Examples of geometric transformations can be found throughout the real world. Think about some places where you might use or se transformations. Give at least three examples for each type of transformation. Make use of the Internet, books, magazines, newspapers, and everyday life experiences to come up with your examples. ​

To find the volume of the solid generated by revolving the given region about the y-axis, we can use the method of cylindrical shells.

First, we need to sketch the region and the axis of rotation to visualize the solid. The region is a parabolic shape that extends from y = -4 to y = 4, and the axis of rotation is the y-axis.

Next, we need to set up the integral that represents the volume of the solid. We can slice the solid into thin cylindrical shells, each with radius r = x and height h = dy. The volume of each shell is given by:

dV = 2πrh dy

where the factor of 2π accounts for the full revolution around the y-axis. To express r in terms of y, we can solve the equation x = v5y2 for x:

x = v5y2
r = x = v5y2

Now we can integrate this expression for r over the range of y = -4 to y = 4:

V = ∫-4^4 2πr h dy
 = ∫-4^4 2π(v5y2)(dy)
 = 80πv5

Therefore, the volume of the solid generated by revolving the given region about the y-axis is 80πv5 cubic units.
To find the volume of the solid generated by revolving the region enclosed by x = √(5y²), x = 0, y = -4, and y = 4 about the y-axis, we can use the disk method.

The disk method involves integrating the area of each circular disk formed when the region is revolved around the y-axis. The area of each disk is A(y) = πR², where R is the radius of the disk.

In this case, the radius is the distance from the y-axis to the curve x = √(5y²), which is simply R(y) = √(5y²).

So the area of each disk is A(y) = π(√(5y²))² = 5πy²

Now, we can find the volume by integrating A(y) from y = -4 to y = 4:

Volume = ∫[A(y) dy] from -4 to 4 = ∫[5πy² dy] from -4 to 4

= 5π∫[y²2 dy] from -4 to 4

= 5π[(1/3)y³] from -4 to 4

= 5π[(1/3)(4³) - (1/3)(-4³)]

= 5π[(1/3)(64 + 64)]

= 5π[(1/3)(128)]

= (5/3)π(128)

= 213.67π cubic units

The volume of the solid generated by revolving the region enclosed by x = √(5y²), x = 0, y = -4, and y = 4 about the y-axis is approximately 213.67π cubic units.

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Answer:

a^4 - b^4

Step-by-step explanation:

a(a - b) + b(a - b) + (a - b)^2  -->  Given

a^2 - ab + b(a - b) + (a - b)^2  -->  Distributive Property

a^2 - ab + ba - b^2 + (a - b)^2  -->  Distributive Property

a^2 - ab + ba - b^2 + a^2 - b^2  -->  Distributive Property

a^4 - ab + ba - b^2 - b^2  -->  Combine Like Terms

a^4 + 0 - b^2 - b^2  -->  Combine Like Terms (a * b = b * a)

a^4 - b^4 -->  Combine Like Terms

The cost to alter each dress is $25 and each suit is $30 based on the given set of relations.

Let us represent the dresses as x and suit as y. Forming the equation for both customers.

Cost of one dress × number of dress +

Cost of one suit × number of suit = total cost

2x + y = 80 : equation 1

x + 3y = 115 : equation 2

Multiply equation with 1

6x + 3y = 240 : equation 3

Subtract equation 2 from equation 3

6x + 3y = 240

- x + 3y = 115

5x = 125

x = 125/5

x = $25

Keep the value of x in equation 2 to find the value of y

25 + 3y = 115

3y = 115 - 25

3y = 90

y = 90/3

y = $30

Hence, the altering cost of each is $25 and $30.

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The complete question is -

A tailor charges set amounts for alterations on dresses and suits. One customer has 2 dresses and 1 suit altered for a total of $80. Another customer has 1 dress and 3 suits altered for a total of $115. How much does it cost to alter each dress?

The slope of the line that is represented by the given data in the question is 3/4.

To find the slope of a line, we need to use the formula:

slope = (change in y) / (change in x)

We can choose any two points on the line and use their coordinates to calculate the change in y and the change in x. Let's choose the points (-9, 4) and (7, 16) from the given data.

Change in y = 16 - 4 = 12

Change in x = 7 - (-9) = 16

Plugging these values into the slope formula, we get:

slope = 12 / 16 = 3 / 4

We can also interpret this slope as the rate of change of y with respect to x. For every increase of 1 in x, y increases by 3/4. Similarly, for every decrease of 1 in x, y decreases by 3/4.

