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.
How does the volume of the larger cube compare to the volume of the smaller cube? Use pencil and
paper. Use this and other examples to describe what happens to the volume of a cube if you triple the length
of the edges.
Answer:
Option C, The volume of the length cube is 3 times the volume of the smaller cube
Step-by-step explanation:
21.8 / 7.8 = 2.8
the volume of the larger would be 21.8^3 = 10360.23
the volume of the larger would be 7.8^3 = 474.55
4X+8<28
ayudenme por favor
Answer:
x < 5.
Step-by-step explanation:
Creo que esto es correcto. :')
What is true about the net of a pyramid?
It will always contain exactly 1 triangle.
It will always contain exactly 2 triangles.
It can contain any number of triangles greater than or equal to 3.
It can contain any number of triangles greater than or equal to 4.
A 3D shape's net is what it looks like after it's flattened out. A net can be folded into a three-dimensional shape. The correct option is C.
What is the net of a 3-D shape?A 3D shape's net is what it looks like after it's flattened out. A net can be folded into a three-dimensional shape. For the same 3D shape, there may be multiple potential networks. You can make a net out of paper and fold it into the desired shape.
The net of the square pyramid has a square and 4 triangles. For a pyramid with a base as a polygon having n sides, the number of triangles is n.
Hence, it can be concluded that a pyramid can contain any number of triangles greater than or equal to 3. Thus, the correct option is C.
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in the figure at the right DC = 20 cm and AB =36 cm . what is FE? a.16cm b.56cm c.28cm d.46cm
Answer:
however dc =20 cm is
Step-by-step explanation:
so this isnt contrary to 50
Answer:
If it's all about trapezoid and FE is the median:
SOLUTION:
m = \frac{1}{2}
2
1
[(20) + (38)]
m = \frac{1}{2}
2
1
(58)
m = \frac{58}{2}
2
58
m = 26
ANSWER: 26 cm
Step-by-step explanation:
What segment is the altitude to side AC?
Segment CR
Segment BT
Segment BL
30 people are running for an office to be chosen randomly. You plan to use a 6-sided dice and a spinner. How many sections need to be on the spinner so each person has their own shape and number?
Select one:
a.
8
b.
6
c.
5
d.
24
The number of sections on the spinner so that each person has heir own shape and number is 5
How to determine the number of sections?The given parameters are:
People, n = 30
Die = 6 sided
Let the sections on the spinner be s.
The equation to calculate s is represented as:
People = Die * s
This gives
30 = 6 * s
Divide both sides by 6
s = 5
Hence, the number of sections on the spinner so that each person has heir own shape and number is 5
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The total numbers of hours several different athletes spent training over the last year are shown below. 155, 175, 190, 210, 213, 243, 340, 380, 390, 457, 670 Which measure BEST shows the center of this data set? mean median mode range
Answer:
range is 515, mode:All values are equal, Median is 243 and the mean is 311.18
mean:to find the mean you need to add up all you numbers so all of these add up to be 3423 and then you divide by the amount of numbers which it 11 so 11 / 3423 = 311.18 so the answer is the mean is 311.18
median: to find it you need to put all the numbers smallest to biggest so it would be in the order 155, 175, 190, 210, 213, 243, 340, 380, 390, 457, 670 and then you pick the one in the middle and that would be 243
mode:All values appeared just once so there is not one
range: to find the range to need to find the differences in the smallest to the biggest number so smallest one is (155) biggers (670) so 155 - 670 = 515 so the range is 515
Hope This Helped
10 points, helpppp please!
Answer:
64
Step-by-step explanation:
Important angle relationships that will help us answer this question :
Supplementary angles : Supplemtary angles are angles that for a straight line. The sum of these angles is 180 degrees.Corresponding angles : corresponding angles are angles that are on the same side of the traversal as well as the same side of the intersecting lines. These angles are congruent ( equal to each other ) if the two lines that they fall on are parallel ( which they are )Opposite exterior angles - These angles are angles on the very outside of the lines and are on opposite sides of the transversal. These angles are congruent as wellFor more clarification kindly view the attached images :)
Using these angles relationships, we can easily find the answer two different ways.
