In how many ways can a math team of 9 students be chosen from a math club which consists of 14 seniors and 8 juniors if the team must consist of 3 seniors and 6 juniors?

Planet 3 has the most circular orbit among the three, with an eccentricity of 0.2.

To figure out which circle is generally round among the three planets, we can think about their erraticism. The unconventionality of oval estimates how extended or round it is. A completely round circle has an unconventionality of 0, while higher whimsy values demonstrate more lengthened circles.

The recipe to ascertain capriciousness (e) is given by:

e = c/a

Where 'c' addresses the distance between the middle and one of the foci, and 'a' addresses the length of the semi-significant hub.

Since the sun is at one concentration and the other spotlight is on the positive x-pivot, we can expect that the distance between the middle and the other center is equivalent to the semi-significant hub (a).

Presently we should analyze the unconventionality of the three circles:

Planet 1:

Semi-significant hub (a) = 200 million kilometers

Distance to center (c) = 200 million kilometers

eccentricity (e) = c/a = 200/200 = 1

Planet 2:

Semi-significant pivot (a) = 150 million kilometers

Distance to center (c) = 100 million kilometers

Whimsy (e) = c/a = 100/150 ≈ 0.67

Planet 3:

Semi-significant pivot (a) = 250 million kilometers

Distance to center (c) = 50 million kilometers

eccentricity (e) = c/a = 50/250 = 0.2

Contrasting the eccentricities, we can see that Planet 3 has the least unpredictability of 0.2, demonstrating the most roundabout circle among the three planets.

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The rectangle has a length of 7 feet and width of 4 feet. The area of a rectangle is 28 ft²

An equation is an expression that shows the relationship between two or more numbers and variables.

The question is not complete. Let us assume the area of a rectangle is 28 ft², and the length of the rectangle is 5 ft less than three times the width. Hence:

Length = l, width = w

Area = lw = 28    (1)

l = 3w - 5

But:

lw = 28;

Substitute l = 3w - 5

(3w - 5)(w) = 28

3w² - 5w - 28 = 0

w = 4 ft.

l = 3(4) - 5 = 7 feet

The rectangle has a length of 7 feet and width of 4 feet

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The probability that the coin will land heads at least twice when tossed three times is 3/8.

The question is asking for the probability that a coin will land heads at least twice when tossed three times.
To find this probability, we can use the concept of binomial probability.
Determine the total number of possible outcomes when tossing a coin three times. Since each toss has two possible outcomes (heads or tails), the total number of possible outcomes is [tex]2^3[/tex] = 8.
Determine the number of favorable outcomes where the coin lands heads at least twice. There are three possible scenarios where the coin can land heads at least twice: HH, HHT, HTH, where H represents heads and T represents tails. So, there are 3 favorable outcomes.
Calculate the probability by dividing the number of favorable outcomes by the total number of possible outcomes. Probability = favorable outcomes / total outcomes = 3 / 8.

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The expression [tex]\sqrt{3}[/tex]/ -27 no longer has an actual-quantity root due to the fact the square root characteristic and most other root functions are defined most effectively for non-negative real numbers.

To recognize the actual-wide variety root of this expression, we need to evaluate it.

The square root of 3 ([tex]\sqrt{3}[/tex]) is an irrational number, about equal to 1.732. When we divide this price by -27, the result is about -0.064.

In terms of actual-number roots, there may be no rectangular root or any other root of a negative quantity. The square root function and most different root functions are described handiest for non-bad real numbers. Therefore, in this situation, the expression [tex]\sqrt{3}[/tex]/ -27 does now not have a real-variety root.

It's worth noting that there are complex number roots for bad numbers. Complex numbers enlarge the real wide variety of gadgets by means of introducing the imaginary unit (i), allowing for the square root of negative numbers. However, within the context of this question, which specially asks for the real-range root, the answer is that there's no actual-variety root for the expression [tex]\sqrt{3}[/tex]/ -27.