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To simplify the expressions and arrange the terms by their degree, we can write:

$\longrightarrow\sf\textbf\:f(x)\:= -x^4\:+\:2x^2\:-\:5x\:+\:3$

$\longrightarrow\sf\textbf\:=\:-x^4 + 2x^2 - x - 4x + 3$

$\longrightarrow\sf\textbf\:=\:-x(x^3 - 2x + 1) - 4x + 3$

$\longrightarrow\sf\textbf\:g(x)\:=\:x^4 + 4x^2 + 2x + 1$

$\longrightarrow\sf\textbf\:x^4 + 4x^2 + 2x + 1 - 2 + 2$

$\longrightarrow\sf\textbf\:x^4 + 4x^2 + 2x - 1 + 2$

$\longrightarrow\sf\textbf\:x^4 + 4x^2 + 2x - 1 + 2$

$\longrightarrow\sf\textbf\:x^4 + 4x^2 + 2x + 1 - 2$

$\longrightarrow\sf\textbf\:(x^2 + 1)^2 - 2$

Therefore, we can express the simplified forms of ${\sf{\textbf{f(x)}}}$ and ${\sf{\textbf{g(x)}}}$ as:

$\longrightarrow\sf\textbf\:f(x)\:=\:-x(x^3 - 2x + 1) - 4x + 3$

$\longrightarrow\sf\textbf\:g(x)\:=\:(x^2 + 1)^2 - 2$

[tex]\huge{\colorbox{black}{\textcolor{lime}{\textsf{\textbf{I\:hope\:this\:helps\:!}}}}}[/tex]

[tex]\begin{align}\colorbox{black}{\textcolor{white}{\underline{\underline{\sf{Please\: mark\: as\: brillinest !}}}}}\end{align}[/tex]

[tex]\textcolor{blue}{\small\textit{If you have any further questions, feel free to ask!}}[/tex]

[tex]{\bigstar{\underline{\boxed{\sf{\textbf{\color{red}{Sumit\:Roy}}}}}}}\\[/tex]

Pythagorean Theorem: a^2 + b^2 = c^2

---a and b are the legs of the triangle

---c is the hypotenuse/diagonal

a = 6

b = 1

c = ?

(6)^2 + (1)^2 = c^2

36 + 1 = c^2

37 = c^2

c = 6.0827

c (rounded) = 6.1

Answer = 6.1 meters

The value of the investment at the end of 6 years is $6,359.77 if the interest is compounded annually, $6,416.52 if it is compounded quarterly, and $6,437.70 if it is compounded monthly.

[tex]A = P(1 + r/n)^{nt}[/tex]

where A is the final amount, P is the initial investment, r is the annual interest rate, n is the number of times the interest is compounded per year, and t is the number of years.

A. If the interest is compounded annually, we have:

A = [tex]5,500(1 + 0.0155/1)^{1*6}[/tex] = $6,359.77

B. If the interest is compounded quarterly, we have:

A = [tex]5,500(1 + 0.0155/4)^{4*6[/tex] = $6,416.52

C. If the interest is compounded monthly, we have:

A = [tex]5,500(1 + 0.0155/12)^{12*6[/tex] = $6,437.70

Therefore, the value of the investment at the end of 6 years is $6,359.77 if the interest is compounded annually, $6,416.52 if it is compounded quarterly, and $6,437.70 if it is compounded monthly.

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The theoretical probability of rolling a one or two on a number cube is 2/5 or 0.4.

To find the theoretical probability of rolling a one or two on a number cube, we need to determine the number of outcomes that correspond to rolling a one or two, and divide that by the total number of possible outcomes.

From the table, we can see that Alexandria rolled a one or two a total of 24 times out of 60 rolls. This means that the probability of rolling a one or two is: P(1 or 2) = 24/60

Simplifying the fraction by dividing both the numerator and denominator by the greatest common factor, we get: P(1 or 2) = 4/10

This can be further reduced to: P(1 or 2) = 2/5

Therefore, the theoretical probability of rolling a one or two on a number cube is 2/5 or 0.4.

In summary, the theoretical probability is the expected probability of an event occurring, based on mathematical reasoning. Here, we used the number of favorable outcomes to calculate the probability of rolling a one or two, and expressed the answer as a fraction in simplest form.

