If n is an integer and n > 1, then n! is the product of n and every other positive integer that is less than n. for example, 5! = 5 x 4 x 3 x 2 x 1a. Write 6! in standard factored formb. Write 20! in standard factored formc. Without computing the value of (20!)2, determine how many zeros are at the end of this number when it is written in decimal form. Justify your answer

Answer:

if u want total price then,

=Rs.35×80=Rs.2800

The people who are included in the survey conducted by Bureau of Labor Statistics are (a) certain unpaid workers, (b) part-time workers and (c) workers on vacation, the correct option is (d).

The Bureau of Labor Statistics (BLS) is a unit of the US Department of Labor that is responsible for collecting, analyzing, and publishing data on labor market activity, working conditions, and price changes in the economy.

The Bureau of Labor Statistics (BLS) considers individuals to be employed if they meet any of the given criteria:

(i) Worked at least one hour as paid employees

(ii) Worked at least 15 hours as unpaid workers in a family-owned enterprise

(iii) Were temporarily absent from their regular jobs due to illness, vacation, strike, etc.

⇒ This means that certain unpaid workers, such as family members working in a family-owned business, would be considered employed by the BLS.

⇒ Part-time workers, who work less than 35 hours per week but are paid for their work, are also included in the employed category.

⇒ The Workers on vacation are considered employed by the BLS because they are temporarily absent from their job.

Therefore, all the options are correct , which is Option(d).

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The given question is incomplete, the complete question is

Who of the following are included in the Bureau of Labor Statistics "employed" category?

(a) certain unpaid workers

(b) part-time workers

(c) workers on vacation

(d) All of the above are correct.

The probability that a randomly selected bottle is correctly placed AND is a Plastic #4 bottle is 0.25 or 25%

What is Conditional probability?

Conditional probability is the probability of an event occurring given that another event has occurred or is known to have occurred. It is denoted by P(A|B), which reads as "the probability of A given B."

The formula for conditional probability is:

P(A|B) = P(A and B) / P(B)

where P(A and B) is the probability of both events A and B occurring, and P(B) is the probability of event B occurring.

The total number of Plastic #2 bottles is 8 (correctly placed) + 5 (incorrectly placed) = 13.

The total number of Plastic #4 bottles is 5 (correctly placed) + 2 (incorrectly placed) = 7.

The probability that a randomly selected bottle is correctly placed AND is a Plastic #4 bottle is given by:

(number of Plastic #4 bottles correctly placed) / (total number of bottles)

So the probability is:

5/20 = 1/4 = 0.25

Therefore, the probability that a randomly selected bottle is correctly placed AND is a Plastic #4 bottle is 0.25 or 25%.

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Cold fronts generally move faster than warm fronts because cold air is denser and thus, moves more quickly. Precipitation with a cold front typically covers a broader area, especially during the winter.

On the other hand, warm fronts move more slowly as they are characterized by the gradual lifting of warm air over colder air. Cold fronts also typically have steeper slopes than warm fronts. This is because the leading edge of a cold front is more abrupt.

With a steep rise in the cold air mass. In contrast, the leading edge of a warm front has a gentler slope as the warm air gradually rises over the colder air.

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Answer:

(1, 2) and (-1, -2). or. (±1, ±2)

Step-by-step explanation:

[tex]{ \sf{f(x) = \frac{4x}{ {x}^{2} + 1 } }} \\ [/tex]

- Simply, a critical number or critical point is gotten by differentiating the function.

From Quotient rule;

[tex]{ \sf{ {f}^{l}(x) = \frac{4( {x}^{2} + 1) - (2x)(4x)}{ {( {x}^{2} + 1)}^{2} } }} \\ \\ { \sf{f {}^{l}(x) = \frac{ {4x}^{2} + 4 - {8x}^{2} }{ {( {x}^{2} + 1) }^{2} } }} \\ \\ { \sf{f {}^{l}(x) = \frac{4(1 - {x}^{2}) }{ {( {x}^{2 } + 1) }^{2} } }}[/tex]

Then equate this derivative to zero;

[tex]{ \sf{0 = \frac{4(1 - {x}^{2} )}{ {( {x}^{2} + 1) }^{2} } }} \\ \\ { \sf{4(1 - {x}^{2} ) = 0}} \\ \\ { \sf{4 - {4x}^{2} = 0}} \\ \\ { \sf{4 {x}^{2} = 4}} \\ \\ { \sf{x = \sqrt{1} }} \\ \\ { \sf{ \underline{ \: x = \pm 1 \: }}}[/tex]

