Find the volume of the solid

By mathematical induction, the statement, n^(n) * (2n - 1) = n^2 - 4 is true for all positive integers n ≥ 3.

Base case: For n = 3, we have:

3^(3) * (2(3) - 1) = 27 * 5 = 135

3^(2) - 4 = 9 - 4 = 5

So the statement is true for n = 3.

Inductive step: Assume that the statement is true for some arbitrary positive integer k ≥ 3. That is,

k^(k) * (2k - 1) = k^2 - 4

Now we want to show that the statement is true for k+1. That is,

(k+1)^(k+1) * (2(k+1) - 1) = (k+1)^2 - 4

First, let's simplify the left-hand side:

(k+1)^(k+1) * (2(k+1) - 1) = (k+1) * k^k * (2k+1) * 2

= 2(k+1) * k^k * (2k+1)

= 2k^k * (2k+1) * (k+1) * 2

= 2k^k * (2k+1) * (2k+2)

= 2k^k * (4k^2 + 6k + 2)

= 8k^(k+2) + 12k^(k+1) + 4k^k

Now let's simplify the right-hand side:

(k+1)^2 - 4 = k^2 + 2k + 1 - 4

= k^2 + 2k - 3

Now we want to show that the left-hand side is equal to the right-hand side. So we need to show that:

8k^(k+2) + 12k^(k+1) + 4k^k = k^2 + 2k - 3

Let's first isolate the k^2 and 2k terms on the right-hand side:

k^2 + 2k - 3 = (k^2 - 4) + (2k + 1)

= k^k * (2k - 1) + (2k + 1)

Now we can substitute in our inductive hypothesis:

k^k * (2k - 1) + (2k + 1) = k^k * (k^2 - 4) + (2k + 1)

= k^(k+2) - 4k^k + 2k + 1

= k^(k+2) + 2k^(k+1) - 4k^k + 2k - 2k^(k+1) + 1

= 8k^(k+2) + 12k^(k+1) + 4k^k - 6k^(k+1) + 2k - 2

So we have shown that:

8k^(k+2) + 12k^(k+1) + 4k^k = k^2 + 2k - 3

Therefore, by mathematical induction, the statement is true for all positive integers n ≥ 3.

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i dont know

what  the hell

Answer:The second equation in the system, 4 + 3y = 5, simplifies to 3y = 1, which means y = 1/3.

Substituting this value of y into the first equation gives:

-9x - 2(1/3) = -16

Multiplying through by 3 to eliminate the fraction gives:

-27x - 2 = -48

Adding 2 to both sides gives:

-27x = -46

Dividing both sides by -27 gives:

x = 46/27

Therefore, the solution to the system of equations is x = 46/27 and y = 1/3.

Step-by-step explanation:

The joint pmf of X and Y is given by P(X = x, Y = y) = 1/15 for x,y = 1,2,...,5.

Since X and Y are independent, their joint pmf is simply the product of their marginal pmfs. The marginal pmf of X is given by P(X = x) = ∑y P(X = x, Y = y) = 1/15 ∑y 1 = 1/3, since there are three values of y for each x. Similarly, the marginal pmf of Y is P(Y = y) = 1/3. Therefore, the joint pmf of X and Y is P(X = x, Y = y) = P(X = x)P(Y = y) = (1/3)×(1/3) = 1/15 for x,y = 1,2,...,5.

Joint probability mass function (pmf) is a function that describes the probability distribution of two or more random variables. It assigns probabilities to all possible combinations of values that the random variables can take. For example, if X and Y are two random variables, the joint pmf P(X=x, Y=y) gives the probability of X=x and Y=y occurring together. The joint pmf satisfies the following properties:

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The events are independent, the probability of selecting any combination of r days is the product of the probabilities of selecting each day, which is the same as the distribution of the unordered set of r days when N(n) = r.