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The ratio of hours spent at soccer practice to hours spent at a birthday party can be represented as 2 for every 3

To provide an accurate ratio, I would need the specific number of hours spent at both soccer practice and the birthday party.

Once you provide that information, you can create the ratio by putting the two numbers in the form 2 for every 3.

For example, if you spent 3 hours at soccer practice and 2 hours at a birthday party, the ratio would be 3:2. This means that for every 3 hours spent at soccer practice, you spent 2 hours at the birthday party.

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The table shows how many hours Sara spent at several activities one Saturday.

Activity "Hours

Soccer practice 2

Birthday party 3

Science project 1

What is the ratio of hours spent at soccer practice to hours spent at a birthday party?

Choose 1 answer:

1 for every 2

B

2 for every 1

2 for every 3

3 for every 2

Answer: 161.2

Step-by-step explanation:

Plug into formula

6.9² = 13²+17.8²-2(17.8)(13) cos C      >simplify numbers

-438.23 = -2(17.8)(13) cos C               >Divide both sides by  -2(17.8)(13)

cos C=.947                                         > use   [tex]cos^{-1}[/tex] C  to solve for angle

<C=180-18.75 = 161.2                          > neded to subtract from 180 for this                    

                                                                one

Answerx = 131.25

Step-by-step explanation:f x varies directly as T, then we can use the formula for direct variation:

x = kT

where k is the constant of proportionality.

To find k, we can use the given values:

x = 105 when T = 400

105 = k(400)

k = 105/400

k = 0.2625

Now that we have the value of k, we can use the formula to find x when T = 500:

x = kT

x = 0.2625(500)

x = 131.25

Therefore, when T = 500, x is equal to 131.25.

Answer:

0.80x400

Step-by-step explanation:

The given ordered pairs (4, 0.5), (2.5, 4.5), (0.5,3), and (2, 0) are plotted on the coordinate plane as shown in the graph below.

From the question, we are to plot the given ordered pairs on the coordinate plane

To plot the given ordered pairs, we will determine the location of the point on the coordinate plane

We will look at the first number in the ordered pair (the x-coordinate) and find that value on the x-axis. Also, we will look at the second number (the y-coordinate) and find that value on the y-axis.

Now, we will plot the point where the x-coordinate and y-coordinate intersect. The point is represented by a dot.

The ordered pairs are plotted on the coordinate plane as shown in the graph below.

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Using the linear model, we can predict that a 100 mg dose of the experimental drug would cause a change in the patient's blood pressure of 540 mmHg.

An equation is a mathematical statement that expresses the equality of two expressions. It consists of two expressions, one on the left side and one on the right side, which are connected by an equals sign (=). Equations are fundamental to mathematics, and are used to solve many problems. In addition, equations can also be used to describe physical laws, such as Newton's law of gravity.

10 + 2(20) + 3(30) + 4(100)

= 10 + 40 + 90 + 400

= 540 mmHg

The linear model suggests that a 100 mg dose of the experimental drug would cause a change in the patient's blood pressure of 540 mmHg. This can be seen by using the linear equation 10 + 2x + 3x + 4x. Here, the first coefficient of 10 represents the change in blood pressure for a 10 mg dose, the second coefficient of 2 represents the change in blood pressure for each additional 10 mg dose, the third coefficient of 3 represents the change in blood pressure for each additional 20 mg dose, and the fourth coefficient of 4 represents the change in blood pressure for each additional 30 mg dose.

For example, if the patient was given a 40 mg dose, the equation would be 10 + 2(20) + 3(30), which would yield a change in blood pressure of 140 mmHg. Similarly, if the patient was given a 50 mg dose, the equation would be 10 + 2(20) + 3(30) + 4(10), which would yield a change in blood pressure of 190 mmHg.

Therefore, using the linear model, we can predict that a 100 mg dose of the experimental drug would cause a change in the patient's blood pressure of 540 mmHg.

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The linear model predicts that a 100 mg dose of the investigational drug will raise the patient's heart rate by 540 mmHg.

A mathematical statement that expresses the equality of two expressions is known as an equation. It comprises of two expressions that are joined together by the equals sign (=), one on the left side and one on the right. Equations are essential to mathematics and are frequently used to resolve issues. Moreover, equations can be utilised to explain natural laws like Newton's law of gravity.

10 + 2(20) + 3(30) + 4(100)

= 10 + 40 + 90 + 400

= 540 mmHg

The linear model predicts that a 100 mg dose of the investigational drug will raise the patient's heart rate by 540 mmHg.

