2}\right)\right)[/tex]
where k ∈ {0, 1}. So the two square roots of z are
[tex]z^{1/2} = \begin{cases}2\left(\cos\left(-\dfrac\pi6\right) + i\sin\left(-\dfrac\pi6\right)\right) & \text{for }k=0 \\\\ 2\left(\cos\left(\dfrac{5\pi}6\right) + i\sin\left(\dfrac{5\pi}6\right)\right) & \text{for }k=1\end{cases}[/tex]
[tex]z^{1/2} = \begin{cases}2\left(\dfrac{\sqrt3}2 - \dfrac12 i\right) & \text{for }k=0 \\\\ 2\left(-\dfrac{\sqrt3}2 + \dfrac12 i\right) & \text{for }k=1\end{cases}[/tex]
[tex]\boxed{z^{1/2} = \begin{cases}\sqrt3 - i & \text{for }k=0 \\ -\sqrt3 + i & \text{for }k=1\end{cases}}[/tex]
A line that includes the points ( – 8,9) and ( – 10,d) has a slope of 6. What is the value of d?
Answer:
The value of d is -3.
Step-by-step explanation:
We are given that a line that includes the two points (-8, 9) and (-10, d) have a slope of 6. We want to determine the value of d.
Recall that the formula for slope is:
[tex]\displaystyle m = \frac{\Delta y}{\Delta x}[/tex]
We know that the slope is 6. Substitute:
[tex]\displaystyle (6) = \frac{(d) - (9) }{(-10) - (-8)}[/tex]
Solve for d:
[tex]\displaystyle \begin{aligned} 6 & = \frac{d - 9}{-10 + 8} \\ \\ 6 & = \frac{d - 9}{-2} \\ \\ -12 & = d - 9 \\ \\ d & = -3 \end{aligned}[/tex]
In conclusion, the value of d is -3.
100 points and brainleist to whoever gets these right
Answer:
the answer is 8 that is
4/1/2
4÷1/2
4×2/1=
8
Answer:
#14/(1/2) = 4*2 = 8#2(12.034) + 5/8 = 12.034 + 0.625 = 12.659#3(2/3)/(1/6) = (2/3)*6 = 4#4(3.406) + 3 4/7 = 3.406 + 3.571 = 6.977 (4/7 is rounded to the nearest thousandths)#53/5 = Rational0 = Whole & Integer & Rational -25/5 = -5 = Whole & Integer & RationalI need help ASAP I’ll give brainliest!!
Cody buys a baseball bat for $164.50 and a baseball glove for $37.25. He estimates the total cost of the bat and glove by rounding each cost to the nearest $10. Drag numbers to the boxes to show how Cody rounds the costs.
164.50+37.50= 201.75
gtu
Joseph is reading a 32 page book. he reads 6 pages in 15 minutes. if he continues to read at the same rate how long it takes to read the entire book
Answer:
80 minutes
Step-by-step explanation:
Time taken to read 1 page = 15/6 = 2.5 minutes
time taken to read 32 page = 2.5 * 32 = 80 minutes
10x - 80 = 16x - 12
what is x?
Help me please help me help please
Answer:
y = 4x + 4
y = rise/run + y-intercept
After your meal you splurge on dessert. The cost is $4.49. What would be a good estimate for the dessert with 10% tax?
Answer:
$4.59 would be a good estimate for the dessert with 10% tax.
Step-by-step explanation:
10% = 0.10
0.10 + 4.49
Choose the number sentence that illustrates the distributive property of multiplication over addition.
A. 2x(5+ 6) = (2 x 5) + 6
B. 2x (5 + 6) = (2 x 5) + (2 x 6)
C. 2x(5 + 6) = (2+5)* (2 +6)
Answer:
One number sentence shows the distributive property of multiplication over addition. ... (5 + 4) × 3 = (5 × 3) + (4 × 3) is an example of the distributive property. (5 − 4) × 3 = 1 × 3. 1 × 3 is not equal to (1 × 3) − (1 × 3).
Step-by-step explanation:
A
someone help me if you can :)
Answer:
the answer is 126,321,000
PLEASE HELP!!!!!
will mark brainliest!!!!!
Answer:
Table:
1: f(x) = 5
2: f(x)= 6
5: f(x) = 9
Points: (1,5), (2,6), and (5,9)
Step-by-step explanation:
1. The function f(x) = x + 4 means that 4 is added to the input, x, to get the output, y.
[tex]f(x) = 1 + 4 = 5[/tex] [tex]f(x) = 2 + 4 = 6[/tex] [tex]f(x) = 5 + 4 = 9[/tex]Therefore, when x = 1 then f(x) = 5, x = 2 then f(x) = 6, and x = 5 then f(x) = 9.
2. Now, to find the points, we must know that x is the x-coordinate and f(x) is the y-coordinate of the point. Because of this, the points are:
[tex](1,5)[/tex] [tex](2,6)[/tex] [tex](5,9)[/tex]Therefore, the points are (1,5), (2,6), and (5,9).