First way :
Creating an equation : We want to find the measure of angle MPN. It's important to note that this angle falls on a line made up of other angles ( angle mpe and angle npe ) forming a straight line ( line e )
Using our angle relationships we know that angles that form a straight line are supplementary and add up to 180 degrees.
So we can say angle mpe + angle npe + angle MPN = 180. ( we can use this equation to solve for angle MPN )
Here we are given that angle mpe is a right angle ( indicated by the little square ) meaning it has a measure of 90 degrees.
So we can further simplify our equation saying that angle npe + 90 + angle MPN = 180 ( by plugging in angle mpe = 90 )
Now if you didn't know we can solve an equation ( find the missing angle ) if there is a single variable ( in this case we want the variable to be angle MPN ). This means that in order to solve for angle MPN we just find the measure of the other angle, angle npe.
Finding angle npe
The angle under "x" is corresponding to angle mpE so we can say the angle under x = angle npe.
The angle under x also appears to be supplementary to the angle with a measure of 153 degrees. So we can say that angle under x + 154 = 180
Subtracting both sides by 153 we get that angle under x = 180 - 154 = 26 degrees.
Because the angle under x and angle npe are corresponding and equal to each other, angle npe also has a measure of 26 degrees.
Finishing creating the equation and solving for angle MPN : So we have angle npe + angle mpe + angle MPN = 180
We know angle mpe = 90 and angle npe = 26
So we can say 90 + 26 + angle MPN = 180
Now we can solve for angle MPN
90 + 26 + angle MPN = 180
==> combine like terms
116 + angle MPN = 180
==> subtract 116 from both sides
angle MPN = 64 degrees
Second way :
We can also solve for angle MPN using the angle relationship " opposite interior angles "
The angle with a measure of 154 degrees and angle ePN are opposite exterior angles meaning they are congruent. ( so angle ePN = 154 )
Angle ePN is made up of angle mpe and angle MPN. This means that angle mpe + angle MPN = 154
We've already identified that angle mpe = 90 degrees because it's a right angle so we can say angle angle MPN + 90 = 154
Subtracting both sides by 90 we get that angle MPN = 154 - 90 = 64 degrees
And we are done!
HELP PLEASE HURRY
g(x)
3
f(x) = 4x + 5
-32
#12
Which function has the greatest y-intercept?
3/
h(x)=3 sin(2x + 1)-2
Answer:
pls write your answer in order i dont know if they are together or separate
What are the coordinates of the point that is 2/5 of the way from a to b
Answer:
the cords for a should be (-3,0) and the cords for b should be (7,15) sorry if im incorrect
Find the value of x. (Giving 30 points please hurry)
Answer: Hope this helps you!
x = 4
Step-by-step explanation:
4+(x-3) = 2x-3
remove parentheses (equals to 4+1(x-3) = 2x-3 which is equal to the same equation without parentheses)
4+x-3 = 2x-3
Simply
1+x = 2x-3
subtract x from both sides
1 = 1x -3
add 3 to both sides
4 = 1x
divide by 1
4 = x
Brainliest?
Graph the line whose x-intercept is 8 and whose y-intercept is -9.
Answer: attached image
Pepper Jackie is putting 742 hard-boiled eggs into cartons for a community
farm fundraiser. Each carton holds 12 eggs. How many cartons of eggs can she
make?
Answer:
About 61 cartons with a reminder with a remainder of 10
Step-by-step explanation:
12 * 61 = 732 → 742 - 732 = 10 eggs
hope this helps
Calculate the first and second
differences for the function f(x)=
3x² +5
xy First Differences
Second
Differences
05
18
217
332
453
Answer:
Hi,
Step-by-step explanation:
[tex]\begin{array}{|c|c|c|c|}x&3x^2+5&\Delta_1x&\Delta_2x\\--&--&--&--\\0&5&-&-\\1&8&3&-\\2&17&9&6\\3&32&15&6\\4&53&21&6\\--&--&--&--\\\end{array}[/tex]
complete the proof and tell me how to come up with the answer please ASAP
Step 4 is the reason SSS
In one country, approximately 178,200 of 990,000 residences are rentals. Determine the percentage of residences in the country that are rentals.
