letter d is 7 someone help me please!!!!

If the determinant of a matrix A is zero, then A is singular, which means that A is not invertible.

This is because the determinant of a matrix represents the scaling factor of the transformation that the matrix represents. If the determinant is zero, it means that the transformation does not preserve the orientation of space and therefore does not have an inverse transformation.

In the case of A^3, the determinant of A^3 is equal to the cube of the determinant of A. Therefore, if det(A^3) = 0, then det(A)^3 = 0, which implies that det(A) = 0. Hence, A is singular and cannot be invertible.

Geometrically, this means that the transformation represented by A^3 collapses the space onto a lower-dimensional subspace, such as a line or a plane, and does not have an inverse that can restore the original space. Therefore, the linear system represented by A^3 is dependent, and the columns of A^3 do not span the full space.

In summary, if det(A^3) = 0, then A is not invertible because the transformation represented by A^3 collapses the space onto a lower-dimensional subspace and does not have an inverse transformation that can restore the original space.

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The statement says that a manager at Claire's makes $500 a week give or take $100 and a doctor at New York Presbyterian makes $5,000 a week give or take $100.

We want to find out which person would have had a more significant change to his or her salary if $100 was taken away from each of them relatively.

We will assume that the $100 given or take on the salaries are standard deviations. We will use the formula:

Coefficient of variation = (standard deviation / mean) x 100

Coefficient of variation of the manager's salary = (100 / 500) x 100 = 20%

Coefficient of variation of the doctor's salary = (100 / 5000) x 100 = 2%

Since the coefficient of variation is higher for the manager's salary than for the doctor's salary, it means that the $100 taken away from the manager will be more significant than the $100 taken away from the doctor.

The manager's salary varies more as a percentage of the mean salary than the doctor's salary.

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The best function that would model the progression of the points in the scatter plot is an exponential function (option D).

To determine which function best models the progression of the points in the scatter plot, we can analyze the pattern of the data. Let's examine the options:

A. A linear function describes a straight line. Looking at the scatter plot, we can see that the points do not form a straight line, so a linear function is not the best choice.

B. A quadratic function represents a curve that opens upwards or downwards. The scatter plot does not exhibit a clear quadratic pattern, so a quadratic function is unlikely to be the best choice.

C. A square root function represents a curve that increases at a decreasing rate. There is no clear indication of a square root pattern in the scatter plot, so a square root function may not be the best choice.

D. An exponential function represents a curve that increases or decreases at an increasing rate. When examining the scatter plot, we can observe that the points show a clear trend of exponential growth. As the x-values increase, the corresponding y-values grow at an increasing rate. Therefore, an exponential function is likely the best choice to model the progression of the points.

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The solution to the first-order differential equation y' = 2y^3/x^2 is y = ±√(x/(4 - 2C1x)), where C1 is the constant of integration.

To solve the first-order differential equation y' = 2y^3/x^2, we can separate the variables and integrate both sides.

Start by rearranging the equation to isolate the variables:

dy/y^3 = 2/x^2 dx

Now, we can integrate both sides:

∫(dy/y^3) = ∫(2/x^2) dx

Integrating the left side:

∫(dy/y^3) = ∫2/x^2 dx

-1/(2y^2) = -2/x + C1

Multiplying both sides by -1/2:

1/(2y^2) = 2/x - C1

To simplify, we can take the reciprocal of both sides:

2y^2 = 1/(2/x - C1)

2y^2 = x/(4 - 2C1x)

Now, solve for y:

y^2 = x/(4 - 2C1x)

y = ±√(x/(4 - 2C1x))

So, the solution to the first-order differential equation y' = 2y^3/x^2 is y = ±√(x/(4 - 2C1x)), where C1 is the constant of integration.

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Answer:

--------------------

After using 2 9/12 yards she has:

To subtract the mixed numbers, first subtract the whole numbers:

Then, subtract the fractions:

Finally, combine the whole number and fraction:

The relative maximum of the function is at the point (1, 4) on the grid.

To determine the relative maximum of the given parabola, we need to examine its shape and position on the grid.