(a) The distribution of N(n) is a binomial distribution, since the events are independent and occur with a fixed probability p. Therefore, N(n) follows a Binomial distribution with parameters n and p:

N(n) ~ Binomial(n, p)

(b) The distribution of T1 is a geometric distribution, as it represents the number of trials until the first success (event occurs) in a sequence of independent Bernoulli trials with probability p. Therefore, T1 follows a Geometric distribution with parameter p:

T1 ~ Geometric(p)

(c) The distribution of Tr is a negative binomial distribution, as it represents the number of trials until the rth success (event occurs) in a sequence of independent Bernoulli trials with probability p. Therefore, Tr follows a Negative Binomial distribution with parameters r and p:

Tr ~ Negative Binomial(r, p)

(d) Given that N(n) = r, the unordered set of r days on which events occurred has the same distribution as a random selection (without replacement) of r of the values 1, 2, ..., n. This is because each event occurs independently and with a fixed probability p. When you select r days randomly (without replacement), the probability of each day being selected is p, and the probability of each day not being selected is (1-p). Since the events are independent, the probability of selecting any combination of r days is the product of the probabilities of selecting each day, which is the same as the distribution of the unordered set of r days when N(n) = r.

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To find the common ratio and the first four terms of the geometric sequence {(9^n+2)/(3)}, let's first rewrite the given expression to make it easier to understand i.e. Term a_n = (9^n+2)/3

Now, let's find the first four terms:
a_1 = (9^(1)+2)/3 = (9+2)/3 = 11/3
a_2 = (9^(2)+2)/3 = (81+2)/3 = 83/3
a_3 = (9^(3)+2)/3 = (729+2)/3 = 731/3
a_4 = (9^(4)+2)/3 = (6561+2)/3 = 6563/3

The first four terms are:
a_1 = 11/3
a_2 = 83/3
a_3 = 731/3
a_4 = 6563/3

To find the common ratio, divide the second term by the first term (or any consecutive terms):
Common ratio = a_2 / a_1 = (83/3) / (11/3) = 83/11 = 3

So, the common ratio is indeed 3, and the first four terms are 11/3, 83/3, 731/3, and 6563/3.

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The final speed of the sled in the second run will be more than 9.00 m/s and the speed of the sled at the bottom of the hill after the second run is 9.00 m/s.

Explanation;-

A) When the sled slides down the ice-covered hill without friction, the only force acting on it is gravity. The initial speed of the sled in the first run is 0 m/s, and its final speed is 7.50 m/s. In the second run, the sled starts with a speed of 1.50 m/s. Since there is no friction and the same force (gravity) is acting on the sled, the change in speed should be the same in both runs. Therefore, the final speed of the sled in the second run will be more than 9.00 m/s.

B) To find the speed of the sled at the bottom of the hill after the second run, we can determine the change in speed from the first run and add it to the initial speed of the second run. The change in speed in the first run is 7.50 m/s - 0 m/s = 7.50 m/s. Now, we add this change in speed to the initial speed of the second run: 1.50 m/s + 7.50 m/s = 9.00 m/s. So, the speed of the sled at the bottom of the hill after the second run is 9.00 m/s.

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a. The set of strings {1, 11, 111, 1111, 11111, ....} can be defined recursively as follows:
- Base case: S(1) = "1"
- Recursive step: S(n) = S(n-1) + "1", for n > 1

b. The function f(n) = n + 1/3, n = 1, 2, 3, ... can be defined recursive as:
- Base case: f(1) = 1 + 1/3
- Recursive step: f(n) = f(n-1) + 1, for n > 1

Recursion is the process of calling itself. This process provides a way to break complex problems into simpler processes that are easier to solve. Recursion can be a bit confusing. The best way to determine how it works is to experiment with it.

a. The recursive definition of the set of strings {1, 11, 111, 1111, 11111, ....} is as follows:
- The base case is the string "1".
- For any string in the set, we can obtain the next string by appending another "1" to the end. In other words, if s is a string in the set, then s + "1" is also in the set.

b. The recursive definition of the function f(n) = n + 1/3, n = 1, 2, 3, ... is as follows:
- The base case is f(1) = 4/3.
- For any n > 1, we can obtain f(n) by adding 1/3 to f(n-1). In other words, f(n) = f(n-1) + 1/3.

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In the long run:

(a) The proportion of customers that enter the system is π1 + π12 = 9/23.

(b) The proportion of customers that visit server 2 is π2 + π12 = 7/23.