Answer:
18%
Step-by-step explanation:
178 200 / 990 000 x 100% = 18%
Which points do the graph of f and g have in g(x)=log2x and g(x)=log10x
Logarithmic are inverse of exponential functions. The point where graph of f and g intersect is (0, ∞)
Logarithmic functionLogarithmic are inverse of exponential functions. Given the following functions;
g(x)=log2x and g(x)=log10x
The point where the graphs intersect is the point where g(x) = g(x)
Equate
log2x = log10x
2x = 10x
The value of x that made them equal is 0
Determine the value of g(x)
g(0) = log2(0)
g(0) = ∞
Hence the point where graph of f and g intersect is (0, ∞)
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Here is a growing pattern of sequences:
Answer:
A, C, F
Step-by-step explanation:
The first figure is drawn using 8 squares. The next uses 11 squares, and the one shown in Step 3 uses 14 squares. The numbers of blocks follow the sequence ...
8, 11, 14
which begins with 8 and increases by 3 from one step to the next. This constant increase from one term to the next makes it an arithmetic sequence.
__
explicit formThe general term, an, of an arithmetic sequence can be written in terms of its first term, a1, and its common difference, d, as ...
an = a1 +d(n -1)
When the first term is a1=8, and the common difference is d=3, this becomes ...
an = 8 +3(n -1)
which simplifies to
an = 8 +3n -3 = 3n +5
Written in functional form, these explicit equations are ...
f(n) = 8 +3(n -1) . . . . . matches choice A
f(n) = 3n +5 . . . . . . . . matches choice F
__
recursive formThe recursive form of the definition of a sequence starts with the first term, then tells you how to obtain the next term from that. Here, the first term is ...
f(1) = a1 = 8
and each term is 3 more than the last:
f(n) = 3 +f(n-1) . . . . . . . . . 3 is added to the previous term
Together, this is ...
f(n) = 8, f(n) = 3 +f(n -1) . . . . . matches choice C
The chess club offered free ice pops to 200 students. Only
170 students accepted the treat. What percentage of students accepted an ice pop?
Answer:
Below
Step-by-step explanation:
100 * 170/200
= 170/2
= 85%.
A study of the amount of time it takes a specialist to repair a mobile MRI shows that the mean is 8.4 hours and the standard deviation is 1.8 hours. If 32 broken mobile MRIs are randomly selected, find the probability that their mean repair time is less than 8.9 hours.
Using the normal distribution, it is found that there is a 0.0582 = 5.82% probability that their mean repair time is less than 8.9 hours.
Normal Probability DistributionThe z-score of a measure X of a normally distributed variable with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex] is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
The z-score measures how many standard deviations the measure is above or below the mean. Looking at the z-score table, the p-value associated with this z-score is found, which is the percentile of X.By the Central Limit Theorem, the sampling distribution of sample means of size n has standard deviation [tex]s = \frac{\sigma}{\sqrt{n}}[/tex].In this problem, the parameters are given as follows:
[tex]\mu = 8.4, \sigma = 1.8, n = 32, s = \frac{1.8}{\sqrt{32}} = 0.3182[/tex]
The probability that their mean repair time is less than 8.9 hours is the p-value of Z when X = 8.9, hence:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
By the Central Limit Theorem
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{8.9 - 8.4}{0.3182}[/tex]
Z = 1.57
Z = 1.57 has a p-value of 0.9418.
1 - 0.9418 = 0.0582.
0.0582 = 5.82% probability that their mean repair time is less than 8.9 hours.
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Patrick buys a wrench with a listed price of $22. Which of these coupons will deduct the most from Patrick's purchase price?
A. Save $5 + 10%
B. Save $10
C. 15% off
D. Save 25%
Answer:
B: Save $10
Step-by-step explanation:
A: $14.80, B: $12, C:$18.70, D: $16.50
PLEASE HELP!! 50 POINTS!!
Answer:
Part A:
In order to find a coterminal angle, or angles of the given angle, simply add or subtract 360 degrees of the terminal angle as many times as possible.
