a) (1 + i)^101 simplifies to i times 2^(101/2).
b) Log(e^(i5π)) simplifies to 2iπ.
a) To calculate (1 + i)^101, we can use De Moivre's theorem, which states that for any complex number z = r(cosθ + isinθ), the nth power of z is given by z^n = r^n(cos(nθ) + isin(nθ)).
In this case, we have (1 + i) = √2(cos(π/4) + isin(π/4)).
Applying De Moivre's theorem, we raise √2 to the 101st power and multiply the angle by 101:
(1 + i)^101 = (√2)^101 * (cos(101π/4) + isin(101π/4))
Simplifying, we have:
(1 + i)^101 = 2^(101/2) * (cos(25π/2) + isin(25π/2))
We get:
(1 + i)^101 = 2^(101/2) * (0 + i)
Therefore, (1 + i)^101 simplifies to i times 2^(101/2).
b) To calculate Log(e^(i5π)), where Log is the principal logarithm, we need to apply the properties of logarithms and exponentials.
Using Euler's formula, e^(ix) = cos(x) + isin(x), we have e^(i5π) = cos(5π) + isin(5π) = -1 + 0i = -1.
Applying the principal logarithm, Log(e^(i5π)) = Log(-1).
Since -1 is a complex number, we can express it in polar form as -1 = e^(iπ + iπ). Therefore, Log(-1) = iπ + iπ = 2iπ.
Hence, Log(e^(i5π)) simplifies to 2iπ.
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which properties are being used to solve the question
Answer:
Both commutative and associative
If the answer isn’t an integer express in simplest radical form
Answer:
[tex]x = \frac{20}{ \sqrt{3} } \\ y = \frac{40}{ \sqrt{3} } [/tex]
Step-by-step explanation:
[tex]for \: \: x \\ \tan(60) = \frac{20}{x} \\ \sqrt{3} = \frac{20}{x} \\ x = \frac{20}{ \sqrt{3} } \\ \\ for \: y \: \\ \sin(60) = \frac{20}{y} \\ \frac{ \sqrt{3} }{2} = \frac{20}{y} \\ \sqrt{3} \: y = 20 \times 2 \\ y = \frac{40}{ \sqrt{3} } [/tex]
I hope that is useful for you :)
What is vertical asymptote for the given equation
f(x) = log(x - 5) + 2
Need help with this question thank you!
Answer:
Y= 3/5-4
Step-by-step explanation:
What is slope-intercept form? Slope intercept form is an equation for graphing. It is y=mx+b. The "m" stands for the slope, or how much coordinates it takes to move to the next point. The "b" stands for the y-intercept, or where the line is perfectly on the y-line.
Step One: The first part of the equation is the slope, so let's figure that out. Since this one a graph, it is easy to tell the slope. The points are nicely marked, so we don't have to guess which points are visible. We are going up 3 jumps and then across 5 jumps. The slope is always "rise/run" where the "rise" is the y and "run" is x. So, the slope is 3/5x.
Step Two: For the last part, we need to know the y-intercept. It is clearly shown on the graph: -4. Because it is negative, we remove the plus sign because a plus with a negative is subtraction.
Step Three: We can combine both parts and that is the answer: y=3/5x-4.
A person places $4630 in an investment account earning an annual rate of 5.7%,
compounded continuously. Using the formula V = Pert, where Vis the value of the
account in t years, P is the principal initially invested, e is the base of a natural
logarithm, and r is the rate of interest, determine the amount of money, to the nearest
cent, in the account after 9 years.
Answer:
75.935
Step-by-step explanation:
If V = Pe^rt and we are given P (the principle investment) =4630 ,
a rate of 5.7 % which as a decimal = 0.057 (% divided by 100 = decimal)
we can substitute this information in and get V=4630 e^(.057t).
then it says what will it be in 9 years? so we put 9 in for "t" and get
v = 4630 e^(0.057 * 9) simplify using a calculator and we will get our answer.
what is the volume of the cylinder?
5 yd
4 yd
cubic yards.
Volume
Answer:
volume=1yd x 1yd = q cubic yard
Which expression is equivalent to (-3)x+(3)(-12)?
Answer:
Step-by-step explanation:
(-3)x + (3)(-12)
= -3x -36
or -3(x + 12)
Answer:
Step-by-step explanation:
(-3)x + 3(-12) = (-3) (x + 12)
(-3) is both the two terms. So we can take (-3) as common
Which of the following assumptions is not true for multiple linear regression?