The state space of the system is S = {10, 1, 2, 12}, where:

State 10 represents that both servers are free.

State 1 represents that server 1 is busy, but server 2 is free.

State 2 represents that server 2 is busy, but server 1 is free.

State 12 represents that both servers are busy.

The transition rates between the states are as follows:

From state 10, transitions to state 1 and state 2 can occur with rates 2 and 0, respectively.

From state 1, transitions to state 10 and state 12 can occur with rates 4 and 2, respectively.

From state 2, transitions to state 10 and state 12 can occur with rates 2 and 2, respectively.

From state 12, transitions to state 1 and state 2 can occur with rates 0 and 2, respectively.

To find the steady-state probabilities, we can set up the balance equations:

λπ10 = 4π1 + 2π12

2π10 = 2π2 + λπ10

4π1 = 2π12 + μπ10

2π2 = 2π12 + λπ10

where λ = 2 is the arrival rate, and μ = 4 and ν = 2 are the service rates of servers 1 and 2, respectively.

Solving the system of equations, we get:

π10 = (4λ(ν + λ))/(4λ(ν + λ) + μ(ν + λ) + μλ) = 8/23

π1 = (2μλ)/(4λ(ν + λ) + μ(ν + λ) + μλ) = 8/23

π2 = (2λ(ν + λ))/(4λ(ν + λ) + μ(ν + λ) + μλ) = 6/23

π12 = (μ(ν + λ))/(4λ(ν + λ) + μ(ν + λ) + μλ) = 1/23

Therefore, in the long run:

(a) The proportion of customers that enter the system is π1 + π12 = 9/23.

(b) The proportion of customers that visit server 2 is π2 + π12 = 7/23.'

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The value of the given integral after reversing the order of integration is:

∫[-∞,+∞]cos([tex]4x^2[/tex])dx

We need to reverse the order of integration of the given integral:

∫∫cos([tex]4x^2[/tex])dxdy

The limits of integration for x are not given, so we assume that the limits are from -∞ to +∞. For y, we assume the limits are from 0 to 1.

To reverse the order of integration, we write the integral as:

∫∫cos([tex]4x^2[/tex])dydx

Now, we integrate with respect to y first, keeping x as a constant:

∫∫cos([tex]4x^2[/tex])dydx = ∫[0,1]∫[-∞,+∞]cos([tex]4x^2[/tex])dydx

Integrating with respect to y, we get:

∫[0,1]∫[-∞,+∞]cos([tex]4x^2[/tex])dydx = ∫[-∞,+∞]cos([tex]4x^2[/tex])∫[0,1]dydx

The integral of y from 0 to 1 is simply (1-0) = 1. So we get:

∫[-∞,+∞]cos([tex]4x^2[/tex])∫[0,1]dydx = ∫[-∞,+∞]cos([tex]4x^2[/tex])dx

This integral cannot be evaluated analytically, so it remains in this form.

Therefore, the value of the given integral after reversing the order of integration is:

∫[-∞,+∞]cos([tex]4x^2[/tex])dx

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Measures of central tendency, measures of variation, and crosstabulation are all types of descriptive statistics.

Descriptive statistics summarize and describe the main features of a data set, including the typical or central values (measures of central tendency) and the spread or variability of the data (measures of variation). Crosstabulation, also known as contingency tables, is a way to summarize the relationship between two variables by displaying their frequency distributions in a table format.
Measures of central tendency, measures of variation, and crosstabulation are types of descriptive statistics. Descriptive statistics are used to summarize and describe the main features of a dataset in a simple and meaningful way.

Central tendency refers to the measures that help identify the center or typical value of a dataset, such as mean, median, and mode. Variation measures describe the spread or dispersion of data, including range, variance, and standard deviation. Crosstabulation is a method of organizing data into a table format to show the relationship between two categorical variables.

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The ladder slipped approximately 289 cm down the wall.

To determine how far down the wall the ladder slipped, we can consider the ladder as the hypotenuse of a right triangle formed with the wall.

Initially, the ladder forms a right triangle with the wall and the ground, where the base (foot of the ladder) is 80 cm away from the wall. Let's denote the distance the ladder slipped down the wall as d cm.