Using the linear equation 10 + 2x + 3x + 4x, this may be observed. In this case, the first coefficient of 10 denotes the change in blood pressure for a dose of 10 mg, the second coefficient of 2, the change for each additional dose of 10 mg, the third coefficient of 3, the change for each additional dose of 20 mg, and the fourth coefficient, the change for each additional dose of 30 mg.

For instance, if the patient received a dose of 40 mg, the equation would be 10 + 2(20) + 3(30), resulting in a 140 mmHg change in blood pressure. The calculation would be 10 + 2(20) + 3(30) + 4(10) if the patient received a 50 mg dose, which would result in a 190 mmHg change in blood pressure.

As a result, we can infer from the linear model that a 100 mg dose of the experimental medication would result in a 540 mmHg change in the patient's blood pressure.

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Without a specific set of scatterplots to examine, I can provide some general guidelines for labeling scatterplots based on their association:

1. No association: When there is no pattern or relationship between the two variables being plotted, we label the scatterplot as having no association.

2. Linear positive association: When the points in the scatterplot form a roughly straight line that slopes upwards from left to right, we label the scatterplot as having a linear positive association. This means that as the value of one variable increases, the value of the other variable also tends to increase.

3. Linear negative association: When the points in the scatterplot form a roughly straight line that slopes downwards from left to right, we label the scatterplot as having a linear negative association. This means that as the value of one variable increases, the value of the other variable tends to decrease.

4. Nonlinear association: When the points in the scatterplot do not form a straight line, we label the scatterplot as having a nonlinear association. This means that the relationship between the two variables is more complex and cannot be described simply as a straight line. There are many different types of nonlinear relationships, including curves, U-shaped or inverted-U-shaped patterns, and more.

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To find the values of a and b, we need to ensure that the function is differentiable at x = 1. Thus, the function defined by f(x) = x^2 + 3x + a, x <= 1 and bx + 2, x >= 1 differentiable at x = 1 are a = 3 and b = 5.

First, we need to check that the function is continuous at x = 1. Since the function has different definitions for x <= 1 and x >= 1, we need to check that the limit of the function as x approaches 1 from both sides is the same.

Limit as x approaches 1 from the left (x <= 1):

f(x) = x^2 + 3x + a

lim x->1- f(x) = lim x->1- (x^2 + 3x + a) = 1^2 + 3(1) + a = 4 + a

Limit as x approaches 1 from the right (x >= 1):

f(x) = bx + 2

lim x->1+ f(x) = lim x->1+ (bx + 2) = b + 2

For the function to be continuous at x = 1, these two limits must be equal.

4 + a = b + 2

a = b - 2

Now we need to check that the derivative of the function at x = 1 exists and is equal from both sides.

Derivative of the function for x <= 1:

f(x) = x^2 + 3x + a

f'(x) = 2x + 3

f'(1) = 2(1) + 3 = 5

Derivative of the function for x >= 1:

f(x) = bx + 2

f'(x) = b

f'(1) = b

For the function to be differentiable at x = 1, these two derivatives must be equal.

5 = b

Substituting b = 5 into the equation we found earlier for a, we get:

a = 5 - 2 = 3

Therefore, the values of a and b that make the function defined by f(x) = x^2 + 3x + a, x <= 1 and bx + 2, x >= 1 differentiable at x = 1 are a = 3 and b = 5.
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The perimeter and area of the shape is 18cm² and 12cm respectively.

The perimeter of a shape is the total measurement of all the edges of a shape. Area is defined as the total space taken up by a flat (2-D) surface or shape of an object.

The perimeter of the shape = 4+4+5+5 = 8 +10

= 18cm.

The area of the shape is = b×h

the base = 4cm and height is 3cm

A = 4× 3

= 12cm²

therefore the perimeter and the area of the shape is 18cm² and 12cm² respectively.

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Answer:

To estimate the positive solution to the equation x⁴ – x = 3 using Newton's Method, we can start by taking the derivative of the equation, which is 4x³ - 1. Then we can use the formula X1 = X0 - f(X0) / f'(X0), where X0 = 1, f(X0) = 1⁴ - 1 - 3 = -3, and f'(X0) = 4(1)³ - 1 = 3. Plugging these values into the formula, we get:

X1 = 1 - (-3) / 3

X1 = 2

Now we can repeat the process using X1 as our new X0:

X2 = X1 - f(X1) / f'(X1)

X2 = 2 - (2⁴ - 2 - 3) / (4(2)³ - 1)

X2 ≈ 1.7708

Therefore, the positive solution to the equation x⁴ – x = 3, rounded to four decimal places, is approximately 1.7708.