Group of answer choices
o The independent variables are not correlated.
o The residuals are normally distributed.
o The relationship between dependent and independent variables is linear.
o There will be a multi-collinearity effect.
The assumption which is not true for multiple linear regression is that there will be a multi-collinearity effect. What is multiple linear regression? Multiple linear regression is a regression method where the response variable is dependent on more than one independent variable. Multiple linear regression is an extension of linear regression, where one response variable is dependent on one independent variable. It helps to determine the nature of the relationship between dependent variables and independent variables in the form of linear equations. It is used to find out the relationship between variables and make predictions based on that relationship. Assumptions of multiple linear regression1. Linearity, It is assumed that the relationship between the independent variables and dependent variables is linear. This implies that as the value of the independent variable changes, the change in the dependent variable is constant.2. Independence, There should be independence among the observations. This implies that there should not be any relationship between the residuals of the observations.3. Homoscedasticity implies that the variance of residuals should remain constant across all levels of the independent variable.4. Multicollinearity, It is the condition where the independent variables are highly correlated with each other. If there is perfect collinearity, it will not be possible to obtain the estimates of regression coefficients.5. Normality, The residuals should follow a normal distribution with mean zero and constant variance. The residuals should be normally distributed and centered on zero. The assumptions that are true for multiple linear regression are: The independent variables are not correlated. The residuals are normally distributed. The relationship between dependent and independent variables is linear. The assumption which is not true for multiple linear regression is that there will be a multi-collinearity effect. Hence, the correct option is D) There will be a multi-collinearity effect.
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2. Find mzR and mzS.
89°
R
157 T
Р
Answer:
m∠R = 123°
m∠S = 91°
Step-by-step explanation:
The measure of an inscribed angle is 1/2 the measure of the intercepted arc
m of arc QRS = 2 m∠P = 2(57) = 114
Now m of arc SPQ = 360 - 114 = 246
So, m∠R = 1/2(246) = 123
m of arc RSP = 2 m∠Q = 2(89) = 178
m of arc PQR = 360 - 178 = 182
m∠S = 1/2(182) = 91
I need help with this , 7x-3y+4x-5y+8x-6y
Answer:
19x−14y
Steps:
7x-3y+4x-5y+8x-6y
=19x-3y-5y-6y
= 19x−14y
plz mark as brainliest
Answer:
19x-14y
Step-by-step explanation:
rearrange
7x+4x+8x-3y-5y-6y
solve the coefficient and add the variables
help me please asap subtract with renaming
Answer:
1. 1/4
2. 43/10 or 4 3/10 either are correct.
3. 67/12 or 5 7/12 either are correct.
Step-by-step explanation:
I hope this helped.
Find the nth Maclaurin polynomial for the function.
f(x) = (4x − 5)/(x + 1), n = 4
The nth Maclaurin polynomial for the function f(x) = -5 - x + 4x² - 5x³ + 4x⁴ + ...
The nth Maclaurin polynomial for the function f(x) = (4x − 5)/(x + 1), we need to expand the function as a power series centered at x = 0 (the Maclaurin series) and then truncate it to the desired degree.
First, let's find the Maclaurin series expansion of f(x):
f(x) = (4x − 5)/(x + 1)
To expand this as a power series, we'll use the concept of geometric series expansion. We'll start by expressing f(x) as a product of two functions:
f(x) = (4x − 5)×(x + 1)⁻¹
Now, let's find the power series expansion for each factor separately.
1.The power series expansion for (4x - 5) centered at x = 0 is given by:
(4x - 5) = -5 + 4x
2.The power series expansion for (x + 1)⁻¹ centered at x = 0 is given by the geometric series formula:
(x + 1)⁻¹ = 1 - x + x² - x³ + x⁴ - ...
Note that this expansion is valid when |x| < 1 since it's based on a geometric series.
Now, we'll multiply these two expansions together to get the expansion for f(x). We'll keep terms up to the fourth degree (n = 4).
f(x) = (4x - 5) × (x + 1)⁻¹ = (-5 + 4x) × (1 - x + x² - x³ + x⁴)
Expanding this expression:
f(x) = -5 + 4x - 5x + 4x² - 5x³ + 4x⁴ + ...