Substitute for x in f(x)

For x = 1

[tex]{ \sf{f(1) = \frac{4(1)}{ {(1)}^{2} + 1} = \frac{4}{2} = 2 }} \\ [/tex]

For x = -1

[tex]{ \sf{f( - 1) = \frac{4( - 1)}{ {( - 1)}^{2} + 1 } = \frac{ - 4}{2} = - 2 }} \\ [/tex]

Therefore points are;

(1, 2) and (-1, -2)

The distance  from Link to the Octorok  is 10.63 units.

We know that the distance between two points (x₁, y₁) and (x₂, y₂) is given by the formula below:

distance = √( (x₂ - x₁)² + (y₂ - y₁)²)

Here we want to find the distance from Link to the Octorok so Link can attack, so we need to get the distance between the points (-4, -5) and (3, 3).

The distance will be:

distance = √( (3 + 4)² + (3 + 5)²)

distance = √( (7)² + (8)²)

distance = √113

distance = 10.63

The distance is 10.63 units.

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Answer:

[tex]xy^8[/tex]

Step-by-step explanation:

Notice if you have the same base you can ADD the exponent, for example:

[tex]x^{-6} x^{7} =x^{-6+7}=x^{1 }=x[/tex]

[tex]y^{6} y^{2} =y^{6+2}=y^{8 }\\[/tex]

so the answer is

[tex]xy^8[/tex]

Answer:

Step-by-step explanation:

Answer:

[tex]\begin{array}{|c|c|c|c|}\cline{1-4}\vphantom{\dfrac12}\sf Location&AB&BC&AC\\\cline{1-4}\vphantom{\dfrac12} A(3,4),\;B(1,1),\;C(5,1)&3.61&4&3.61\\\cline{1-4}\vphantom{\dfrac12} A(4,5),\;B(2,1),\;C(7,3)&4.47&5.39&3.61\\\cline{1-4}\end{array}[/tex]

[tex]\begin{array}{|c|c|c|c|}\cline{1-4}\vphantom{\dfrac12}\sf Location&m \angle A&m \angle B&m \angle C\\\cline{1-4}\vphantom{\dfrac12} A(3,4),\;B(1,1),\;C(5,1)&67.38^{\circ}&56.31^{\circ}&56.31^{\circ}\\\cline{1-4}\vphantom{\dfrac12} A(4,5),\;B(2,1),\;C(7,3)&82.87^{\circ}&41.63^{\circ}&55.49^{\circ}\\\cline{1-4}\end{array}[/tex]

Step-by-step explanation:

Place points A, B and C on the coordinate grid.

Alternatively, type the following into the input field as 3 separate inputs:

Triangle 1

Triangle 2

Use the Segment tool to join each pair of points.

Alternatively, type Segment( <Point>, <Point> ) into the input field (replacing <Point> with the letter name of the point) to create a segment between two points.

Record the length of each side.

[tex]\begin{array}{|c|c|c|c|}\cline{1-4}\vphantom{\dfrac12}\sf Location&AB&BC&AC\\\cline{1-4}\vphantom{\dfrac12} A(3,4),\;B(1,1),\;C(5,1)&3.61&4&3.61\\\cline{1-4}\vphantom{\dfrac12} A(4,5),\;B(2,1),\;C(7,3)&4.47&5.39&3.61\\\cline{1-4}\end{array}[/tex]

Use the Angle tool to measure each angle in the resulting triangle.

Alternatively, type Angle(Polygon(A, B, C)) into the input field to create all interior angles.

Record the measure of each angle.

[tex]\begin{array}{|c|c|c|c|}\cline{1-4}\vphantom{\dfrac12}\sf Location&m \angle A&m \angle B&m \angle C\\\cline{1-4}\vphantom{\dfrac12} A(3,4),\;B(1,1),\;C(5,1)&67.38^{\circ}&56.31^{\circ}&56.31^{\circ}\\\cline{1-4}\vphantom{\dfrac12} A(4,5),\;B(2,1),\;C(7,3)&82.87^{\circ}&41.63^{\circ}&55.49^{\circ}\\\cline{1-4}\end{array}[/tex]

Note: All measurements have been given to the nearest hundredth (2 decimal places).