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Answer:

  see attached

Step-by-step explanation:

You want the steps to simplify the sum of two rational expressions.

In general, the steps will be ...

See the attachment for the order for the given sum.

__

Additional comment

You can also look for factors in the combined numerator that will cancel denominator factors. And, you can expand the denominator to standard form.

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The solution to the system of equations is:

a = 2

b = -1

c = -3

To solve the system of equations using an inverse matrix, we can represent the system in matrix form:

[A] [X] = [B]

where:

[A] = coefficient matrix

[X] = variable matrix

[B] = constant matrix

The coefficient matrix [A] is:

| 1 2 1 |

| 0 1 1 |

|-3 0 1 |

The variable matrix [X] is:

| a |

| b |

| c |

The constant matrix [B] is:

| 14 |

| 1 |

| 6 |

To find [X], we need to calculate the inverse of [A] and multiply it by [B]:

[X] = [A]⁻¹ [B]

First, we find the inverse of [A]. If the inverse exists, the product [A]⁻¹ [A] should be the identity matrix [I]:

[A]⁻¹ [A] = [I]

Next, we can find the inverse of [A]:

| -1/3 2/3 -1/3 |

| 1/3 -1/3 2/3 |

| 1/3 -1/3 -1/3 |

Now, we can multiply [A]⁻¹ by [B]:

[X] = [A]⁻¹ [B]

| a | | -1/3 2/3 -1/3 | | 14 |

| b | = | 1/3 -1/3 2/3 | * | 1 |

| c | | 1/3 -1/3 -1/3 | | 6 |

Multiplying the matrices, we get:

| a | | 2 |

| b | = |-1 |

| c | |-3 |

Therefore, the solution to the system of equations is:

a = 2

b = -1

c = -3

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The required equation of the circle is x² + y² - 18x + 56 = 0.

The equation of a circle is calculated using the formula -

(x - h)² + (y - k)² = r²

Keep the values in formula where h and k are the coordinates of the centre of the circle. Moreover, r represents the radius of the circle.

(x - 9)² + (y - 0)² = 5²

(x - 9)² + y² = 25

Expanding the x-axis coordinates of the equation

x² + 81 - 18x + y² = 25

x² + y² - 18x + 81 - 25 = 0

Subtracting the relevant values in the equation

x² + y² - 18x + 56 = 0

Hence, the required equation of the circle is x² + y² - 18x + 56 = 0.

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The r-value that represents the most moderate correlation is given as follows:

0.18.

A correlation coefficient is a statistical measure that indicates the strength and direction of a linear relationship between two variables.

The coefficients can range from -1 to +1, with -1 indicates a perfect negative correlation, 0 indicates no correlation, and +1 indicates a perfect positive correlation.

Hence the most moderate correlation is the value with the absolute value closest to zero, hence it is given as follows:

0.18.

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F(-2) simplifies to 2 to evaluate the function f(t) = t/(t + 1) at the value t = -2, we substitute t = -2 into the function and perform the calculation:

f(-2) = (-2)/((-2) + 1)

simplifying the denominator:

f(-2) = (-2)/(-1)

when dividing by a negative number, we can simplify by multiplying both the numerator and denominator by -1:

f(-2) = -2/1 = -2

so, f(-2) equals -2.

f(-2) = -2/(-1)

to evaluate the function f(t) = t/(t + 1) at the value t = -2, we substitute t = -2 into the function:

f(-2) = (-2)/((-2) + 1) = (-2)/(-1) = 2 the function f(t) = t/(t + 1) represents a rational function that is undefined when the denominator (t + 1) is equal to zero. in this case, when t = -2, the denominator is not zero, and the function evaluates to -2.

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The linear factor that represents the 13-ft length is x - 5. The dimensions of the tank are 13 ft x 4 ft x 3 ft.