Using the Pythagorean theorem, we have:

(80 cm)^2 + d^2 = (300 cm)^2

Simplifying the equation, we get:

6400 cm^2 + d^2 = 90000 cm^2

Rearranging the equation and solving for d, we have:

d^2 = 90000 cm^2 - 6400 cm^2

d^2 = 83600 cm^2

Taking the square root of both sides, we find:

d ≈ √83600 cm

d ≈ 289 cm

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Relationship between convex loss functions and their solutions in a data set with m data points and n variables, where m > n. The statement is: Different loss functions would return the same sets of solutions as long as they are convex.

Assuming a data set has m data points and n variables, where m > n, different convex loss functions may not necessarily return the same sets of solutions. While convex loss functions guarantee a global minimum and are easier to optimize, they can have different properties and lead to different optimal solutions.

The choice of loss function depends on the problem you are trying to solve and the desired properties of the solution.

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n is indeed sane according to the given definition.

Based on the provided information, we want to prove that if 3 divides n (3 | n), then n is sane. Here's the proof:

Suppose n is an integer such that 3 | n. This means that n = 3k for some integer k. We are given that an integer n is sane if 3 divides (n^2 + 2n). We need to show that n is sane.

Let's consider the expression n^2 + 2n:

[tex]n^2 + 2n = (3k)^2 + 2(3k) = 9k^2 + 6k = 3(3k^2 + 2k)[/tex]Since both 3k^2 and 2k are integers, their sum (3k^2 + 2k) is also an integer. Therefore, we can see that 3 divides (n^2 + 2n), as the expression is equal to 3 times an integer.

So, n is indeed sane according to the given definition.

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Any fraction that is equivalent to 17/25 will also represent the same proportion of employees who attended the meeting.

An part of a whole is a fraction. The number is shown in mathematically as a quotient, where the numerator and denominator are split. Both are integers numbers . A fraction appears in the numerator or denominator of a complex fraction. The numerator of a proper fraction is less than the denominator.

A mathematical comparison of two numbers is known  as a proportion. According to proportion, two sets of provided numbers are said to be directly proportional to one another if they increase or decrease in the same ratio. "::" or "=" are symbols used to indicate proportions.

To  determine the Fraction of employee who attended the meeting:

Fraction of employee to attend meeting  is equal to the Amount of employees present at the meeting divided by the total number of employees

= [tex]\frac{153}{225}[/tex]

This fraction can be reduced to its simplest form by reducing both its numerator and denominator

In this,  largest common factor = 9;than we get:

[tex]\frac{153}{225}[/tex] = (153 ÷ 9) / (225 ÷ 9) = 17 / 25

Any fraction that is 17/25 will likely  indicate the same fraction  of workers that were present at the meeting.

For example ,68/100,34/50,51/75,204/300

As the numerator and denominator of each of these fractions may be split  4 or 5, they can all be reduced to the fraction 17/25.

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The correct answer is C. The sampling distribution of the proportion approximately follows the normal distribution when sampling with replacement or without replacement from extremely large populations when nx and n(1-x) are each at least 5.

The sampling distribution of the proportion is the distribution of proportions obtained from multiple random samples taken from a population. The central limit theorem states that for large sample sizes, the sampling distribution of the proportion will be approximately normally distributed, regardless of whether sampling is done with replacement or without replacement.

In option A, it is mentioned that nx and n(1-x) should be at least 10. This is a more conservative threshold and may not always be necessary for approximation to a normal distribution. Option C, on the other hand, states that nx and n(1-x) should be at least 5. This is a commonly used threshold in statistics and is generally considered sufficient for approximation to a normal distribution for large populations.

Therefore, option C is the correct answer as it includes both sampling with replacement or without replacement and allows for nx and n(1-x) to be at least 5 for approximation to a normal distribution in most cases.