The parabola is described as opening downward, which means it has a concave shape and its vertex represents the highest point on the graph.

The vertex of the parabola is given as (1, 4), which means the highest point of the parabola occurs at x = 1 and y = 4. In other words, the parabola reaches its maximum value of 4 when x equals 1.

Since the vertex is the highest point of the parabola and no other point on the graph is higher, we can conclude that the relative maximum of the function is at the point (1, 4) on the grid.

This means that for any other point on the graph, the y-coordinate value will be lower than 4. The parabola opens downward from the vertex, and as we move away from the vertex along the x-axis in either direction, the y-values of the points on the parabola decrease. Therefore, the relative maximum occurs only at the vertex.

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To find the number of terms needed to calculate the sum of the series with a desired accuracy, we need to determine the smallest integer value of n for which the absolute error of the partial sum is less than 0.0003.

The series given is \sum_{n=1}^\infty (-1)^{n-1}\frac{9}{n^4}. To find the sum to a desired accuracy, we can calculate the partial sums of the series and check the absolute error.

Let's denote the partial sum of the series with n terms as S_n. To find the absolute error, we need to calculate the difference between the actual sum (which is unknown since the series is infinite) and S_n.

We continue calculating S_n by adding more terms until the absolute error becomes smaller than 0.0003. This means we need to find the smallest value of n for which |actual sum - S_n| < 0.0003.

By incrementally increasing the value of n, we compute the partial sums S_n and check the absolute error. Once we reach a value of n that satisfies |actual sum - S_n| < 0.0003, we have found the number of terms needed to achieve the desired accuracy.

Note that since the series converges (alternating series with decreasing terms), the partial sums will approach the actual sum as n increases. Thus, by adding more terms, we can improve the accuracy of the approximation.

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By integrating the joint PDF over the specified region.

To compute the probability P(X ≤ 1.25, Y ≥ 0.75) using the given joint PDF, we need to integrate the joint PDF over the specified region.

The region of interest is defined as 1 ≤ x ≤ 1.25 and 0.75 ≤ y ≤ 10.

The joint PDF, FXY(x, y), is given by FXY(x, y) = (x - cy) for 1 ≤ x ≤ 2 and 0 ≤ y ≤ 10, and 0 otherwise.

To compute the probability, we integrate the joint PDF over the specified region:

P(X ≤ 1.25, Y ≥ 0.75) = ∫∫[FXY(x, y)]dydx

Breaking down the integral into two parts:

∫[∫[FXY(x, y)]dy]dx

First, integrate the inner integral with respect to y:

∫[FXY(x, y)]dy = ∫[(x - cy)]dy = xy - (cy[tex]^2[/tex])/2

Next, integrate the outer integral with respect to x over the given range:

∫[xy - (cy[tex]^2[/tex])/2]dx = ∫[(xy - (cy)[tex]^2[/tex]/2)]dx

Evaluate the integral over the range 1 ≤ x ≤ 1.25:

= [∫[(xy - (cy[tex]^2[/tex])/2)]dx] evaluated from 1 to 1.25

Substitute the limits of integration into the expression:

= [(1.25y - (cy[tex]^2[/tex])/2) - (y - (cy[tex]^2[/tex])/2)]

Simplifying the expression:

= 1.25y - (cy[tex]^2[/tex])/2 - y + (cy[tex]^2[/tex])/2

= 0.25y

Finally, evaluate the integral with respect to y over the range 0.75 ≤ y ≤ 10:

∫[0.25y]dy = (0.25/2)y[tex]^2[/tex] evaluated from 0.75 to 10

Substitute the limits of integration into the expression:

= (0.25/2)(10^2 - (0.75)^2)

= (0.25/2)(100 - 0.5625)

= (0.25/2)(99.4375)

= 12.4296875

Therefore, P(X ≤ 1.25, Y ≥ 0.75) is approximately equal to 12.4296875.

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The matrix A will be singular when its determinant is equal to zero. To determine the value(s) of λ for which A is singular, we need to calculate the determinant of A and find the values of λ that make the determinant zero.