Is x-10 a factor of f(x)=5x^3+60x^2+109x+90
Answer:
No
Step-by-step explanation:
Given values are
f(x) = 5x^3+60x^2+109x+90
x = -10.
Given Polynomial is 5x^3+60x^2+109x+90 .
= (90)+(5.x^3)+(60.x^2)+(109.x)
By putting x = (-10) we can rewrite it as
= (90)+(5.(-10)^3)+(60.(-10)^2)+(109.(-10))
= (90)+(-5000)+(6000)+(-1090)
= 0
The remainder of given polynomial is 0.
Find a10, for the geometric sequence an = 3(4)^n-1.
A. 262,144
B. 786,432
C. 1,048,579
D. 3,145,728
help me please with the question I will mark BRAINLIEST
Answer:
4^126^78^57^3Step-by-step explanation:
apply the indices law
multiple means additiondivision means subtraction you are suppose to add the powers or subtract where necessary4^9x 4^3= 4^56^5x6^2=6^78^6÷8=8^57^8÷7^6=7^2Mr. Alcazar bakes cookies in batches of 60 and batches of 90. If he bakes 6 batches of 60 cookies, how many cookies does he bake?
Answer:
Mr. Alcazar baked 810 cookies.
Step-by-step explanation:
Given that Mr. Alcazar bakes 6 batches of 60 cookies and five batches of 90 cookies, to determine the amount of cookies that he cook, the following calculation must be performed:
(6 x 60) + (5 x 90) = X
360 + 450 = X
810 = X
Therefore, according to the calculation that has been made, Mr. Alcazar baked 810 cookies.
what is 13q - 9q + 5 = 17
the answer is: q=3
start by combining like terms (13q - 9q) which would leave you with 4q + 5 = 17. subtract 5 from both sides and you're left with 4q = 12. divide by 4 on both side and you get q = 3
Answer:
q=3
Step-by-step explanation:
13q - 9q + 5 = 17
13q - 9q = 17 - 5
4q = 12
4q/4 = 12/4
q = 3
What kind of fraction is 14/6
Yes, It is an improper fraction
Find the x-intercept and y-intercept of the line. 7x+5y=-15
Answer:
The x-intercept is (-2 1/7, 0).
The y-intercept is (0, -3).
Step-by-step explanation:
To find the y-intercept, substitute x with zero.
To find the x-intercept, substitute y with zero.
Finding the y-intercept by substituting x with zero:
7*0 + 5y = -15
5y = -15
y = -3
The y-intercept is (0, -3).
Finding the x-intercept by substituting y with zero:
7x + 5*0 = -15
7x = -15
x = -15/7
x = -2 1/7
The x-intercept is (-2 1/7, 0).
See photo please help
Answer:
2/3 is right one . option no B
Please Help giving 18 points
Answer:five inches
Step-by-step explanation:
Have a great day
Hi! I am giving the community a chance, will give brainliest if you answer this question for 20 points
Answer:
Step-by-step explanation:
Lol, so you start by looking at two of your fingers.
You pull up two more fingers.
To find the answer, you add those two numbers (two and two), which gives you four.
Final Answer: 'Four' in words or 4 in numbers.
Hope you found this helpful, or at least amusing.
Shureka Washburn has scores of 62,81,76, and 59 on her algebra tests. a. Use an inequality to find the scores she must make on the final exam to pass the course with an average of 72 or higher, given that the final exam counts as two tests. b. Explain the meaning of the answer to part (a).
Answer:
54
Step-by-step explanation:
Below is the graph of a function, y = f (x).
(a) Draw a secant line on the graph above from x = 0 to x = 4. (copy the graph if you are writing your answers on a separate paper)
(b) What is the average rate of change of f (x) from x = 0 to x = 4? Show how you get your answer.
Answer:
See answer I left in previous post
Step-by-step explanation:
What is the equation of a line parallel to y=-4x-7 that passes through 2,29
What percent is 11÷88
Answer:
hello
Step-by-step explanation:
it must be 0,125
hope it helps
OMG I NEED HELP BADLY RN
n which quadrant does the point (6 , -1) lie?
A.
Quadrant I
B.
Quadrant II
C.
Quadrant IV
D.
Quadrant III
Answer:
C) Quadrant 4
Step-by-step explanation:
Hope this helps! Pls give brainliest!
You have 8 cups of flour. A recipe calls for ⅔ cup of flour. Another recipe calls for ¼ cup of flour. How much flour do you have left after making the recipes?
Answer:
7 1/12 cups of flour
Step-by-step explanation:
2/3 + 1/4 = (8 + 3)/12 = 11/12
8 - 11/12 = 85/12 = 7 1/12 cups
Which of the following is true?
A. |−8| < 6
B. |−6| < |−8|
C. |8| < |−6|
D. |−6| < −8
Answer:
B. |-6| < |-8|
Step-by-step explanation:
I used a calculator, pls let me know if im incorrect
Find the unit price $3.30 for 15 bananas
Answer:
0.22 cents each
Step-by-step explanation: