Compute the indicated power by using de Moivre's theorem.
[tex]\sqrt[4]{1-i\sqrt{3} }[/tex]
please hurry
I assume you know that there are n total n-th roots for any non-zero complex number.
(If you're looking for "the" principal root, you'll have to specify which branch you're calling the principal branch of the n-th root function.)
Start by writing [tex]1 - i\sqrt3[/tex] in exponential form. We have modulus/magnitude
[tex]|1 - i\sqrt3| = \sqrt{1^2 + (-\sqrt3)^2} = \sqrt4 = 2[/tex]
and [tex]1-i\sqrt3[/tex] lies in the fourth quadrant of the complex plane, so its argument is
[tex]\arg(1-i\sqrt3) = \tan^{-1}\left(-\sqrt3\right) = -\dfrac\pi3[/tex]
Then the exponential form is
[tex]1 - i \sqrt3 = 2 \exp\left(-i\dfrac\pi3\right)[/tex]
(where [tex]\exp(x) = e^x[/tex], if you're not familiar with the notation)
By de Moivre's theorem, we have the fourth roots
[tex]\sqrt[4]{1 - i\sqrt3} = \sqrt[4]{2} \exp\left(i \dfrac{-\frac\pi3+2k\pi}4\right)[/tex]
where [tex]k\in\{0,1,2,3\}[/tex], so that we have a choice of
[tex]\sqrt[4]{1 - i\sqrt3} = \begin{cases} \sqrt[4]{2} \exp\left(-i\dfrac\pi{12}\right) \\\\ \sqrt[4]{2} \exp\left(i\dfrac{5\pi}{12}\right) \\\\ \sqrt[4]{2} \exp\left(i\dfrac{11\pi}{12}\right) \\\\ \sqrt[4]{2} \exp\left(i\dfrac{17\pi}{12}\right) = \sqrt[4]{2} \exp\left(-i\dfrac{7\pi}{12}\right) \end{cases}[/tex]
(I rewrote the exponent to the last root just to be consistent about having each argument between -π and π radians)
If you want these in rectangular form (a + bi), sorry, that's where I draw the line; it can be done with simple trigonometry and algebra, but it's rather tedious, and the exponential forms are far more compact.
Which table represents a linear function?
X
X
y
1
1
1
2
2
1
2
3
4
34
O
122
1/7/2
O
y
1
121314
X
1
2
3
4
X
1
2
3
14
O
y
7
9
13
21
y
0
6
16
30
the third one
Step-by-step explanation:
Factorise completely
12y² + 4y
Step-by-step explanation:
[tex]12y^{2} + 4y \\4y( \frac{12y^{2} }{4y} + \frac{4y}{4y} ) \\ 4y(3y + 1)[/tex]
Step-by-step explanation:
12y²+4y
divide both sides by 4
therefore,
3y²+y
y(3y+1)
Represent -5 / 4 and 5 / 4 on number line
Answer:
Image included:
Step-by-step explanation:
Use the hundredth grids to model the question.
Two 10 by 10 grids are shown. In the first grid, 8 columns and 9 squares are shaded brown and 1 column and 1 square are shaded blue. The second grid is entirely shaded blue.
Which is the sum of 0.89 and 1.11?
A.
1.00
B.
1.89
C.
1.90
D.
2.00
The sum of 0.89 and 1.11 will be 2. Then the correct option is D.
What is Algebra?The analysis of mathematical representations is algebra, and the handling of those symbols is logic.
Two 10 by 10 grids are shown.
In the first grid, 8 columns and 9 squares are shaded brown and 1 column and 1 square is shaded blue.
The second grid is entirely shaded blue
Then the sum of 0.89 and 1.11 will be
⇒ 0.89 + 1.11
⇒ 2.00
Then the correct option is D.
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kantabai bought 1/2 kg tea and 5 kg sugar from a shop .she paid rupess 50 as return fare for rickshaw .total expense was rupess 700 .then she realised that by ordering online the goods can be bought with free home delivery at the same price .so next month she placed the order online for 2 kg tea and 7 kg sugar she paid rupess 880 for that .find the rate of sugar and tea per kg
Answer:
40 Rs per kg.
Step-by-step explanation:
solution
Let the rate of tea be x Rs per kg and that of sugar be y Rs per kg.
When Kantabai bought the items by going to the shop,
3/2x+5y+50=700
⇒3x+10y=1300....(I)
When Kantabai bought the items online then
2x+7y=880......(II)
Multiplying (I) with 2 and (II) with 3 we get
6x+20y=2600......(III)
6x+21y=2640.......(IV)
eq.(IV)−(III)
y=40
Putting the value of y =40 in (II)
2x+7(40)=880
⇒2x=880−280=600
⇒x=300
Thus, tea is at 300 Rs per kg and sugar is 40 Rs per kg.
Select ALL the correct answers. Observe the expression below and select the true statement(s). The "6" in the third term is a coefficient. The "7" in the third term is a coefficient. The "9" in the second term is a constant. The "y" in the third term is a factor. The "4" In the first term is a constant. The "x" in the first term is an exponent. 4r + 9 = y(6x + 7)
An expression is defined as a set of numbers, variables, and mathematical operations. The statements that are true are 1, 3, and 5.
What is an Expression?In mathematics, an expression is defined as a set of numbers, variables, and mathematical operations formed according to rules dependent on the context.
For the given expression 4x + 9 - y(6x + 7), the following things are can be written,
4 and 6 are coefficients.9 and 7 are constants.x and y are variables.Thus, the statements that are true are:
The 6 in the third term is a coefficient.The 9 in the second term is a constant.The y in the third term is a factor.Hence, the statements that are true are 1, 3, and 5.
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^ What is the product of 3x(x²+4)? ´0x²+3x+4 O 3x³+4 O 3x³ + 12x O 3x² + 12x
Answer:
3x³ + 12x
Step-by-step explanation:
→ Multiply 3x by x²
3x³
→ Multiply 3x by 4
12x
answer this question asap
Answer:
the first option
Step-by-step explanation:
f(c)= 1/3p + 5/3
make r the subject of the formula in V= ⅓π r²h
Answer:
switch the position
1/3πr²h=v
multiply all through by 3
3•1/3πr²h=3•v
3 will cancel out 3
πr²h=3v
divide both sides by πh
r²=3v/πh
root both sides
r=±√3v/πh
Recently, bats have been disappearing from the wild due to a contagious fungus. To monitor the current population in a certain region, Troy collects 78 bats, marks them, and releases them. Later, Troy collects 600 bats, and finds that 39 of them are marked. To the nearest whole number, what is the best estimate for the bat population?
Answer:
1200 bats
Step-by-step explanation:
78 - 39 = 39
600 x 2 = 1200(bats).
Mika bought 9 rolls of film to take 180 pictures on a field trip. Some rolls had 36 exposures and the rest had 12 exposures. How many of each type did Mika buy
The number of each type Mika bought =x = 6.8 rolls; y= 2.2 rolls.
Calculation using ratioThe number of rolls of film Mika bought = 9 rolls
The number of rolls that had 36 exposures = x
The number of rolls that had 12 exposures = y
The total number of exposures = 36 + 12 = 48
To find the exposure by one roll = 48/9 = 5.3
If one roll = 5.3 exposure
x roll = 36 exposure
Make x roll the subject of formula,
x roll = 36/5.3 = 6.8 rolls
If one roll = 5.3 exposurey roll = 12 exposure
Make y roll the subject of formula,
y = 12/5.3 = 2.2 rolls
Therefore, the number of each type Mika bought = x = 6.8 rolls; y= 2.2 rolls.
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help i need it quick!!
Answer:
47°
Step-by-step explanation:
45+88= 133
180-133=47
what type of function is this?
Answer:
It's a sech x function
Step-by-step explanation:
Explanation is in attached file
find the approximate volume of this cone. use 3.14 for pi.
if cos a =0.8290 then a < is approximately
Answer:
34 degrees
Step-by-step explanation:
Just reverse the cos function ( you will need a calculator)
arc cos (.8290) = 34 degrees
this is sometimes written as cos^-1 (.8290) <====it is the same thing
1. Triangle STU is shown on the coordinate plane below. Triangle STU is transformed using the rule (x, y) -> (x+4, y-2). In the image of the transformation, triangle S'T'U', what is the x-coordinate of S'?
Using translation concepts, considering S(2,-1), the x-coordinate of S' is given by 6.
What is a translation?A translation is represented by a change in the function graph, according to operations such as multiplication or sum/subtraction in it's definition.
In this problem, the rule is given by:
(x, y) -> (x+4, y-2).
Considering S(2,-1), 4 is added to the x-coordinate in the translation, hence:
2 + 4 = 6.
The x-coordinate of S' is given by 6.
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Question 25 of 25
Over what interval is the function in this graph increasing?
AV
M
<-10
10
A. 6≤x≤ 10
B. -4≤x≤1
C. -10≤x≤-4
D. 1≤x≤6
NEED HELP ASAP
The correct option is option B : -4 ≤ x ≤ 1
What is Increasing and decreasing function?
Increasing and decreasing functions are functions whose graphs go upwards and downwards respectively as we move towards the right-hand side of the x-axis. Increasing and decreasing functions are also called non-decreasing and non-increasing functions.
Here, from graph
as we can observe from x = -4 graph starts increases upto x = 1 and after that graph becomes constant.
Thus, between x belongs -4 to 1 the graph is increasing.
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Mr Ellis invests £800 into an account that pays 3.4% compound interest per year how much will be in the account after two years
The amount after 2 years will be 855.3248
What is Interest?Interest, in its most simple form, is calculated as a percent of the principal.
Given:
P= £800 , r= 3.4%, t= 2 year
For, compounded annually
=800*(0.034+1)²
=800* 1.034*1.034
=800* 1.069156
=855.3248
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According to the box-and-whisker plot shown above, what are the following:
6. The median?
7. The first quartile?
8. The third quartile?
9. The minimum value?
10. The maximum value?
I need help i put the plot i need help 6 threw 10
Answer:
6. 80
7. 60
8. 100
9. 40
10. 110
Step-by-step explanation:
A box and whisker plot is a visual representation of the 5-number summary of a data set. The features of the plot are, left-to-right, the minimum, first quartile, median, third quartile, maximum.
You are being asked to identify the numerical value associated with each of these features.
__
The left whisker extends to the minimum of 40.
The left end of the box identifies the first quartile as 60.
The line in the middle of the box is the median, 80.
The right end of the box identifies the third quartile as 100.
The right whisker extends to the maximum of 110.
__
Additional comment
We hope you can rearrange the order of these numbers to suit the order in which the questions ask for them.
DUE TODAY: PLEASE HELP
Use transformations to prove that the two figures are similar. Shape A is the original image
(Show work please, how it is similar or not)
Answer:
See below ~
Step-by-step explanation:
We need to prove that the figures are similar by applying the various kinds of transformations.
===============================================================
Question A :
⇒ Compare the side lengths of A and B
⇒ 2 : 5 (for all sides)
⇒ 1 : 2.5
⇒ Hence, Shape A has been dilated with a scale factor of 2.5 to form Shape B
=============================================================
Question B :
⇒ Compare the side lengths of A and B
⇒ A : B = 2 : 6
⇒ A : B = 1 : 3
⇒ Shape A has been dilated with a scale factor of 1/3 to form Shape B
Which operation should be completed first to find the value of the expression below? 12 + 30/ (6 - 3) + 1
12 + 30
30 ÷ 6
3 + 1
6 - 3
The operation that should be completed first to find the value of the expression is 6 - 3
How to determine the first operation?The expression is given as:
12 + 30/(6 - 3) + 1
When solving an expression;
All expressions in the bracket must be first evaluated
This means that the operation that should be completed first is 6 - 3
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a negative number raised to an odd power is ____ negative ?
Answer:
always negative?
Step-by-step explanation:
-2^3=-8
Answer:
It is negative
Step-by-step explanation:
since you are multiplying the negative no. by itself an odd no. of times, your answer will be negative, b/c a negative no. multiplied an odd no. of times will be negative, and if it is multiplied an even no. of times it will be positive.
A family reunion planning committee with 8 members plans to elect 3 officers - a president, treasurer, and historian. if each office is to be held by one person and no person can hold more than one office, in how many ways can those offices be filled
Using the permutation formula, it is found that these offices can be filled in 336 ways.
There are different roles, hence the order in which the people are chosen is important, hence the permutation formula is used.
What is the permutation formula?The number of possible permutations of x elements from a set of n elements is given by:
[tex]P_{(n,x)} = \frac{n!}{(n-x)!}[/tex]
3 people will be chosen from a set of 8, hence the number of ways is given by:
[tex]P_{8,3} = \frac{8!}{5!} = 336[/tex]
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Use the scenario to answer the question.
Bryan is asked to find the interquartile range for the data set {5,6,6,8,10,11,14,16,17}.
His work follows.
Step A: Find the median of the data set. The median, or Q2, is 10.
Step B: Find the value of Q1, or the median of the lower half of the data set, {5,6,6,8}.
