What is (2-2i√3)4 equivalent to?
Drag a value to each box to correctly express the answer in polar form and then in rectangular form.
Answers
a. (2 - 2i√3)⁴ in polar form is 256(cos(-4π/3) + isin(-4π/3)) = 256cis(-4π/3)
b. (2 - 2i√3)⁴ in rectangular form is -128 + 128√3
To answer the question, we need to know what complex numbers are
What are complex numbers?Complex numbers are numbers of the form z = x + iy
a. Complex numbers in polar formComplex numbers in polar form z = r(cosθ + isinθ) where
r = √(x² + y²) and θ = tan⁻¹(y/x)Given that z = (2 - 2i√3)⁴ =
So,
x = 2 and y = -2√3So, converting to polar form
r = √(x² + y²)
= √[2² + (-2√3)²]
= √[4 + 4(3)]
= √[4 + 12]
= √16
= 4
θ = tan⁻¹(y/x)
θ = tan⁻¹(-2√3/2)
θ = tan⁻¹(-√3)
θ = -π/3
So, z = r(cosθ + isinθ)
= 4(cos(-π/3) + isin(-π/3))
Powers of complex numbersA complex number z raised to power n is zⁿ = rⁿ(cosnθ + isin(nθ)]
z⁴ = (2 - 2i√3)⁴
= r⁴(cos4θ + isin4θ)
= 4⁴(cos(4 × -π/3) + isin(4 × -π/3))
= 256(cos(-4π/3) + isin(-4π/3))
= 256cis(-4π/3)
(2 - 2i√3)⁴ in polar form is 256(cos(-4π/3) + isin(-4π/3)) = 256cis(-4π/3)
b. Complex numbers in rectangular formThe complex number z = r(cosθ + isinθ) in rectangular form is z = x + iy where
x = rcosθ and y = rsinθGiven that z⁴ = 256(cos(-4π/3) + isin(-4π/3)) in rectangular form,
x = rcosθ
= 256(cos(-4π/3)
= 256cos(-4 × 60°)
= 256cos(-240)
= 256cos(240)
= 256 × -1/2
= -128
y = rsinθ
= 256sin(-4π/3)
= 256sin(-4 × 60°)
= 256sin(-240)
= -256sin240
= -256 × -√3/2
= 128√3
So, z⁴ = x + iy
= -128 + 128√3
So, (2 - 2i√3)⁴ in rectangular form is -128 + 128√3
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Related Questions
Audrey takes a sheet of paper and makes a diagonal cut from one corner to the opposite
corner, making two triangles. The cut she makes is 87 inches long and the width of the paper
is 60 inches. What is the paper's length?
Answers
[tex]\large{\underline{\underline{\pmb{\frak {\color {blue}{Solution:}}}}}}[/tex]
Let us take the width of the paper as base, the cut she made be hypotenuse and the paper's length be perpendicular.
Here,
Hypotenuse (H) = 87 inch
Base (B) = 60 inch
Perpendicular (P) = [To be calculated]
As, we can use Pythagoras theorem to find. So by using Pythagoras theorem :
H² = P² + B²
[tex] {87}^{2} = {P}^{2} + {60}^{2} \\ \\ \implies \: 7569 = {P}^{2} + 3600 \\ \\ \implies \: 7569 - 3600 = {P}^{2} \\ \\ \implies3969 = {P}^{2} \\ \\ \implies \: P = \sqrt{3969} \\ \\ \implies \: P = 63[/tex]
The length of the paper is 63 inches.
[tex] \boxed{ \frak \pink{BrainlyDamurai}}[/tex]
Nashville is 595 feet above sea level. You board an airplane and the pilot ascended 39000 feet. Write the two integers that represent this scenario.
Answers
To solve this problem, let's simply find the difference in the distance between the two locations.
EquationTo express this integer, we need to define the locations with variables and then calculate the difference.
Nashville = A = 595ftPilot = B = 39000ftThe difference between the two location can be written as
[tex]Pilot - Nashville = B - A = 39000 - 595 = 38405ft[/tex]
The two integers that represent this scenario are 595 and 39000 respectively.
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f(x) = 4x² + 5x - 3
g(x) = 4x³
3x² + 5
Find (f+g)(x).