Step-by-step explanation:

The positive solution to the equation x⁴ – x = 3, estimated using Newton's Method with x₀ = 1 and x₂ as the final estimate, is approximately 1.5329, rounded to four decimal places.

To use Newton's Method to estimate the positive solution to the equation x⁴ – x = 3, we need to find the derivative of the function f(x) = x⁴ – x. This is given by:

f'(x) = 4x³ - 1

We can then use the formula for Newton's Method:

x(n+1) = x(n) - f(x(n)) / f'(x(n))

where x(n) is the nth estimate of the solution.

Starting with x₀ = 1, we can plug this into the formula to get:

x₁ = 1 - (1^4 - 1 - 3) / (4(1^3) - 1) ≈ 1.75

We can then repeat this process using x₁ as the new estimate, to get:

x₂ = 1.75 - (1.75^4 - 1.75 - 3) / (4(1.75^3) - 1) ≈ 1.5329

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The equation that can be used to determine the number of people (x) who can go to the amusement park is:

207.50 = 5 + (33.75 × x).

To determine the number of people (x) who can go to the amusement park, we need to create an equation using the given information. We know that they have $207.50 to spend, parking costs $5, and each ticket costs $33.75 per person (including tax).

We can represent the total cost of the trip as the sum of the cost for parking and the cost of the tickets for x number of people. The equation would be:

Total Cost = Cost of Parking + (Cost of Tickets per Person × Number of People)

Since we know the total cost is $207.50, the cost of parking is $5, and the cost of tickets per person is $33.75, we can plug in these values:

207.50 = 5 + (33.75 × x)

This equation can be used to determine the value of the variable x, the number of people who can go to the amusement park. To find the value of x, simply solve the equation by isolating the variable x.

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The radius of the lake is approximately 17.89 feet.

To find the radius of the lake, we can use the information given and apply the properties of tangents to circles.

Since point B is the point of tangency, the line segment AB is tangent to the circle. A radius drawn to the point of tangency, in this case from the center of the lake (point O) to point B, will be perpendicular to the tangent line (line AB).

Now, let's use the given measurements. The distance from the duck (point C) to the point of tangency (point B) is 16 feet, and the distance from the duck (point C) to the edge of the lake in the direction of A (line AC) is 8 feet. We can form a right-angled triangle OBC with the given information.

Since OB is perpendicular to AB, we have a right-angled triangle with legs CB and OC. Using the Pythagorean theorem, we can find the length of the hypotenuse, which is the radius of the lake:

OC^2 + CB^2 = OB^2
(8 feet)^2 + (16 feet)^2 = OB^2
64 + 256 = OB^2
320 = OB^2
OB = √320
OB ≈ 17.89 feet

So, the radius of the lake is approximately 17.89 feet.

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The answer is not 9 > 6, as that is a comparison between two numbers, but rather g ≤ 6, as it sets a limit on the number of goldfish that can be in the tank.

The correct inequality that represents the situation is g ≤ 6. The problem states that the maximum number of goldfish that can live in a small tank is 6, meaning that the number of goldfish must be less than or equal to 6.

The symbol ≤ represents "less than or equal to", while the symbol > represents "greater than".

Therefore, the answer is not 9 > 6, as that is a comparison between two numbers, but rather g ≤ 6, as it sets a limit on the number of goldfish that can be in the tank.

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A. The proportion we can set up is: c/a = d/b, and B. x = (c * b) / a. This gives us the value of x in terms of the lengths of the other segments.

A) The corresponding sides in similar triangles are proportional, so we can use this fact to set up a proportion between the sides of triangles BED and BCA. Let's call the length of segment BC "a", the length of segment AC "b", the length of segment BE "c", and the length of segment DE "d".
The proportion we can set up is:
c/a = d/b
This is because we know that triangle BED is similar to triangle BCA, so the ratio of the lengths of their corresponding sides must be the same.

B) We can now use the proportion to solve for x, which is the length of segment DE. We can start by cross-multiplying the proportion:
c * b = d * a
Then, we can isolate for x by dividing both sides by the coefficient of x:
x = (c * b) / a
This gives us the value of x in terms of the lengths of the other segments.