Part B:Simply put, coterminal angles start at 0° and end at the same place, though in different directions and multiple times around (you're probably going to want to make that sound a little more formal). Since each time around is 360°, boom
Part C: 75° is in Quadrant II, right? If I go one more time around, 75 + 360 = 435°. I can also go "backward. Starting at 0 and going clockwise, I'm going 75 fewer degrees than all the way around, so 360-75 = 285° but since I'm going "backward" 75 - 360 = -285°. One more is on you.
Step-by-step explanation:
Based on the work done in checking his solution, does it appear that x = 9/2 is the
solution? Explain why or why not
be detailed, giving out brainliest to.
Answer:
Yes x = 9/2 is the correct solution.
Step-by-step explanation:
You have substituted x = 9/2 into both sides of the equation and they have both simplified to the same value.
1.
Solve the equation below.
a.
x = 1
x = -2
x = 3
x = -4
b.
C.
d.
2x + 3 = 5
Answer:
a) x = 1
Step-by-step explanation:
Given: 2x + 3 = 5
1. Subtract 3 from both sides:
2x + 3 - 3 = 5 - 3
⟹ 2x = 2
2. Divide both sides by 2:
2x ÷ 2 = 2 ÷ 2
⟹ x = 1
3. Check your work:
2(1) + 3 = 5
2 + 3 = 5
⟹ 5 = 5 ✔
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Using the information in the diagram, what is the height of the tree to the nearest foot?
45 ft
54 ft
60 ft
135 ft
The relationship between the corresponding sides of similar triangles indicates that the height of the tree is 60 feet. The correct option is therefore;
60 ftWhat are similar triangles?Similar triangles are triangles in which the ratio of the corresponding sides of the triangles are equivalent for the three sides of the triangle.
The location of the tree divides the hypotenuse side and the base of the triangle formed by the building into two parts of the same length, therefore;
The length of the hypotenuse side to the building = 2 × 108 ft = 216 ft
The length of the base to the building = 2 × 90 ft = 180 ft
The ratio of corresponding sides of similar triangles and the trigonometric ratio of sines, indicates that we get;
Height of the tree/108 = 120/216
Height of the tree = 108 × (120/216) = 60 feet
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A butterfly is 12 inches long and 13 inches wide. Lashay makes a scale drawing
of the butterfly that is 1 inch long. What is the width of his drawing?
Answer:
1.08 inches
Step-by-step explanation:
13 / 12 , to keep aspect ratio the same
Which pair of angles shares ray A and F as a common side?
The pair of angles that share a ray as a common side in the given diagram is: ∠FAB and ∠FAD.
What are Adjacent Angles?Adjacent angles are angles that lie adjacent or next to each other sharing a common vertex and have the same ray as a common side.
In the diagram attached below, ∠FAB and ∠FAD share a common vertex, A, and a ray FA as a common side.
Therefore, the pair of angles is: ∠FAB and ∠FAD.
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How do u find the volume of 10m ,12m ,and 15m room
Answer:
1800 meter cube
Step-by-step explanation:
you know volume = W× h× b = meter cube
10 m × 12m× 15m = 1800 meter cube
Ted is cooking a stew. A stew for 4 people requires 300g potato, 180g meat and 500g root veg. He wants to cook the stew for 6 people. He has the following ingredients: 480g potato 300g meat 750g root veg Does he have enough to cook for 6 people?
A water tank is built in the form of a right rectangular prism, with dimensions that measure 17.6 ft by 12.6 ft by 10.4 ft. How many cubic feet of water would it hold when it’s a quarter full?
Volume is a three-dimensional scalar quantity. The amount of water that this tank will hold till it is a quarter full is 576.576 ft³.
What is volume?A volume is a scalar number that expresses the amount of three-dimensional space enclosed by a closed surface.
The total volume of the right rectangular prism will be,
Total Volume = 17.6ft × 12.6ft × 10.4ft
= 2306.304 ft³
Since the total volume of the right rectangular prism is 2306.304 ft³, the water it will hold will be equal to its volume.
Now, the amount of water that this tank will hold till it is a quarter full is,
Amount of water = Total volume /4
= 2306.304ft³ / 4
= 576.576ft³
Hence, the amount of water that this tank will hold till it is a quarter full is 576.576 ft³.
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