The polynomial 2x³ + 9x² + 4x - 15 represents the volume of the rectangular holding tank, where the depth is (x - 1) ft and the length is 13 ft. If we assume that the length is the greatest dimension, then the volume of the tank can be expressed as follows:

(length)(width)(depth) = 13x²(x - 1) = 13x³ - 13x²

Comparing this to the given polynomial, we can see that the linear factor that represents the 13-ft length is x - 5.

The dimensions of the tank can then be found by solving the equation 13x² - 13x = 2x³ + 9x² + 4x - 15. This equation can be solved as follows:

2x³ - 13x² - 13x + 15 = 0

x(2x² - 13x - 15) = 0

(x - 5)(2x + 3) = 0

x = 5 or x = -\frac{3}{2}

Since the length is the greatest dimension, the value of x must be 5. Therefore, the dimensions of the tank are 13 ft x 4 ft x 3 ft.

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The correct answer is B. The drop of 15 percentage points on an economics test exceeded the marginal benefit from a second hour of gymnastics.

This conclusion can be drawn by comparing the marginal benefit and marginal cost of Kate's decision to continue gymnastics for an additional hour. Marginal benefit refers to the additional benefit or satisfaction gained from an extra unit of an activity, while marginal cost refers to the additional cost or sacrifice incurred from that same unit. In this case, Kate's marginal benefit from the first hour of gymnastics was high enough to justify the time and effort invested. However, when she decided to extend her workout for a second hour and cut her study time, the marginal cost of sacrificing study time resulted in a significant drop in her economics grade. This drop of 15 percentage points on the economics test outweighed the marginal benefit she expected to gain from the extra hour of gymnastics.

The decision to continue gymnastics for the first hour had a positive marginal benefit that exceeded its marginal cost. Kate likely felt the physical and mental benefits of exercising, which enhanced her overall well-being and potentially improved her focus and energy levels for studying. However, the decision to extend the workout for a second hour resulted in a significant drop in her economics grade, indicating that the marginal cost of sacrificing study time was higher than the marginal benefit gained from the additional hour of gymnastics. This suggests that Kate should have allocated more time to studying rather than spending an extra hour on gymnastics, considering the impact it had on her academic performance.

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If a histogram of the amount of time spent each day watching Netflix were created, it would be skewed to the right.

This is because most people watch Netflix for only a short while each day, whereas a select few spend a considerable amount of time doing so, and hence the minimum possible value we can articulate, but the maximum time one can spend watching Netflix per day can vary upto possibly 24 hours.

Now, a histogram is typically right-skewed when the minimum possible value has a limit imposed on it, but not the maximum. As a result, the histogram would be skewed to the right and have a long tail on the right.

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Since the value of x is not provided in the question, we cannot determine the exact measure of angle 9 without more information. On the other hand, the measure of angle 10 is given as 2. Therefore, we know that m∠ 10 = 2.

The measure of angle 9 is given by the expression 2x - 4, where x represents an unknown value. The measure of angle 10 is stated as 2, without any variable involved. To find the measure of angle 9, we need to substitute the given value of x into the expression 2x - 4 and simplify the equation. The answer will be a numerical value representing the measure of angle 9.

In summary, we need to substitute the value of x into the expression 2x - 4 to determine the measure of angle 9. The measure of angle 10 is already given as 2.

Given that m∠ 9 = 2x - 4, we need to substitute the value of x into this expression to find the measure of angle 9. Since the value of x is not provided in the question, we cannot determine the exact measure of angle 9 without more information.

On the other hand, the measure of angle 10 is given as 2. Therefore, we know that m∠ 10 = 2.

To solve for the measure of angle 9, we would need the value of x. Without it, we cannot calculate the precise measure of angle 9. However, if the value of x were given, we could substitute it into the expression 2x - 4 to find the numerical value representing the measure of angle 9.