As a result, both varieties of trees will be the same height after 9 years. We can change both equations to a = 9 to verify this: Height of the type A tree is 36 + 8(9) or 36 + 72 inches, whereas the height of the type B tree is 18 + 10(9) or 18 + 90 inches.

Trigonometry includes heights and distances, and it has numerous uses in practical daily life. It utilised to compute height of towers, structures, mountains, etc., and distance between any two objects such celestial bodies others. ,sys,s tos.as to .... and.

Let's use "a" to denote the number of years after planting the trees.

A type A tree will reach the following height after "a" years:

Height of type A tree = 36 + 8a

After "a" years, the height of a type B tree will be:

Height of type B tree = 18 + 10a

We must set the two types of trees' heights equal to one another and solve for "a" to determine when they will reach the same height:

36 + 8a = 18 + 10a

Subtracting 8a from both sides, we get:

36 = 18 + 2a

Subtracting 18 from both sides, we get:

18 = 2a

Dividing both sides by 2, we get:

a = 9

Therefore, after 9 years, both types of trees will be the same height.

To check this, we can substitute a = 9 into both equations:

Height of type A tree = 36 + 8(9) = 36 + 72 = 108 inches

Height of type B tree = 18 + 10(9) = 18 + 90 = 108 inches

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Question:

You have two types of trees to plant in your yard: type A trees are 36 inches tall and grow 8 inches per year, while type B trees are 18 inches tall and grow 10 inches per year. At what point in time will the trees be the same height? How tall will the trees be at that time?

Answer:

D.

Step-by-step explanation:

Answer:

Here is a diagram and transition matrix for this case:

Diagram:

        +---(0.65)---> State 1 (work)

        |

Start ---+

        |

        +---(0.35)---> State 2 (procrastinate)

                  |

                  +---(0.15)---> State 1 (work)

                  |

                  +---(0.85)---> State 2 (procrastinate)

Transition matrix:

         |  State 1  |  State 2  |

----------+-----------+-----------+

State 1   |   1.00    |   0.00    |

----------+-----------+-----------+

State 2   |   0.15    |   0.85    |

----------+-----------+-----------+

In the transition matrix, the rows represent the starting state and the columns represent the ending state. The entries in the matrix represent the probabilities of transitioning from the starting state to the ending state. For example, the entry in row 1 and column 2 (0.00) represents the probability of transitioning from State 1 to State 2, which is 0.00.

the definite integral of f(x) over the interval [5, 7] is (-5 / 600).

How to find?

The given function is:

f(x) = 1 / (x² + 10x + 25)

To find the definite integral of this function over the interval [5, 7], we can use the following steps:

Rewrite the function using partial fraction decomposition:

f(x) = 1 / (x² + 10x + 25)

= 1 / [(x + 5)²]

Using partial fraction decomposition, we can write this as:

f(x) = A / (x + 5) + B / (x + 5)²

where A and B are constants to be determined. Multiplying both sides by the common denominator (x + 5)², we get:

1 = A(x + 5) + B

Setting x = -5, we get:

1 = B

Setting x = 0, we get:

1 = 5A + B

= 5A + 1

Solving for A, we get:

A = 0

Therefore, the partial fraction decomposition is:

f(x) = 1 / [(x + 5)²]

= 0 / (x + 5) + 1 / (x + 5)²

Use the formula for the definite integral of a power function:

∫ xⁿ dx = (1 / (n + 1))× x²(n + 1) + C

where C is the constant of integration.

Using this formula, we can find the antiderivative of the function 1 / (x + 5)²:

∫ 1 / (x + 5)² dx = -1 / (x + 5) + C

Evaluate the definite integral over the interval [5, 7]:

∫[5,7] 1 / (x + 5)² dx

= [-1 / (x + 5)] [from 5 to 7]

= (-1 / 12) - (-1 / 10)

= (-5 / 600)

Therefore, the definite integral of f(x) over the interval [5, 7] is (-5 / 600).

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Answer:

A and B are not mutually exclusive

Step-by-step explanation:

A and B are not mutually exclusive because P(A and B) > 0. If A and B were mutually exclusive, then they would have no outcomes in common and the probability of their intersection would be zero. However, in this case, they do share some outcomes, since P(A and B) is greater than zero.

In response to the stated question, we may state that As a result, the overall probability of an accurately picked drive-thru order across all chains is roughly 0.929, or 92.9%.