Answers
Answer:
[tex]4x^3+7x^2+5x+2[/tex]
Step-by-step explanation:
Given:
[tex]f(x)=4x^2+5x-3[/tex]
[tex]g(x)=4x^3+3x^2+5[/tex]
[tex]\begin{aligned}\implies (f+g)(x)=f(x)+g(x) & = (4x^2+5x-3) + (4x^3+3x^2+5) \\& = 4x^2+5x-3+4x^3+3x^2+5 \\& = 4x^3+4x^2+3x^2+5x-3+5 \\& = 4x^3+7x^2+5x+2 \\\end{aligned}[/tex]
Write the following ratio using two other notations.
[tex] \frac{1}{3} [/tex]
Answers
Answer:
1:3
1 to 3
Step-by-step explanation:
The other notations a ratio [tex]\frac{a}{b}[/tex] can be written in are a:b and a to b.
From what we are given, a = 1 and b = 3, so in the form a:b, the ration would be 1:3 and in the form a to b it would be 1 to 3.
HELP PLEASEEE GEOMETRY
Answers
Simplify: square root of 81+18[tex]\sqrt{7}[/tex]+7
Answers
[tex] \sqrt{81 \: + \: 18 \sqrt{7} \: + \: 7} [/tex]
Factor the indicated expression:[tex] \sqrt{(9 \: + \: \sqrt{7} ) ^{2} } [/tex]
Simplified the index, the root and also the exponent using the number 2.[tex] \boxed{ \bold{9 \: + \: \sqrt{7} }}[/tex]
MissSpanishLeslie buys a new pair of sneakers for
$43 and several pairs of laces, I, that
cost $6 each. Write an algebraic
expression to represent the total
amount she will spend.
Answers
Answer:
$43 + $6l
Step-by-step explanation:
Hello!
She spends $43 and bus several $6 laces.
We don't know how many she buys, but we know that we can represent that number using l.
That means she spends $43 + $6l.
The algebraic expression is $43 + $6l.
We can determine the price if we know the number of laces she buys. We can simply plug in the number of laces she bought, and we can plug it in for l in the equation.
Answer:
43= 6(x)
Explanation:
43 is the total, and each of them are 6$ each. X is the variable that we don't know.
You have been given data about gas prices in 15 states. Comparing the two data sets below, what does the larger standard deviation, range, and variance say about gas prices in 2022 compared to gas prices in 2018?
Gas Prices in 2018 Gas Prices in 2022
Max: $2.42
Min: $2.03
Mean: $2.098
Median: $2.06
Mode: $2.04
Variance: 0.0121
Standard Deviation: $0.11
Max: $5.87
Min: $4.48
Mean: $4.83
Median: $4.72
Mode: $4.72
Variance: 0.0142
Standard Deviation: $0.38
Answers
Comparing the two data sets shown below
What is standard deviation and variance?Variance is the average squared deviations from the mean, while standard deviation is the square root of this number.
Both the mean and the median can be used to describe where the “center” of a dataset is located.
So, the mean of data in 2022 is more than mean of data in 2018.
It’s best to use the mean when the distribution of the data values is symmetrical and there are no clear outliers.
As the variance of 2022 more than in variance in 2018.
A high variance indicates that the data points are very spread out from the mean, and from one another.
As the standard deviation of 2022 more than in standard deviation in 2018.
The high standard deviation indicates data are more spread out.
The bigger the range, the more spread out the data.
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please answer, much appreciated!!!!!
Answers
Answer:
i don't understand your question
Step-by-step explanation:
please help !!!! and please answer every question not just one part.
Answers
Answer:
1) C. x-1 tells us that it has decreased by a value
2) original number of apples
3) 4(x-1)=28. Divide by 4, then take the 1 over
What is the value of x? Do not round your answer.
Answers
Answer:
x = 3
Step-by-step explanation:
Setup a proportion
[tex]\frac{CB}{CA} =\frac{DB}{DA} \\\frac{4}{3} =\frac{x}{2.25} \\3x=9\\\frac{3x}{3} =\frac{9}{3} \\x=3[/tex]
Simplify the expression to a + bi form:
√4+ √−4+√36 + √−4
Answers
Answer:
8 + 4i
Step-by-step explanation:
according to the bi form, the solution to the expression √4+ √−4+√36 + √−4 would be 8 + 4i
For what value of x does 4 superscript x baseline = (one-eighth) superscript x 5?