The determinant of a matrix can be found by applying the rule of expansion along a row or column. In this case, we can use the first column to calculate the determinant:

det(A) = 1 * (1 * 4 - λ * 0) - 0 - (5 * (1 * 0 - λ * λ))

= 1 * (4 - 0) - 0 - (5 * (0 - λ^2))

= 4 - 5λ^2.

To make the determinant equal to zero, we solve the equation 4 - 5λ^2 = 0. Rearranging the equation, we have 5λ^2 = 4. Dividing both sides by 5, we get λ^2 = 4/5.

Taking the square root of both sides, we find λ = ±(2√5)/5. Therefore, the value(s) of λ for which the matrix A will be singular are λ = ±(2√5)/5.

In conclusion, the answer is f. none of the above, as none of the given options match the correct value(s) of λ for which the matrix A is singular

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Thus, the original series converges by the Integral Test.

To determine if the series converges or diverges using the Integral Test, we will evaluate the corresponding improper integral:
∫(1 to infinity) xe^(-3x) dx

To solve this integral, we use integration by parts, where u = x and dv = e^(-3x) dx. Then, du = dx and v = -1/3 e^(-3x).
Using the integration by parts formula, we get:
∫(1 to infinity) xe^(-3x) dx = -1/3 x e^(-3x) | (1 to infinity) - ∫(1 to infinity) (-1/3 e^(-3x) dx)

Now we evaluate the remaining integral:
∫(1 to infinity) (-1/3 e^(-3x) dx) = (-1/3) ∫(1 to infinity) e^(-3x) dx = (-1/9) [e^(-3x)] (1 to infinity)

Evaluating the limits, we have:
-1/3 [(-1/3)e^(-3)(infinity) - (-1/3)e^(-3)(1)] - (-1/9)[0 - e^(-3)]

Which simplifies to:
(-1/3)(-1/3)e^(-3) - (-1/9)e^(-3) = (1/9)e^(-3)

Since the integral converges, the original series also converges by the Integral Test.

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A line segment with an endpoint at (1, 6) and a length of 1.2 units can be drawn by extending the segment from the endpoint in the positive x-direction.

To draw the line segment, we start with the endpoint at (1, 6) on the coordinate plane. We then extend the segment from this point by moving 1.2 units in the positive x-direction.

To determine the endpoint of the line segment, we add the length of the segment, which is 1.2 units, to the x-coordinate of the starting point. Since the starting point has an x-coordinate of 1, adding 1.2 units results in an x-coordinate of 2.2. Therefore, the endpoint of the line segment is (2.2, 6), as the y-coordinate remains the same.

Using a ruler or a straightedge, we can now draw a line connecting the starting point (1, 6) to the endpoint (2.2, 6). This line segment will have a length of 1.2 units and will extend in the positive x-direction from the starting point.

Overall, by extending the line segment 1.2 units from the starting point (1, 6) in the positive x-direction, we can accurately represent a line segment with the desired endpoint and length.

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The amount of fertilizer required for a 200 square-foot garden is 5 pounds.

According to the given data, the directions say to use 4 pounds of fertilizer for 160 square feet of soil. Then, for 1 square foot of soil, Omar needs 4/160 = 0.025 pounds of fertilizer.So, to find the amount of fertilizer needed for 200 square feet of soil, we will multiply the amount of fertilizer for 1 square foot of soil with the area of Omar's garden.i.e., 0.025 × 200 = 5 pounds of fertilizer.
So, Omar needs 5 pounds of fertilizer for a 200 square-foot garden.

Therefore, the amount of fertilizer required for a 200 square-foot garden is 5 pounds.

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The effective annual growth rate of the quantity with a half-life of 14 years is 4.9%.

To find the effective annual growth rate of a nominal exponential growth/decay, we can use the formula:

Effective annual growth rate = (1 + r)^n - 1

where r is the nominal annual growth rate (expressed as a decimal) and n is the number of compounding periods per year.