Probability theory is an area of mathematics that calculates the likelihood of an occurrence or a proposition being true. A risk is a number in the range of 0 and 1, whereas 1 implies certainty and a probability of roughly 0 indicates how likely an event seems to be to occur. Probability is a mathematical expression of the chance or chances that a given event will occur. Probabilities can alternatively be stated as integers between 0 and 1 or as % from 0% to 100%. the ratio of occurrences among equally likely choices that result in a certain event in comparison to all other outcomes.

Using the data in the table, we can compute the likelihood of a correct drive-thru order for each fast food chain, as well as the overall chance of an accurate order across all chains.

Divide the number of accurate orders by the total number of orders to find the chance of a randomly picked order being accurate at each chain:

P(accurate order) = 1246 / 1300 = 0.958 for McDonald's

P(accurate order) = 1020 / 1100 = 0.927 Taco Bell

P(accurate order) = 708 / 800 = 0.885 for Burger King

P(accurate order) = 940 / 1000 = 0.94 for Wendy's

P(adequate overall order) = 0.3 * 0.958 + 0.25 * 0.927 + 0.2 * 0.885 + 0.25 * 0.94 = 0.929

As a result, the overall likelihood of an accurately picked drive-thru order across all chains is roughly 0.929, or 92.9%.

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The area of the living room is 24.7 m², and the area of the game room is 37.05 m².

Area is the measure of the two-dimensional space occupied by a figure or an object. It is usually expressed in square units such as cm2, m2, or in2. It is used to measure the size of a figure or object, and can also be used to calculate the amount of material required for a project.

The area of a living room measuring 4.75 m by 5.2 m can be calculated using the formula A = l × w, where A is the area, l is the length and w is the width. In this case, A = 4.75 m × 5.2 m = 24.7 m².

To calculate the area of the game room, we must multiply the area of the living room by 1.5. Therefore, the area of the game room is 1.5 × 24.7 m² = 37.05 m².

The area of the living room is 24.7 m², and the area of the game room is 37.05 m².

The area of a room can be used to determine how much furniture can fit in the room or how much floor space is available for activities. Knowing the area of a room is also important for calculating the cost of painting, wallpapering, carpeting, and other flooring materials.

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The asymptotes of the function f(x) = (2x² - 5x + 3)/(x - 2) are given as follows:

The function for this problem is defined as follows:

f(x) = (2x² - 5x + 3)/(x - 2)

The vertical asymptote is the value of x for which the function is not defined, hence it is at the zero of the denominator, and thus it is given as follows:

x - 2 = 0

x = 2.

The oblique asymptote is at the quotient of the two functions, hence:

(mx + b)(x - 2) = 2x² - 5x + 3

mx² + (b - 2m) - 2b = 2x² - 5x + 3.

Hence the values of m and b are given as follows:

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The 1,262 students who received a merit scholarship, the amount they received varied per student and totaled an average of $3,458 ($454). That amount is 78.2% of the full tuition cost of $4,400.

Merit is a term used to describe the quality of something or someone that makes them worthy of recognition or respect. It is an indicator of worthiness and is usually based on a person's ability, effort, or accomplishments. Merit is often used when evaluating an individual or a group for a promotion, hiring, or award. Merit is subjective, as different people have different standards for what merits recognition.

Therefore, 21.8% of the students who received a merit scholarship did not receive enough to cover full tuition. Therefore, the percentage of students who received a merit scholarship and did not receive enough to cover full tuition is 21%.

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As per the axial force, the support reaction at A is equal in magnitude to the axial force that is applied at the other end of the bar.

To determine the support reaction at A, we will draw a free-body diagram of the bar.

Since the bar is fixed at A, the support reaction at that end will be a vertical force, and we will label it as RA.

Using Newton's second law of motion, we can write the equation of equilibrium for the bar in the vertical direction:

ΣFy = 0

where ΣFy is the sum of all the forces in the vertical direction. Since there are only two forces acting on the bar, the axial force and the support reaction, we can write:

-FA + RA = 0

where FA is the axial force that is applied at one end, and RA is the support reaction at the other end.

From this equation, we can solve for RA:

RA = FA

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Complete Question:

Determine the support reaction at A

The axially loaded bar is fixed at A and loaded as shown. Draw a free-body diagram and determine the support reaction at A.

The population of interest is the group of students who submitted an application to the college in 2017.