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The area of the surface S, where S is the part of the plane 2x + y + 2z = 10 that lies inside the cylinder x^2 + y^2 = 16, needs to be calculated.

the area of the surface S, we first need to determine the region of the plane 2x + y + 2z = 10 that lies inside the cylinder x^2 + y^2 = 16. The given equation of the plane represents a flat surface in three-dimensional space, while the equation of the cylinder represents a circular tube.

By substituting the equations, we can find the points of intersection between the plane and the cylinder. Once we have the points of intersection, we can calculate the area of the surface S within that region. This can be done by integrating over the region or by approximating the area using numerical methods.

The calculation of the area involves working with three-dimensional geometry and finding the intersection between a plane and a cylinder. Depending on the specific values of x, y, and z in the region, the calculation may involve solving equations, using integration techniques, or applying numerical methods.

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Zelda's utility, based on her utility function u(x,y) = min{0.2x, 1.5y}, when consuming a basket with (x,y) = (20.5, 26), is 3.9. In another scenario where the consumer prefers to consume exactly 1 unit of x with each unit of y, the price of x is $15, the price of y is $5, and the consumer has an income of $250, the optimal amount of good x* to consume is 11.0.

To determine Zelda's utility when consuming a basket with (x,y) = (20.5, 26), we evaluate her utility function u(x,y) = min{0.2x, 1.5y}. In this case, 0.2x = 0.2 * 20.5 = 4.1, and 1.5y = 1.5 * 26 = 39. Since the minimum of these two values is 4.1, Zelda's utility is 3.9.

In the second scenario, where Zelda prefers to consume exactly 1 unit of x with each unit of y, we need to determine the optimal amount of good x* to consume given the prices of x and y and her income. The consumer's goal is to maximize utility while staying within her budget constraint. The consumer's budget constraint is given by the equation: p_x * x + p_y * y = income, where p_x and p_y are the prices of x and y, respectively.

In this case, the price of x is $15, the price of y is $5, and Zelda's income is $250. Plugging in these values into the budget constraint, we have 15x + 5y = 250. Since Zelda prefers to consume exactly 1 unit of x with each unit of y, we can substitute y = x into the equation, resulting in 15x + 5x = 250. Simplifying, we get 20x = 250, and solving for x, we find x* = 250 / 20 = 12.5.

Therefore, the optimal amount of good x* for Zelda to consume is 12.5 units.

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The factored form of the quartic function is:

f(x) = k(x - 0)(x - 6)

Here, we have,

A quartic function is a polynomial function of degree 4.

To find a quartic function with the given zeros, we can start by writing the equation in factored form using the zero-product property.

Since the zeros are x = 0 and x = 6, we can write the factors as (x - 0) and (x - 6).

The factored form of the quartic function is:

f(x) = k(x - 0)(x - 6)

To determine the value of the constant k, we need more information.

For example, if we know the value of f(1), we can substitute it into the equation and solve for k.

Without additional information, we cannot determine the specific quartic function.

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It will take Joey's family approximately 21 months to save enough money, compounded monthly at a 5% interest rate, to finance their vacation trip to the amusement park.

To determine the time it will take to save enough money, we can use the formula for the future value of an ordinary annuity:

FV = P * [(1 + r)^n - 1] / r,

where FV is the future value, P is the monthly payment, r is the monthly interest rate, and n is the number of periods.

Given:

Desired future value (FV) = $5000,

Monthly payment (P) = $240,

Monthly interest rate (r) = 5% / 12 = 0.05 / 12 = 0.004167.

We need to solve for the number of periods (n) in the formula. Rearranging the formula, we have:

n = log(1 + (FV * r) / P) / log(1 + r),

where log denotes the logarithm with base 10.

Plugging in the given values, we get:

n = log(1 + ($5000 * 0.004167) / $240) / log(1 + 0.004167).

Using a calculator, we find that n is approximately 20.998, which rounds up to 21.