In this case, the quantity has a half-life of 14 years, which means that it decreases by a factor of 2 every 14 years. We can use the formula for exponential decay:

N(t) = N0 * e^(-kt)

where N0 is the initial quantity, t is the time elapsed, and k is the decay constant. Since the half-life is 14 years, we know that:

1/2 = e^(-k*14)

Taking the natural logarithm of both sides, we get:

ln(1/2) = -k*14

Solving for k, we get:

k = ln(2)/14

Now we can use the formula for the nominal annual growth rate:

r = e^(k) - 1

Substituting the value of k, we get:

r = e^(ln(2)/14) - 1

r = 0.0489

This means that the quantity is decreasing at a nominal annual growth rate of 4.89%. To find the effective annual growth rate, we need to know how often the quantity is being compounded. If we assume that it is compounded once a year (i.e. annual compounding), then the effective annual growth rate is:

Effective annual growth rate = (1 + 0.0489)^1 - 1

Effective annual growth rate = 0.0489 or 4.9% (rounded to one decimal place)

Therefore, the effective annual growth rate of the quantity with a half-life of 14 years is 4.9%.

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Option C Ill only is the right option

Let's analyze each option one by one:

I. arccos(sins):

The range of the arcsine function is [-π/2, π/2], and the range of the cosine function is [0, π].

Since the arcsine of any value is always between -π/2 and π/2, it is not possible to have a value outside that range. Therefore, arccos(sins) is not a valid expression.

II. ectan(cot(-7)):

The tangent function has a period of π, which means tan(x) = tan(x + π) for any value of x.

Therefore, tan(-7) is the same as tan(-7 + π) = tan(-7 + 3.14...) = tan(-3.14...), which is defined and equal to 0.

Since the cotangent is the reciprocal of the tangent, cot(-7) = 1/tan(-7) = 1/0, which is undefined. Thus, ectan(cot(-7)) is not a valid expression.

III. arcsin(csc):

The cosecant function (csc) is the reciprocal of the sine function, so csc(x) = 1/sin(x).

The arcsine function (arcsin) is the inverse of the sine function, so arcsin(sin(x)) = x for any x within the range of the arcsin function.

Therefore, arcsin(csc) simplifies to arcsin(1/sin), and since these functions are inverses of each other, arcsin(1/sin) = x.

The value of x depends on the specific value of sin, which is not provided.

Therefore, arcsin(csc) is a valid expression, but we cannot determine a specific value without knowing the value of sin.

Therefore, the correct answer is C. III only.

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Hence, there were 149 children and 230 adults who swam at the public pool that day.

Let the number of children who swam at the public pool that day be 'c' and the number of adults who swam at the public pool that day be 'a'.

Given that the total number of people who swam that day is 379.

Therefore,

c + a = 379   ........(1)

Now, let's calculate the total revenue for the day.

The cost for a child is $1.50 and for an adult is $2.25.

Therefore, the revenue generated by children = $1.5c and the revenue generated by adults = $2.25

a. Total revenue will be the sum of revenue generated by children and the revenue generated by adults. Hence, the equation is given as:$1.5c + $2.25a = $741.0  ........(2)

Now, let's solve the above two equations to find the values of 'c' and 'a'.

Multiplying equation (1) by 1.5 on both sides, we get:

1.5c + 1.5a = 568.5

Multiplying equation (2) by 2 on both sides, we get:

3c + 4.5a = 1482

Subtracting equation (1) from equation (2), we get:

3c + 4.5a - (1.5c + 1.5a) = 1482 - 568.5  

=>  1.5c + 3a = 913.5

Now, solving the above two equations, we get:

1.5c + 1.5a = 568.5  

=>  c + a = 379  

=>  a = 379 - c'

Substituting the value of 'a' in equation (3), we get:

1.5c + 3(379-c) = 913.5  

=>  1.5c + 1137 - 3c = 913.5  

=>  -1.5c = -223.5  

=>  c = 149

Therefore, the number of children who swam at the public pool that day is 149 and the number of adults who swam at the public pool that day is a = 379 - c = 379 - 149 = 230.

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The 9th term of the sequence 128, 32, 8, 2, 1/2 is 0.003.