A sample is a subset of a population that is selected and studied in order to make inferences or conclusions about the population. The sample is usually selected to be representative of the population in some way, so that the conclusions drawn from the sample can be generalized to the population as a whole.

The correct answer is C.

The purpose of the study is to determine why students who are interested in the college decide to enroll or not. Therefore, the population of interest is the group of students who submitted an application to the college in 2017.

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a) The drawing of all data points on a 2D plane is illustrated below.

b) The angle in between the five other points is 27

c) The matrix outer product for X is R = XXT.

d) The matrix outer product can also be calculated via R is (2,27)

The data matrix X is a 2-by-7 matrix, where each column represents a data point with two components. We can visualize each data point as a vector in a two-dimensional plane, where the horizontal axis represents the first component and the vertical axis represents the second component.

To draw all data points on a 2D plane, we can plot each column of X as a vector starting from the origin. Proper labeling of the two axes and providing important tick values will facilitate a precise graphical description of the data.

Next, we can use the inner product formula to calculate the angle between the first data point (2, 27) and the other five data points, respectively.

The inner product, also known as the dot product, is a way to measure the similarity between two vectors by multiplying their corresponding components and summing the products.

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The utility that maximizes the possible utility of both the agents Alice and Bob is equal to option D. $10.

Compare Alice's utility from each option and choose the one that maximizes her utility.

Let us consider each option,

If Alice offers Bob $0, her utility will be,

If XA > XB then

UA (XA , XB )= 0.3x3 + 0.72A

UA(10,0)

= 0.3(10)³ + 0.72(10)

= 307.2

If Alice offers Bob $5, Bob's utility will be,

UB(10,5)

= 0.314 + 0.72(5)

= 0.314 + 3.6

= 3.914

And Alice's utility will be,

UA(5,10)

= -0.32(5) + 1.3(10)

= 11.4

Total utility for both Alice and Bob will be,

UA(5,10) + UB(10,5)

= 11.4 + 3.914

= 15.314

If Alice offers Bob $2, Bob's utility will be,

UB(10,2)

= 0.314 + 0.72(2)

= 1.754

And Alice's utility will be,

UA(2,10)

= -0.32(2) + 1.3(10)

= 12.4

So the total utility for both Alice and Bob will be,

UA(2,10) + UB(10,2)

= 12.4 + 1.754

= 14.154

If Alice offers Bob $10, Bob's utility will be,

UB(10,10)

= 0.314 + 0.72(10)

= 7.514

And Alice's utility will be,

UA(10,10)

= 0.3(10)³ + 0.72(10)

= 307.2

So the total utility for both Alice and Bob will be,

UA(10,10) + UB(10,10)

= 307.2 + 7.514

= 314.714

Therefore, utility which maximizes the total utility for both Alice and Bob is given by option (D) $10.

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The correct standard form of the equation of the parabola is:

[tex]y = -x^2 - 1[/tex].

To find the standard form of the equation of the parabola that passes through the given points (-3, 3) and (1, -1), we can use the general form of the equation of a parabola:

[tex]y = ax^2 + bx + c[/tex] ___________(1)

Substituting the coordinates of the two given points into this equation, we get a system of two equations in three unknowns (a, b, and c):

[tex]3 = 9a - 3b + c[/tex]

[tex]-1 = a + b + c[/tex]

To solve for a, b, and c, we can eliminate one of the variables using subtraction or addition. Subtracting the second equation from the first, we get:

[tex]4 = 8a - 4b[/tex]

Simplifying this equation, we get:

[tex]2 = 4a - 2b[/tex]

Dividing both sides by 2, we get:

[tex]1 = 2a - b[/tex]___________(2)

Now we can substitute this expression for b into one of the earlier equations to eliminate b. Using the first equation, we get:

[tex]3 = 9a - 3(2a - 1) + c[/tex]

Simplifying this equation, we get:

[tex]3 = 6a + c + 3[/tex]

Subtracting 3 from both sides, we get:

[tex]0 = 6a + c[/tex]

Solving for c, we get:

c = -6a __________(3)

Substituting this expression for c into the second equation, we get:

[tex]-1 = a + (2a - 1) - 6a[/tex]

Simplifying this equation, we get:

[tex]-1 = -3a - 1[/tex]

Adding 1 to both sides, we get:

[tex]-3a =0[/tex]

Solving for a, we get:

[tex]a = 0[/tex]

Substituting this value of a into the equation(3) for c, we get:

c = 0

Substituting a = 0 into the equation(2) for b that we found earlier, we get:

[tex]1 = 0 - b[/tex]

Solving for b, we get:

[tex]b = -1[/tex]

Putting the values of a, b and c in (1), we get

[tex]y = -x^2 - 1[/tex]

Therefore, the equation of the parabola that passes through the given points (-3, 3) and (1, -1) is:

[tex]y = -x^2 - 1[/tex]

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Complete question:

A parabola goes through (-3, 3) & (1, -1). A point is below the parabola at (-3, 2). A line above the parabola goes through (-3, 4) & (0, 4). A point on the parabola is labeled (x, y).