Simplifying the terms:
f(x) = -5 - x + 4x² - 5x³ + 4x⁴ + ...
This is the Maclaurin series expansion for f(x). The nth Maclaurin polynomial, we'll truncate the series at the fourth degree (n = 4):
f(x) ≈ -5 - x + 4x² - 5x³ + 4x⁴
Therefore, the fourth Maclaurin polynomial for f(x) is given by:
P₄(x) = -5 - x + 4x² - 5x³ + 4x⁴
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Find the value of x and y.
Answer:
Step-by-step explanation:
Factor the expression completely. 12X+18
Answer:
6(2x+3)
Step-by-step explanation:
There's really no way to explain this. Take the number that all of the numbers in the equation can be divisible by and then single it out and then yeah.
Hope this helped! :)
how do you solve 1/2(6)(h+1)=18
Multiply 6 by 1/2 to get 3
Ow you have :
3( h + 1) = 18
Distribute the 3:
3h + 3 = 18
Subtract 3 from both sides:
3h = 15
Divide both sides by 3:
H = 5
Jericho has m paintings and n drawings. He sold his paintings for $x each and drawings for $y each. His earnings for one day are
represented by the expression given below.
x(m) + y(m- 12)
Which statement best describes the term (m - 12)?
A. It is the number of paintings sold, which was 12 more than the number of drawings sold.
B. It is the number of drawings sold, which was 12 more than the number of paintings sold.
C. It is the number of paintings sold, which was 12 less than the number of drawings sold.
D. It is the number of drawings sold, which was 12 less than the number of paintings sold.
Answer:
b
Step-by-step explanation:
What is the radius and diameter of the following circle?
Answer:
Diameter= 10.2 - Radius=5.1
Step-by-step explanation:
5.1 x 2= 10.2
10.2/2=5.1
Certain standardized math exams have a mean of 100 and a standard deviation of 60. Of a sample of 36 students who take this exam, what percent could you expect to score between 80 and 90?
16
15.85
13.5
12.5
Approximately 12.5% percentage of the students could be expected to score between 80 and 90.
To determine the percentage of students who could be expected to score between 80 and 90, we need to calculate the z-scores for these scores and use the standard normal distribution.
First, we need to calculate the z-scores for 80 and 90 using the formula:
z = (x - μ) / σ
where x is the individual score, μ is the mean, and σ is the standard deviation. Plugging in the values of μ = 100, σ = 60, and x = 80 or x = 90, we can calculate the z-scores.
For x = 80:
z = (80 - 100) / 60 = -0.3333
For x = 90:
z = (90 - 100) / 60 = -0.1667
Next, we need to find the area under the standard normal distribution curve between these two z-scores. This represents the percentage of students who scored between 80 and 90.
Using a standard normal distribution table or calculator, we can find that the area between -0.3333 and -0.1667 is approximately 0.1210 or 12.1%.
Therefore, approximately 12.5% of the students could be expected to score between 80 and 90.
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Practice Problems
1)
Chase is comparing music subscription prices. A 12-month subscription with All Ears costs
$91.80 and a 3-month subscription with Press Play costs $19.35. Which service costs less per
month?
Step-by-step explanation:
[tex]91.8 \div 12 = 7.65 \\ 19.35 \div 3 = 6.45[/tex]
3 months
what is the equation of the line that has a slope of -3 and a y-intercept of -1
Answer:
y=4x+2
Step-by-step explanation:
Tell whether the angles are complementary or supplementary. Then find the value of X.
Answer:
Complementary
x= 60
Step-by-step explanation:
Answer:
The angles are complementary.
X = 60
Step-by-step explanation:
which shows a difference of squares?
a.10 y squared minus 4 x squared
b.16 y squared minus x squared
c.8 x squared minus 40 x + 25
d.64 x squared minus 48 x + 9
The expression that shows a difference of squares is option B, i.e., 16y² − x². We can represent it as (4y - x)(4y + x).Therefore, the correct option is B.
Difference of Squares is a squared quantity that can be represented as the difference of the squares of two variables or numbers. In simple terms, when two squares are subtracted, a difference of squares is produced. The squares of the two quantities, x and y, are subtracted from each other in the Difference of Squares.Here, the expression that shows a difference of squares is option B, i.e., 16y² − x². We can represent it as (4y - x)(4y + x).Therefore, the correct option is B. In this expression, we have the square of two quantities subtracted, which results in the Difference of Squares.Checking the other options, we can say that option A (10y² − 4x²) cannot be considered a difference of squares because there is no squared quantity subtracted from it. Similarly, option C (8x² − 40x + 25) and option D (64x² − 48x + 9) do not show the Difference of Squares as they do not have two squared quantities subtracted from each other.