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The volume is V = (integral from 0 to 0.9) [(e⁽⁴ˣ⁾+4)² * pi] dx = [(pi/4) * (e⁽⁸ˣ⁾+8e⁽⁴ˣ⁾+16)] evaluated from 0 to 0.9.
Substituting in the limits of integration,

we get V = [tex][\pi /4 * e^{8*0.9} +8e^{4*0.9} +16]-[\pi /4(e^{8*0} +8e^{4*0} +16][/tex]
Simplifying, we get

V = [(pi/4) * (e⁷°²+8e³°⁶+16)] - (pi/4) * (17)
Therefore, the volume is approximately 11.24 cubic units.

The volume of an object is a measure of how much space an object occupies. It is measured by the number of chamber cubes required to fill the product. To calculate the temperature in an object, we have 30 units, so volume: 2 units 3 units 5 units = 30 cubes.

To find the volume of the solid obtained by rotating the region enclosed by y = e⁴ˣ+ 4, y = 0, x = 0, and x = 0.9 about the x-axis, you can use the method of disks with the integral:

V = ∫[ (e⁴ˣ + 4)² * pi ] dx, with limits of integration a = 0 and b = 0.9.

To compute the volume, integrate with respect to x:

V = pi * ∫[ (e⁴ˣ+ 4)² ] dx, from 0 to 0.9.

Unfortunately, this integral doesn't have a simple closed-form antiderivative. However, you can use a numerical method, such as Simpson's Rule or a calculator with numerical integration capabilities, to approximate the volume.

The volume will be V ≈ (numerical approximation) cubic units.

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The approximate original size of the amoeba population which is about 33.33 the closest answer choice is (B) 33.

N(2) = [tex]N_0 * e^(^k^2^) = 100[/tex]

N(4) = [tex]N_0 * e^(^k^4^) = 300[/tex]

Dividing the second equation by the first, we get:

N(4) / N(2) = [tex]e^(^k^4^) / e^(^k^2^) = e^(^k^*^2^)[/tex]

Taking the natural logarithm of both sides, we have:

[tex]ln(N(4) / N(2)) = k*2[/tex]

Therefore, we can solve for k:

k = ln(N(4) / N(2)) / 2 = ln(300/100) / 2 = ln(3) / 2

N_0 = [tex]N(2) / e^(^k^*^2^) = 100 / e^(^l^n^(^3^)^/^2^ * ^2^) = 100 / e^l^n^(^3^) = 100 / 3[/tex]

Therefore, the approximate original size of the amoeba population is 100/3, which is about 33.33. The closest answer choice is (B) 33, so that is our answer.

In step 1, we set up two equations using the formula for exponential growth and the information given about the amoeba population. We then used these equations to solve for the value of k, which represents the rate of growth.

In step 2, we used the value of k to find the approximate original size of the amoeba population using the equation for exponential growth with time t=2.

We found that the approximate original size of the amoeba population is 100/3, which is about 33.33.

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lim n -> infinity an = 1.

To find the smallest value of M such that |an - 1| < t for all n > M, we can start by manipulating the inequality:

|an - 1| = |(n+1)/(n+2) - 1| = |n - 1| / |n + 2|

Since we want this expression to be less than t, we can write:

|n - 1| / |n + 2| < t

Multiplying both sides by |n + 2|, we get:

|n - 1| < t|n + 2|

We can split this inequality into two cases: n > 2 and n <= 2. For n > 2, we can drop the absolute values to get:

n - 1 < t(n + 2)

Expanding the right-hand side, we get:

n - 1 < tn + 2t

Solving for n, we get:

n > (1 - 2t) / (1 - t)

For n <= 2, we can drop the absolute values and reverse the inequality to get:

1 - n < t(n + 2)

Expanding the right-hand side, we get:

1 - n < tn + 2t

Solving for n, we get:

n > (1 - 2t) / (1 + t)

Therefore, the smallest value of M is the maximum of the values obtained from these two cases:

M = ceil(max((1 - 2t) / (1 - t), (1 - 2t) / (1 + t)))

Now, let's use the limit definition to prove that lim n -> infinity an = 1. We need to show that for any t > 0, there exists an integer N such that |an - 1| < t for all n > N.