To find the 9th term of the sequence, we need to determine the pattern followed by the sequence. We can see that each term is one-fourth of the previous term. Using this pattern, we can write the general formula for the nth term of the sequence as: a_n = 128*(1/4)^(n-1)

Now we can substitute n = 9 in the formula and simplify to get the 9th term as: a_9 = 128*(1/4)^8 ≈ 0.003

A geometric progression, sometimes referred to as a geometric sequence in mathematics, is a series of non-zero numbers where each term following the first is obtained by multiplying the preceding one by a constant, non-zero value known as the common ratio. For instance, the geometric progression 2, 6, 18, 54, etc. has a common ratio of 3. Similar to that, the geometric series 10, 5, 2.5, 1.25,... has a common ratio of 1/2.

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The given statements

(a) [1+1] we can then conclude that all the means are different from one another is false

(b) [1] the standardized variability between groups is higher than the standardized variability within groups is true

(c) [1+1]the pairwise analysis will identify at least one pair of means that are significantly since there are four groups is true

(d) [1+1] the appropriate α to be used in pairwise comparisons is 0.05 / 4 = 0.0125 since there are four groups is true.

(a) False. Rejecting the null hypothesis that the means of four groups are all the same using ANOVA at a 5% significance level does not necessarily mean that all the means are different from one another. It only indicates that there is at least one group that is significantly different from the others, but it does not provide information about which groups are different.

(b) True. If the null hypothesis is rejected, it means that there is a significant difference between at least one group mean and the overall mean. This implies that the standardized variability between groups is higher than the standardized variability within groups.

(c) True. If the null hypothesis is rejected, it means that there is a significant difference between at least one group mean and the overall mean. Therefore, pairwise analysis will identify at least one pair of means that are significant since there are four groups.

(d) True. When conducting pairwise comparisons, the appropriate α level to be used should be adjusted to account for multiple comparisons. In this case, since there are four groups, the appropriate α level would be 0.05/4 = 0.0125 to control for the family-wise error rate.

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Answer:

its like choosing one of 52 which is a 0,0213 chance

Step-by-step explanation:

The phasor form of a sinusoid represents the amplitude and phase angle of the sinusoid in complex number notation. In this case, the phasor form of [tex]8 sin(20t 57)[/tex] would be 8 ∠57°. The amplitude, 8, is the magnitude of the complex number, and the phase angle, 57°, is the angle of the complex number in the complex plane.

In terms of amplitude and phase angle, a sinusoidal waveform is mathematically represented in phasor form. Electrical engineering frequently employs it to depict AC (alternating current) circuits and signals. A complex number that depicts the magnitude and phase of a sinusoidal waveform is called a phasor. The phase angle is represented by the imaginary component of the phasor, whereas the real part of the phasor represents the waveform's amplitude. Complex algebra can be used to analyse AC circuits using the phasor form, which makes computations simpler and makes it simpler to see how the circuit behaves.

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The surface area of the gift (cube shaped) with a side length of 12 inches indicates that the amount of wrapping paper that Mrs. Hendren need to purchase is therefore;

A cube is a square based prism, with six congruent square faces, and in which the adjacent faces are perpendicular and the frontal faces are parallel.

The side length of the cube shaped box, s = 12 inches

The surface area of the a cube = 6 × s²

The surface area, A, of the cube shaped gift box Mrs. Hendren intends to wrap  for her daughter is therefore;

A = 6 × (12 in)² =  864 in²

Amount of wrapping paper Mrs. Hendren used = 578 square inches

The amount of more wrapping paper she needs = 864 - 578  = 286

The amount of wrapping paper Mrs. Hendren needs to purchase therefore is; 286 square inches

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The most logical order for the steps in correlation analysis is as follows:

1. Make scatter diagram.

2. Calculate a correlation coefficient.

3. Draw a least squares fit line.

The first step in correlation analysis is to create a scatter diagram, which involves plotting the paired data points on a graph. This helps visualize the relationship between the variables and provides an initial understanding of the data distribution.