Therefore, it will take Joey's family approximately 21 months to save enough money, compounded monthly at a 5% interest rate, to finance their vacation trip to the amusement park.

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a.  If we let u = (3 - 2x), then the derivative du/dx would be -2, which is not equal to zero. This indicates that the substitution does not satisfy the requirement for u to be differentiable.

b. Michael's substitution satisfies the requirement for u to be differentiable.

(a) i. Kayla's proposed substitution of u as (3 - 2x) will not work in this case. The reason is that when using the u-substitution method, it is necessary for the chosen u to be differentiable, meaning that its derivative du/dx should exist and be non-zero. However, if we let u = (3 - 2x), then the derivative du/dx would be -2, which is not equal to zero. This indicates that the substitution does not satisfy the requirement for u to be differentiable.

ii. Kayla's idea of always picking the most inside factor as u does not work for all u-substitutions. There can be cases where choosing the most inside factor may not lead to a valid substitution that simplifies the problem or makes integration easier. It is important to consider the properties of the function and choose a suitable substitution accordingly.

(b) i. Michael's proposed substitution of u as 3√3 - 2x will work in this case. If we let u = 3√3 - 2x, then the derivative du/dx would be -2, which is non-zero. Therefore, Michael's substitution satisfies the requirement for u to be differentiable.

ii. Michael's idea of always picking the most complicated factor as u also does not hold true for all u-substitutions. The choice of u depends on various factors, including the structure of the function, simplification possibilities, and making the integration process more manageable. It is not necessarily the case that the most complex factor will always result in a successful substitution.

It is important to consider the specific characteristics of the function and apply appropriate judgment in choosing the substitution u to simplify the problem effectively.

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The square root of the given fraction simplifies to 2. A numerator of a 2 to the power of 1002 plus 2 to the power of 1006

The square root of the fraction (2^1002 + 2^1006) / (17 * 2^998) can be simplified as follows:

To simplify this expression, let's start by breaking down the numerator and the denominator separately.

Numerator:

We have 2^1002 + 2^1006. Notice that both terms have a common factor of 2^1002. Factoring that out, we get:

2^1002 * (1 + 2^4)

Simplifying further, we have:

2^1002 * (1 + 16) = 2^1002 * 17

Denominator:

The denominator is 17 * 2^998.

Now, let's combine the simplified numerator and denominator:

√((2^1002 + 2^1006) / (17 * 2^998))

= √((2^1002 * 17) / (17 * 2^998))

= √(2^1002 / 2^998)

= √(2^4)

= 2

Therefore, the square root of the given fraction simplifies to 2.

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The distributive property of multiplication over addition justifies the statement: If x(y + z) = a, then xy + xz = a.

The distributive property of multiplication over addition states that when you multiply a number (in this case, x) by the sum of two other numbers (y + z), you can distribute the multiplication to each term inside the parentheses. This means you can multiply x by y and then add the result to x multiplied by z. Mathematically, it can be expressed as:

x(y + z) = xy + xz

In the given statement, x(y + z) = a is the given equation. By applying the distributive property, we can rewrite it as xy + xz = a. This shows that the sum of xy and xz is equal to a, which justifies the statement.

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The equation of the circle expressed in the form x²+y²+ax+by+c=0 is (x+3)² + (y+4)² - 100 = 0.

Center of the circle = (-3,-4)Point on the circumference of the circle = P(5,2) We know that the equation of the circle is given by: (x−a)²+(y−b)²=r² where the center of the circle is (a, b) and the radius is r.

Step 1: Find the radius of the circle using the distance formula Distance between the center of the circle and point

P = radius of the circle.

We get

r = √((-3-5)² + (-4-2)²)r = √64+36r = √100 = 10

Step 2:Find the equation of the circle substituting the center and the radius into the equation of the circle

(x−a)²+(y−b)²=r²(x-(-3))² + (y-(-4))² = 10²(x+3)² + (y+4)² = 100(x+3)² + (y+4)² - 100 = 0

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The speed of the train is 45 mi/h hence the correct answer is option G. 45 mi/h.