Answers
An equation is formed of two equal expressions. The value of x for which 4ˣ=(1/8)ˣ⁺⁵ is 3.
What is an equation?An equation is formed when two equal expressions are equated together with the help of an equal sign '='.
The value of x for which 4ˣ=(1/8)ˣ⁺⁵ is,
4ˣ=(1/8)ˣ⁺⁵
(2²)ˣ = (2⁻³)ˣ⁺⁵
2²ˣ = 2⁽⁻³ˣ⁺¹⁵⁾
2x = -3x + 15
5x = 15
x = 3
Hence, the value of x for which 4ˣ=(1/8)ˣ⁺⁵ is 3.
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20 points! Which graph represents two functions that are decreasing on all points across the domain that is common to both functions?
Answers
D(x) is decreasing
Answer:
Its C or D, I would say C tho
Step-by-step explanation:
I took the test and both A and B are wrong
Evaluate the numerical expression.
9 × [(25 − 4) − (4 + 6)]
The value of the expression is
Answers
Answer:
firstly the inner brackets are simplified followed by the outer brackets, and finally the value we got from simplifying brackets is multiplied with 9
Consider function g.
g(x)={(6,-8<=(-2)<-2 ),(0,-2<=(-2)<4),(-4,4<=(-2)<10):}
What are the values of the function when x = -2 and when x = 4?
g(-2) = __
g(4) = __
Answers
By evaluating the piecewise function, we get:
g(-2) = 0.
g(4) = 4.
How to evaluate the piecewise function?A piecewise function is a function that behaves differently in different parts of its domain.
Here, we first want to get g(-2). x = -2 belongs to the second domain, then we use that part:
g(-2) = 0.
For x = 4, it belongs to the third domain, then we have:
g(4) = 4.
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I need help in this question if highly appreciate it if someone can help me solve this problem
Answers
Answer:
The one you chose is correct.
Step-by-step explanation:
The scale factor is 1.5. We know QP is 5. QP is the corresponding side to KJ. So, we multiply 5 with 1.5 and get 7.5 which is equal to 15/2
I found the scale factor by dividing LK which was 9 and QR which was 6.
9/6=1.5.
Hope this helps
What is the slope of the line that passes through the pair of points?
(5, 8), (9,-3)
Answers
Answer:
-11/4
Step-by-step explanation:
slope = [tex]\frac{-3-8}{9-5} =\frac{-11}{4}[/tex]
Hope this helps
Answer:
-11/4
Step-by-step explanation:
slope = (difference in y)/(difference in x)
slope = (-3 - 8)/(9 - 5) = -11/4
A shop has 4 fruits (bananas, oranges, apples, and grapes) and 2 dips (peanut butter and caramel).
Zeke's mom randomly buys 1 fruit and 1 dip for his lunch.
What is the probability that she selects his favorite option: bananas and peanut butter?
Answers
Answer:
total fruits and dips=4+2=6
P(select banana&peanut butter)=banana+peanut /total fruits and d
Step-by-step explanation:
total fruits and dips=4+2=6
P(select banana&peanut)=banana+peanut/total fruits and dips
P(select banana&peanut)=1+1=2/6
=1/3
What is the name of the numeral system we use today?.
Answers
Simplify.
√50
OA. 10√5
OB. 2√5
OC. 5√2
OD. 25√2
Answers
Decomposition in factors of 50: 2 · 5².
[tex]\mathrm{Apply\:the\:laws\:of\:exponents}:\quad \sqrt{ab}=\sqrt{a}\sqrt{b},\:\quad \:a\ge 0 ,\:b\ge 0[/tex]
[tex]\sqrt{2\cdot \:5^2}=\sqrt{2}\sqrt{5^2}[/tex]
[tex]\mathrm{Apply\:the\:laws\:of\:exponents}:\quad \sqrt{a^2}=a,\:\quad \:a\ge 0[/tex]
[tex]\sqrt{5^2}=5[/tex]
[tex]=\sqrt{2}\cdot \:5[/tex]
[tex]\bf{=5\sqrt{2} \ === > \ Answer }[/tex]
Option C
↓
[tex]\boxed{\boxed{\large\displaystyle\text{$\begin{gathered}\sf \bf{MyHeritage} \end{gathered}$}}}[/tex]
Add the polynomials.