Q1=6+62=6
Step C: Find the value of Q3, or the median of the upper half of the data set, {11,14,16,17}.
Q3=14+162=15
Step D: Find the difference between Q3 and Q1.
IQR=15−6=9
At which step did Bryan make a mistake, if he made one?
At Step B, he incorrectly found the value of Q1. He should have found the sum of the values and then divided by 4 to get Q1=6.25.At Step B, he incorrectly found the value of textsf cap q 1. He should have found the sum of the values and then divided by 4 to get
At Step A, he incorrectly found the value of the median. He should not have included both 6 values, so the median is 10+112=10.5.At Step A, he incorrectly found the value of the median. He should not have included both 6 values, so the median is
At Step C, he incorrectly found the value of Q3. He should have included 10 to get Q3=14.At Step C, he incorrectly found the value of textsf cap q 3. He should have included 10 to get
At Step D, he incorrectly found the IQR. He should have found the difference between each quartile to get 10−6=4 and 15−10=5.At Step D, he incorrectly found the IQR. He should have found the difference between each quartile to get 10 minus 6 is equal to 4 and
He did not make a mistake.
Given the steps used to calculate the interquartile range, Bryan did not make any mistake.
What is the interquartile range?
The interquartile range is the difference between the third quartile and the first quartile.
Third quartile = 3/4(n + 1) : 3/4 x 10 = 7.5 term : (14 + 16) / 2 = 15First quartile = 1/4 x (n + 1) 1/4 x 10 = 2.5th term : (6 + 6) / 2 = 6Interquartile range = 15 - 6 = 9To learn more about interquartile range, please check: https://brainly.com/question/3966385
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A perpendicular bisector crosses a line segment.which type of angle does it form with the segment
Answer: A right angle
Step-by-step explanation:
Perpendicular lines form right angles
Answer:
90 degree
Step-by-step explanation:
Find the area of the shaded region in terms of pi
Answer:
[tex]13\frac{1}{3} \pi[/tex]
Step-by-step explanation:
First, find the area of the small circle whose radius is 3cm.
[tex]A = \pi r^{2} \\A = \pi (3^{2})\\A = \pi (9)\\A = 9\pi \\[/tex]
Next, find the area of the large circle. First, you need to find the radius of the large circle by adding the 3 and 4.
Radius of large circle.
3 + 4 = 7 cm
Area of large circle:
[tex]A = \pi r^{2} \\A = \pi (7^{2})\\A = \pi (49)\\A = 49\pi \\\\[/tex]
Now subtract the area of the small circle from the large circle since the shaded area doesn't include the small circle.
[tex]49\pi -9\pi =40\pi[/tex]
Finally find the shaded area. To do this first find what degree of the circle is shade by dividing 120 by 360.
120/360 = 1/3
Now multiply the area by 1/3, or you can divide it by 3
[tex]\frac{40\pi }{3} =13\frac{1}{3} \pi[/tex]
Write the equation for g(x)
The equation of the function, g(x) from the task content is; g(x) = -x/25 + 203/25.
What is the equation of the function g(x)?From observation of the function, f(x), it follows that the slope of f(x) is 25. Hence, it follows that the slope of g(x) is; -1/25 since both functions are perpendicular to each other.
Therefore, the equation for g(x) is;
-1/25 = (y-8)/(x-3)
25y -200 = -x +3
Hence, y = -x/25 + 203/25.
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Assuming that no denominator equals zero, which is equivalent to? r^2-r-6/(r-2)(r-3)
The given expression is [tex]\dfrac{ r^2-r-6}{(r-2)(r-3)}[/tex]. the option C [tex]\dfrac{ (r+2)}{(r-2)}[/tex]is correct.
What is a simplification of an expression?Usually, simplification involves proceeding with the pending operations in the expression.
Simplification usually involves making the expression simple and easy to use later.
The given expression is
[tex]\dfrac{ r^2-r-6}{(r-2)(r-3)}[/tex]
By solving the numerator
[tex]r^2-r-6\\\\r^2 - (3-2)r- 6\\\\(r-3)(r+2)[/tex]
Put the solution in the above equation
[tex]\dfrac{ (r+2)(r-3)}{(r-2)(r-3)}[/tex]
[tex]\dfrac{ (r+2)}{(r-2)}[/tex]
Therefore, the option C is correct.
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If you use 1/8 lb of ham on each sandwich you make, how many
sandwiches
can you make from a 22 lb ham?