Let's assume the speed of the train is x mi/h.

Since the car is traveling 15 mi/h faster than the train, the speed of the car is (x + 15) mi/h.

The distance traveled by train in 2 hours is 2x miles (since speed * time = distance).

The distance traveled by the car in 2 hours is 2(x + 15) miles.

According to the Pythagorean theorem, the square of the hypotenuse of a right triangle is equal to the sum of the squares of the other two sides.

In this case, the hypotenuse represents the distance between the train and the car, which is approximately 150 miles.

Using the Pythagorean theorem, we can set up the equation:

[tex](2x)^2 + [2(x + 15)]^2 = 150^2\\4x^2 + 4(x + 15)^2 = 150^2[/tex]

Simplifying and solving the equation, we find:

[tex]4x^2 + 4(x^2 + 30x + 225) = 22500\\4x^2 + 4x^2 + 120x + 900 = 22500\\8x^2 + 120x + 900 - 22500 = 0\\8x^2 + 120x - 21600 = 0[/tex]

Dividing the equation by 8 to simplify:

[tex]x^2 + 15x - 2700 = 0[/tex]

Now we can solve this quadratic equation using factoring, completing the square, or the quadratic formula. Factoring is the easiest method in this case:

(x + 60)(x - 45) = 0

Setting each factor equal to zero, we have:

x + 60 = 0  or  x - 45 = 0

x = -60  or  x = 45

Since the speed cannot be negative, we discard the solution x = -60.

Therefore, the speed of the train is 45 mi/h hence the correct answer is option G. 45 mi/h.

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(a) The Nash equilibrium for the $20 ultimatum game is a 50-50 split: the proposer offers $10 and the responder accepts. (b) Experimental findings show deviations from the Nash equilibrium due to fairness considerations, strategic behavior, and social preferences. (c) Proposers deviate from Nash equilibrium due to fairness concerns and strategic considerations. (d) To test the explanation, conduct experiments manipulating fairness norms and incentives to observe proposers' behavior.

(a) The Nash equilibrium for the $20 ultimatum game, where offers are made to the nearest $0.50, is a 50-50 split, where the proposer offers $10 and the responder accepts.

(b) The experimental literature has found deviations from the Nash equilibrium in both proposers and responders. These deviations can be attributed to various factors such as fairness considerations, strategic behavior, and social preferences.

(c) The behavior of the proposer deviates from the Nash equilibrium due to factors like fairness concerns and strategic considerations. Proposers may make more generous offers to avoid rejection or to signal fairness and maintain a positive reputation.

(d) To test the explanation that proposers deviate from the Nash equilibrium due to fairness concerns, one could design an experiment where fairness is manipulated. Participants could be exposed to different fairness treatments, and their offers could be observed and compared to those in a control group. Statistical analysis can then be used to determine if there is a significant difference in offers between the fairness treatments and the control group.

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To sketch the graph of the polynomial function y = x³ - 3, we can use a series of transformations to obtain the final graph. Let's break down the process step by step:

1. Start with the graph of the basic cubic function y = x³. This function is symmetric with respect to the origin and passes through the point (0, 0).

2. Apply a vertical shift downward by 3 units. This means that each point on the graph will move 3 units downward. The new function becomes y = x³ - 3. The graph will now pass through the point (0, -3).

3. Analyze the behavior of the function for large positive and negative values of x. As x approaches positive infinity, y approaches positive infinity, and as x approaches negative infinity, y approaches negative infinity. This information gives us an idea of how the graph extends beyond the visible region.

4. Observe that the graph is a smooth curve with no sharp corners or breaks.

Using these steps, we can sketch the graph of the polynomial function y = x³ - 3. The graph will have a shape similar to a basic cubic function, but it will be shifted downward by 3 units. It will pass through the point (0, -3) and exhibit behavior characteristic of a cubic function. The graph extends infinitely in both the positive and negative x directions.