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ABM Services paid a $4.15 annual dividend on a day it closed at a price of $54 per share. What
was the yield?
Answer:
- 22719559.
Step-by-step explanation:
Complete the two-way frequency table below, which shows the relationship between adults' gender and whether these adults buy a truck or a car for their first new vehicle. From a sample of 94 adults, the following data are collected:
Truck Car Total
Female 9 36
Male 18 31
Total 94
What is the probability (rounded to the nearest whole percent) that an adult will buy a car, given that he is a male? Are the events being male and buying a car independent? (1 point)
63%; they are dependent
46%; they are dependent
63%; they are independent
33%; they are independent
I will come back to this question when it has been further clarified.
(Honestly I just really need to delete this answer since I misread the question but it doesn't give me an option to delete it).
a business woman bought a personal computer for $108000.calculate her selling price on the personal computer if she wants to make profit of 25%
Answer:
$135,000
Step-by-step explanation:
108000*1.25
To join a local square dancing group, Jan has to pay a $100 sign-up fee plus $25 per month. Write an equation for the cost (y) based on the number of months.
a. y = 25x + 100
b. y = 100x + 25
c. y = 25 + 100x
d. y = 100 + 25x
The correct answer is option A which is y = 25x + 100.
Given the following:
To join a local square dancing group, Jan has to pay a $100 sign-up fee plus $25 per month
We need to write an equation for the cost (y) based on the number of months.
To solve the above problem, the answer is;a. y = 25x + 100
Explanation; Let's break down the problem
The $100 sign-up fee is a fixed cost that is added only once to the monthly fee which is $25. Thus the equation for the cost (y) based on the number of months can be expressed as; y = 25x + 100 where:y is the cost for the number of monthsx is the number of months
Therefore the correct answer is option A which is;y = 25x + 100.
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To join a local square dancing group,
Jan has to pay a $100 sign-up fee plus $25 per month.
The equation for the cost (y) based on the number of months is
y = 25x + 100,
where x is the number of months.
Option A is the correct equation for the cost based on the number of months.
Writing the equation:
y = 25x + 100
Where:
y = Cost based on the number of months
x = Number of months
Therefore, when Jan has been part of the local square dancing group for 1 month, the total cost will be:
$25 * 1 + $100 = $125
And if Jan has been a part of the group for 4 months, then the total cost would be:
$25 * 4 + $100 = $200
Therefore, option A is the correct answer.
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490 students and 10 teacher will ride the bus to go to the zoo if each bus can seat so passengers how many buses will be needed many zoo trip
Answer:
If you add teacher plus students which is 490+10=500
you get 500 people in total after that you divide 500 by 50 and you get 10 so you will need 10 busses in total hope this helped!
Step-by-step explanation:
You call 5 friends, tomorrow they call 5 each. Write an equation.
Answer:
5+5=10
Step-by-step explanation:
5+5=10
hope this helps. plz mark me as brainliest.
Can someone please help me with math
Answer:
A
Step-by-step explanation:
Hello There!
This is the formula for volume of a cylinder
[tex]V=\pi r^2h[/tex]
where r = radius and h = height
we are given the height and diameter height - 10 ft diameter - 22 ft
we are not give the radius directly but we can easily convert the diameter to the radius by dividing by 2
22/2=11 so the radius is 11
now having found the radius and height we plug in the values into the formula (note: it says leave answer in terms of pi so we will not apply pi to the final answer.)
[tex]V=\pi 11^210\\11^2=121\\121*10=1210\\V=1210\pi[/tex]
so we can conclude that the answer is A
A swimming pool measures 25.8 meters by 11.9 meters. It is filled with water to a depth of 1.1 meters. The pool is emptied by a tube that emits 29.8 liters per second. Estimate the approximate time required to empty the pool
Answer:
11 Seconds
Step-by-step explanation:
Swimming pool volume:
25.8 × 11.9 × 1.1 = 337.722
Time to empty pool:
337.722 ÷ 29.8 = 11.3329530201 ≈ 11 seconds