(3x^2+5x−3)+(4x^2+2x)
7x^2+7x−3
7x^2+5x+5
3x^2+9x−1
x^2+5x−3
Answers
Area of the shaded segment=
Answers
The area of the entire sector is [tex](\pi)(6^{2}) \left(\frac{60}{360} \right)=6\pi[/tex]
The area of the triangle OAB is [tex]\frac{\sqrt{3}}{4}(6^{2})=9\sqrt{3}[/tex].
So, the area of the segment is [tex]\boxed{6\pi-9\sqrt{3}}[/tex]
How to find a fraction from a decimal
Answers
Answer:
0.6= 6/10
[tex]0.6 = 6 \div 10[/tex]
Step-by-step explanation:
To convert a decimal to a fraction
place the decimal number over its place value
For example
in 0.8 the 8is in the tenths place,
so we place 6 over 10 to create the equivalent fraction, 8/10.
example 2
=6.25
625/100
25/4
how many significant figures are there in the number 11
Answers
Answer: the maximum of three
Step-by-step explanation: each number has 2 significant figures. so 10.85 has four significant figures and must be rounded to 11 so it can have a maximum of three
15.23*3.6=54.83
Write an equation for the line through the point (1,3) parallel to
the line y =
1/2x + 10.
Answers
y=3
Answer:
y=1/2x+5/2
Step-by-step explanation:
The slope will not change if the line is parallel.
So start with 1/2x and plug in 1.
You will get 1/2 and you need to get a total of 6/2 to get a y of 3.
since you already have 1/2 the remanding value needed is 5/2 and once you add the two you will get 6/2 which equals a y of 3.
The diagram shows a rectangle. The area of the rectangle is 310 m². Work out the value of w when 5x-9 is the length and 3x+7 is the another length
Answers
Answer:
w = 10
Step-by-step explanation:
First we take the two values for the breadth of the rectangle
So:
5x - 9 = 3x + 7
now we solve this equation as follows:
5x - 3x = 9 + 7
2x = 16
x = 8
now that we have found the value for x, we can substitute it in the equation, 5x - 9,or in the equation, 3x + 7.
when we substitute x in any of these equations, we get
5(8) - 9 = 31
3(8) + 7 = 31
now that we have the value for the breadth we can form the following equation:
31 × w = 310
31w = 310
w = 310/31 = 10
Answer:
10
Step-by-step explanation:
Firstly, we find the value of X
since 5X-9 and 3X+7 are lenghts
5X-9= 3X+7
5X-3X= 7+9
2X = 16
dividing bothsides by 2
2X/2= 16/2
X = 8
Hence,
5X-9= 5(8)-9= 40-9 = 31
3X+7=3(8)+7= 24+7= 31
To find the width
Area= lenght×width
310= 31×w
31w= 310
w= 310/31
W= 10
the simplification lf (x^3+2x-3)-(x^2-2x+4)
Answers
Answer:
x³ - x² + 4x - 7
Step-by-step explanation:
(x³ + 2x - 3) - (x² - 2x + 4)
remove first parenthesis and distribute second parenthesis by - 1
= x³ + 2x - 3 - x² + 2x - 4 ← collect like terms
= x³ - x² + 4x - 7 ← in standard form
A map has the scale 1 : 200,000 and the distance between two villages on the map is 8 cm. What is the distance in metres between the towns in real life?
Answers
The distance in metres between the towns in real life is 1600000 meters
How to determine the distance in real life?The scale ratio is given as:
Scale = 1 cm : 200000 m
The map distance of 8cm implies that:
8 cm : Real = 1 cm: 200000 m
Express as fraction
Real/8 = 200000 m/1
Multiply both sides by 8
Real = 1600000 m
Hence, the distance in metres between the towns in real life is 1600000 meters
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help im being timed plssss
Answers
The value of cos 45 in a right angle triangle is 1/√2.
What is the trigonometry ratio?The trigonometry ratios are ratios that can be used to determine the side and angles of a triangle depending on the known parameter. The basic trigonometry ratios are denoted by the acronym: SOH CAH TOA
i.e.
Sin θ = opp/hypCos θ = adj/hypTan θ = opp/adjFrom the given diagram, we are to find the Cos 45°
i.e.
θ = 45°
Let's recall the angle 45 45 90 theorem that states that right angles have the following values for the sides:
hypotenuse = √2opposite = 1adjacent = 1Since
Cos θ = adj/hyp
Cos 45 = 1/√2
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