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The simplified form of the expression √75 - 4√18 + 2√32 is 5√3 - 4√2.

To simplify the expression √75 - 4√18 + 2√32, we need to simplify each individual square root and then combine like terms.

Let's start by simplifying each square root term:

1. √75:

We can simplify √75 by breaking it down into its prime factors. Since 75 is divisible by 25, we have:

√75 = √(25 × 3)

    = √25 × √3

    = 5√3

2. -4√18:

Similarly, we can simplify √18:

√18 = √(9 × 2)

    = √9 × √2

    = 3√2

Therefore, -4√18 becomes -4(3√2) = -12√2

3. 2√32:

We can simplify √32:

√32 = √(16 × 2)

    = √16 × √2

    = 4√2

Now, we can rewrite the expression with the simplified square root terms:

√75 - 4√18 + 2√32

= 5√3 - 12√2 + 2(4√2)

= 5√3 - 12√2 + 8√2

Next, we combine like terms:

-12√2 + 8√2 = -4√2

Finally, the simplified expression becomes:

5√3 - 4√2

In summary, the simplified form of the expression √75 - 4√18 + 2√32 is 5√3 - 4√2.

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Jacques should have spent 1 hour working on problems and 3.75 hours reading to get the best possible exam score.

Let's denote the increase in exam score by x, and assume that working on 15 problems raises the exam score by x and reading the textbook for 1 hour also raises the exam score by x. We can then set up the following equation:

4 = a(15) + b

where a is the number of hours spent on problem sets, b is the number of hours spent reading the textbook, and 4 represents the total number of study hours available.

To solve for a and b, we can use the information given in the problem: "working on 15 problems raises a student’s exam score by about the same amount as reading the textbook for 1 hour." This means that:

15a = b

Substituting this into our first equation, we get:

4 = a(15) + 15a

Simplifying this equation, we get:

4 = 16a

a = 0.25

So Jacques should spend 0.25 * 4 = 1 hour working on problem sets, and 15 * 0.25 = 3.75 hours reading the textbook.

Therefore, Jacques should have spent 1 hour working on problems and 3.75 hours reading to get the best possible exam score.

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The possible values of 'a' such that √72 + √a simplifies to a single term are when 'a' is equal to 2.

To simplify the expression √72 + √a into a single term, we need to find values of 'a' that allow us to combine the square roots.

First, let's simplify √72:

√72 = √(36 * 2) = √36 * √2 = 6√2

Now, our expression becomes:

6√2 + √a

To combine the square roots into a single term, the radicands (numbers inside the square root) must be the same. In this case, the radicands are 2 and 'a'. To simplify the expression, we need 'a' to be equal to 2.

Therefore, the possible values of 'a' such that √72 + √a simplifies to a single term are when 'a' is equal to 2.

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The rational expression (x² + 10x + 25) / (x² + 9x + 20) simplifies to (x + 5) / (x + 4), with the restriction that x cannot be equal to -4.

To simplify the rational expression (x² + 10x + 25) / (x² + 9x + 20), we can factorize the numerator and denominator and then cancel out any common factors.

Factorizing the numerator:

x² + 10x + 25 = (x + 5)(x + 5) = (x + 5)²

Factorizing the denominator:

x² + 9x + 20 = (x + 4)(x + 5)

Now, we can simplify the expression:

(x² + 10x + 25) / (x² + 9x + 20) = (x + 5)² / (x + 4)(x + 5)

After canceling out the common factor (x + 5) from the numerator and denominator, we are left with:

= (x + 5) / (x + 4)

Therefore, the simplified rational expression is (x + 5) / (x + 4). However, we need to note the restriction on the variable. In this case, x cannot be equal to -4 since it would result in division by zero, which is undefined. So, the restriction is x ≠ -4.

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