Using the expression for an, we can write:

|an - 1| = |(n+1)/(n+2) - 1| = 1/(n+2)

Therefore, we need to find an integer N such that 1/(n+2) < t for all n > N. Solving for n, we get:

n > 1/t - 2

Therefore, we can choose N = ceil(1/t - 2) + 1. Then for any n > N, we have:

n > 1/t - 2

n + 2 > 1/t

1/(n+2) < t

Therefore, lim n -> infinity an = 1.

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The value of Rext is 20,000 Ω (ohms). This can be answered by the concept of resistor-capacitor.

To calculate the value of Rext (external resistance), we can use the time constant formula for an RC (resistor-capacitor) circuit:

τ = Rext × Cext

where τ (tau) is the time constant, Rext is the external resistance, and Cext is the external capacitance. You've provided τ as 0.2 seconds and Cext as 10 µF.

Rearranging the formula to solve for Rext, we get:

Rext = τ / Cext

Plugging in the values:

Rext = 0.2 sec / 10 µF

Since 1 µF = 10⁻⁶ F, we can rewrite Cext as:

Rext = 0.2 sec / (10 × 10⁻⁶ F)

Now, perform the calculation:

Rext = 0.2 sec / (10 × 10⁻⁶ F) = 20,000 Ω

So, the value of Rext is 20,000 Ω (ohms).

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The value of Rext is 20,000 Ω (ohms). This can be answered by the concept of resistor-capacitor.

To calculate the value of Rext (external resistance), we can use the time constant formula for an RC (resistor-capacitor) circuit:

τ = Rext × Cext

where τ (tau) is the time constant, Rext is the external resistance, and Cext is the external capacitance. You've provided τ as 0.2 seconds and Cext as 10 µF.

Rearranging the formula to solve for Rext, we get:

Rext = τ / Cext

Plugging in the values:

Rext = 0.2 sec / 10 µF

Since 1 µF = 10⁻⁶ F, we can rewrite Cext as:

Rext = 0.2 sec / (10 × 10⁻⁶ F)

Now, perform the calculation:

Rext = 0.2 sec / (10 × 10⁻⁶ F) = 20,000 Ω

So, the value of Rext is 20,000 Ω (ohms).

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Answer:

Step-by-step explanation:

0.6

Answer:

Step-by-step explanation:

Answer:

Step-by-step explanation:

in 1 gram dirt, 9.2 it s the answer because you need to drop one gram and add to the answer 9.2

Answer:

0.0075

Step-by-step explanation:

Here we have to find  3/4 of 1 Meter

In such questions, we have to take the product of the two numbers.

    For example, if we have to find 1/4 of 20

   Then, 1/4 of 20 = 1/2 x 20

                              = 5

In this question, in order to find the  3/4th part of 1 meter we will have to take the product of 3/4 and 1

   so,  3/4 of 1 Meter = 3/4 x 1

                                 =3/4

So,  3/4 of 1 meter will be 3/4 meter.

We can use the laws of exponents to rewrite 4^8.869 as:

4^8.869 = 4^8 x 4^0.869

None of the answer choices match this form exactly, but we can simplify some of them to match it.

Option A can be simplified using the product rule of exponents:

4^8 x 4^85/10 x 4^9/100 = 4^8 x 4^8.5 x 4^0.09 = 4^16.59

Option B can be simplified using the power of a sum rule of exponents:

4^8+8/10+6/10+9/1000 = 4^9.025

Option C can be simplified using the sum rule of exponents:

4^8 + 4^8/10 + 4^6/100 = 4^8 x (1 + 0.1 + 0.04) = 4^8 x 1.14

Option D can be simplified using the product rule of exponents:

4^8 x 4^8/10 x 4^6/100 x 4^9/1000 = 4^8 x 4^0.8 x 4^0.06 x 4^0.009 = 4^9.869

Therefore, the answer is option D, 4^8 x 4^8/10 x 4^6/100 x 4^9/1000.

Step-by-step explanation:

First, find the distance from the center point to -1,7    ....this is the radius

    d = sqrt ( 4^2 + 4^2 ) = sqrt 32           then r^2 = 32

then put the circle in standard form    (x-h)^2 + (y-k)^2 = r^2  

   where the center is      (h,k) = (-5,3)

( x+5)^2  + (y-3)^2 = 32