Once the scatter diagram is created, the next step is to calculate a correlation coefficient. This numerical value quantifies the strength and direction of the relationship between the variables. It indicates the degree of linear association between the variables and ranges from -1 to 1, with positive values indicating a positive correlation, negative values indicating a negative correlation, and values close to zero indicating a weak or no correlation.

Finally, after obtaining the correlation coefficient, one can draw a least squares fit line. This line represents the best linear approximation of the relationship between the variables. It is obtained by minimizing the sum of the squared differences between the observed data points and the predicted values on the line. The least squares fit line provides a visual representation of the trend or pattern observed in the data and can help in making predictions or drawing conclusions about the relationship between the variables.

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Answer:

m ∠B = 29°

Step-by-step explanation:

When two angles are complementary, they from a right angle.  Therefore, the sum of the measures of the two angles equals 90°.

Step 1:  We can first find x by setting the sum of the two expressions given for the measures of angles A and B equal to 90:

m ∠A + m ∠B = 90

(8X + 5) + (3x + 8) = 90

(8x + 3x) + (5 + 8) = 90

11x + 13 = 90

11x = 77

x = 7

Step 2:  Now we can plug in 7 for x in 3x + 8 (i.e., the expressions that represents the measure of angle B) to find the measure of angle B:

m ∠B = 3(7) + 8

m ∠B = 21 + 8

m ∠B = 29°

Thus, the measure of angle B is 29°.

Answer:

Solution is in attached photo.

Step-by-step explanation:

Do take note for this question, since we let X be the distance between the start point and the falls, 2X will be the total distance travelled for the round trip.

The value of  t = -10/3 is outside the time interval [0, 4], we can conclude that the particle does return to its initial position.

The acceleration of the particle is given by the derivative of its velocity function: a(t) = v'(t) = 10 + 3t. Substituting t = 5.1, we get a(5.1) = 10 + 3(5.1) = 25.3.

The speed of the particle is given by the absolute value of its velocity function: |v(t)| = |10t + 3t^2|. To find when the speed is 1, we solve the equation |10t + 3t^2| = 1.

This gives us two intervals: (-3, -1/3) and (1/3, 2/3). Since we're only interested in the interval [0, 1.5], we can conclude that the speed is 1 when t = 1/3.

The position function of the particle is given by integrating its velocity function: x(t) = 5t^2 + 3/2 t^3 + 7. Substituting t = 4, we get x(4) = 120 + 48 + 7 = 175.

To determine whether the particle is moving toward or away from the origin, we calculate its velocity at t = 4: v(4) = 10(4) + 3(4)^2 = 58, which is positive.

Therefore, the particle is moving away from the origin at time t = 4.

To determine if the particle returns to its initial position, we need to solve the equation x(t) = 7 for t.

This gives us a quadratic equation: 5t^2 + 3/2 t^3 = 0. Factoring out t^2, we get t^2(5 + 3/2t) = 0.

This has two solutions: t = 0 and t = -10/3. Since t = -10/3 is outside the time interval [0, 4], we can conclude that the particle does return to its initial position.

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Based on the information provided, we have two given values for x: 14 and 15. The range of values for x can be expressed as [14, 15].

However, you also mentioned the value "1620". If this is intended to be part of the range for x, please provide additional clarification or context.

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The correct answer to the given expression is (C) 48√3(cos 120° + i sin 120°).

We can simplify the expression [tex][2√3(cos 120^o + i sin 120^o)]^4[/tex] by using De Moivre's theorem, which states that for any complex number z = r(cos θ + i sin θ), the nth power of z is given by:

[tex]z^n = r^n(cos (n\theta) + i sin (n\theta))[/tex]

Using this formula, we can write:

[tex][2\sqrt3(cos\ 120+ i sin \ 120)]^4 = (2\sqrt3)^4(cos\ 480 + i sin\ 480)[/tex]

Simplifying further:

(2√3)⁴(cos 480° + i sin 480°) = 48(cos 480° + i sin 480°)

Since the cosine and sine functions have a period of 360 degrees, we can add or subtract any multiple of 360 degrees to the angle inside the cosine and sine functions without changing the value of the expression.