What is the correct standard form of the equation of the parabola?

The figure is in the image attached below

Understanding the equilibria, sketching a phase portrait, and determining the stability of equilibria for autonomous equations are important tools for analyzing and understanding the behavior of systems over time.

Autonomous equations are differential equations that do not depend explicitly on time. To find the equilibria of an autonomous equation, we set the derivative of the function to zero and solve for the values of the independent variable that satisfy the equation. These values represent points at which the function does not change over time and are known as equilibrium points.

To sketch a phase portrait for an autonomous equation, we plot the slope field of the function and then draw solutions through each equilibrium point. The resulting graph shows the behavior of the function over time and helps us understand how the solutions behave near each equilibrium point.

The stability of an equilibrium point is determined by examining the behavior of nearby solutions. If nearby solutions move toward the equilibrium point over time, the equilibrium point is stable. If nearby solutions move away from the equilibrium point over time, the equilibrium point is unstable. Finally, if the behavior of nearby solutions is inconclusive, further analysis is needed.

Here is the sketch for [tex]dx/dt = x - x^3[/tex]

       / <--- (-∞)  x=-1  (+∞) ---> \

      /                                \

  <--0-->       x=-1       x=1        0-->

      \                                /

       \ <--- (-∞)  x=1   (+∞) --->  /

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Step-by-step explanation:

2x + ay + 2 = 0

the gradient or slope of the line is the factor m of x in the form

y = mx + b

so let's transform the given equation into that form :

ay = -2x - 2

y = (-2/a)x - 2/a

we know that m = 1.

so,

-2/a = 1

-2 = a

Answer:

The smallest possible average of five distinct positive even integers will occur if we choose the five smallest even integers. Since we want the integers to be distinct, we start with 2 and add the next four even integers:

2, 4, 6, 8, 10

The average of these five integers is:

(2 + 4 + 6 + 8 + 10) / 5 = 30 / 5 = 6

Therefore, the smallest possible average of five distinct positive even integers is 6, which is answer choice C.

Two radicals are said to be radicals if they have the same indices and the radicand.

In mathematics, a radicand is an expression, a number, or a variable that is enclosed in a root symbol. The factor for which we are determining the root is quantity. As we study exponents and roots, the word "radicand" is utilised. Therefore, the radicand in 2 has a value of 2. Following are some radicand examples:

3√(pq) → pq is the radicand

√(a+b) → a + b is the radicand

4√15 → 15 is the radicand

A radical is a symbol used to represent a number's root. It is symbolised as. The word or phrase that appears after the radical symbol is known as the radicand. Hence, it may be claimed that the radical represents the radicand. Alternatively said, the radicand sign is √.

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Complete question:

Type the correct words in the blanks:

Two radicals are said to be radicals if they have the same               and the radicand.

Step-by-step explanation:

35:12

40:17

45:22

50:27

55:32

Answer:

the answer to that is -31

Answer:

hey baby

Step-by-step explanation:

hi thwrw honey i love you lol

The operation we use to simplify a ratio after finding the greatest common factor (GCF) is division.

Option A is the correct answer.

An expression contains one or more terms with addition, subtraction, multiplication, and division.

We always combine the like terms in an expression when we simplify.

We also keep all the like terms on one side of the expression if we are dealing with two sides of an expression.

Example:

1 + 3x + 4y = 7 is an expression.

3 + 4 is an expression.

2 x 4 + 6 x 7 – 9 is an expression.

33 + 77 – 88 is an expression.

We have,

To simplify a ratio after finding the greatest common factor (GCF), we use division.

We divide both terms of the ratio by the GCF.

This reduces the ratio to its simplest form.

Thus,

The operation we use to simplify a ratio after finding the greatest common factor (GCF) is division.

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