Therefore, we can subtract 360 degrees from the angle 480 degrees to get an angle between 0 and 360 degrees:

48(cos 480° + i sin 480°) = 48(cos 120° + i sin 120°)

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To prove the statement "For all real numbers x and y, |x-y| = |y-x|," we can use proof by cases.

Case 1: x ≥ y

In this case, |x-y| = x-y and |y-x| = -(x-y).

So, |x-y| = x-y = -(y-x) = |y-x|.

Case 2: x < y

In this case, |x-y| = -(x-y) and |y-x| = y-x.

So, |x-y| = -(x-y) = y-x = |y-x|.

Since these two cases cover all possible values of x and y, we have proven that |x-y| = |y-x| for all real numbers x and y.

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The equation of the elipse in the graph is the one in option B.

x²/3² + y²/2² = 1

Here we can see the graph of an elipse.

Now, if we define a as the horizontal distance between the center and the edge.

b as the vertical distance between the center and the edge,

(h, k) as the center of the elipse.

Then the equation is:

(x - h)²/a² + (y - k)²/b² = 1

First, notice that the center is at (0, 0).

Also the vertical distance to the edge is 2 units, and the horizontal distance to the edge is 3 units, then the equation for the elipse is:

x²/3² + y²/2² = 1

So the correct option is B

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To determine the convergence of the series, let's analyze the terms and apply the ratio test. Answer : The limit evaluates to 0, which is less than 1.

The series can be written as:

∑(n=1 to ∞) n^2 * (3/8)^n

Using the ratio test, we compute the limit:

lim(n→∞) |(n+1)^2 * (3/8)^(n+1) / (n^2 * (3/8)^n)|

Simplifying the expression inside the absolute value:

lim(n→∞) |(n+1)^2 * (3/8)^(n+1) / (n^2 * (3/8)^n)|

= lim(n→∞) |(n+1)^2 * (3/8) / (n^2 * (3/8))|

Canceling out common terms:

lim(n→∞) |(n+1)^2 / n^2|

Expanding the numerator:

lim(n→∞) |(n^2 + 2n + 1) / n^2|

Taking the limit as n approaches infinity:

lim(n→∞) |1 + 2/n + 1/n^2|

As n approaches infinity, both (2/n) and (1/n^2) tend to zero, leaving us with:

lim(n→∞) |1|

Since the limit evaluates to 1, the ratio test does not provide a definitive answer. In such cases, we need to consider other convergence tests.

Let's try using the root test instead. The root test states that if the limit of the nth root of the absolute value of the terms is less than 1, the series converges.

We compute the limit:

lim(n→∞) [(n^2 * (3/8)^n)^(1/n)]

Simplifying inside the limit:

lim(n→∞) [(n^(2/n) * ((3/8)^n)^(1/n))]

Taking the nth root of the terms:

lim(n→∞) [n^(2/n) * (3/8)^(1/n)]

Since (3/8) is a constant, we can pull it out of the limit:

(3/8) * lim(n→∞) [n^(2/n) / n]

Simplifying further:

(3/8) * lim(n→∞) [(n^(1/n))^2 / n]

Taking the limit as n approaches infinity:

(3/8) * (1^2 / ∞) = 0

The limit evaluates to 0, which is less than 1. Therefore, by the root test, the series converges.

In summary, both the ratio test and the root test confirm that the series converges.

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The exact value of x, a right triangle with two sides of 25, is given as follows:

[tex]x = 25\sqrt{2}[/tex]

The Pythagorean Theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the lengths of the other two sides.

The theorem is expressed as follows:

c² = a² + b².

In which:

The angle of 45º has the same measure for the sine and the cosine, hence the two sides of the triangle have a length of 25.

Then the hypotenuse x is obtained as follows:

x² = 25² + 25²

[tex]x = \sqrt{2 \times 25^2}[/tex]

[tex]x = 25\sqrt{2}[/tex]

The triangle in this problem has a side length of 25 and an angle of 45º, while x is the hypotenuse.

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