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The temperature of City A and City B on that day was 4°C and 3°C respectively.

Let x represent the temperature of city A and y represent the temperature of city B.

Four times the temperature of City A was 7° C more than three times the temperature of City B, hence:

4x = 3y + 7

4x - 3y = 7       (1)

The temperature of City A minus three times the temperature of City B was −5° C, hence:

x - 3y = -5        (2)

From both equations, solving simultaneously:

x = 4, y = 3.

The temperature of City A and City B on that day was 4°C and 3°C respectively.

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using the given similar triangles, the length of the lake is approximately 26.05 meters.

We have,

In the given similar triangles, we have the following information:

Length of the longer side of the larger triangle: AC = 215m

Length of the longer side of the smaller triangle: A'C = 50m

Length of the corresponding shorter side of the smaller triangle: A'B = 112m

Let's denote the length of the lake (the longer side of the smaller triangle) as x.

Now, we can set up a proportion between the sides of the two triangles:

AC / A'C = A'B / x

Substitute the given values:

215 / 50 = 112 / x

Now, solve for x:

215x = 50 * 112

Divide both sides by 215:

x = (50 * 112) / 215

x ≈ 26.05

Thus,

The length of the lake is approximately 26.05 meters.

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The complete question:

To measure the length of a lake, the following similar triangles were built, in which it is necessary to: AC = 215m, A'C= 50m, A'B=112m. What is the length of the lake?

Answer:

x=137

Step-by-step explanation:

sqrt(x+7)=12

x+7=144

x=137

Answer:

Step-by-step explanation:

we can use the laws of exponents, which state that when multiplying terms with the same base, we add their exponents. In this case,both terms have a base of 8, so we can add their exponents of 2/7 and 1/4.

First, let's write 8 as a power of 2: 8 = 2^3. Then we can rewrite the original expression as (2^3)^(2/7) * (2^3)^(1/4). Using the power of a power rule, we can simplify this to 2^(3 * 2/7) * 2^(3 * 1/4).

Next, we can simplify the exponents by finding a common denominator. The smallest common multiple of 7 and 4 is 28, so we can rewrite the exponents as 6/28 and 21/28, respectively. Thus, we have 2^(3 * 6/28) * 2^(3 * 21/28).

Now we can simplify the exponents by multiplying the bases and exponents separately: 2^(18/28) * 2^(63/28). We can simplify the fractions by dividing both the numerator and denominator by 2, giving us 2^(9/14) * 2^(63/28).

Finally, we can add the exponents since we are multiplying terms with the same base: 2^(9/14 + 63/28). We can simplify the exponent by finding a common denominator of 28,

giving us 2^(36/28 + 63/28) = 2^(99/28). This is our final answer, which is an irrational number that is approximately equal to 69.887.

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Answer:

1,400

Step-by-step explanation:

its the same thing as 7 times 200

To draw a line segment with endpoint at 1.6 and length 1.2, draw a number line and mark 1.6. Measure 1.2 units to left of 1.6 and mark the starting point. Connect the starting and endpoint.

To draw a line segment with an endpoint at 1.6 and a length of 1.2, we can follow these steps

Draw a number line and mark the point 1.6.

From the point 1.6, measure a distance of 1.2 units in the direction of the negative numbers.

Mark the endpoint of the line segment at the point where the distance of 1.2 units ends.

Draw the line segment connecting the endpoint at 1.6 to the starting point.

In the diagram, the starting point is marked with at 0.4, and the endpoint is marked with at 1.6, which is 1.2 units away from the starting point. The line segment connecting the starting point to the endpoint is shown as a horizontal line.

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The number of monkeys when number of monkeys varies inversely to the number of clowns is 45.  

Inverse proportion is a mathematical relationship between two variables, in which an increase in one variable causes a proportional decrease in the other variable, and a decrease in one variable causes a proportional increase in the other variable.

In other words, the two variables vary in such a way that their product remains constant.

Here we have

10 monkeys vary inversely when there are 18 clowns.

We can set up the inverse variation equation as:

=> monkey ∝ 1/clown

If k is the constant of proportionality.

=> Monkey (clown) = k

It is given that when there are 10 monkeys, there are 18 clowns, so we can write:

=> (10)(18) = k

Solving for k, we get:

k = (10 x 18) = 180

Now we can use this value of k to find the number of monkeys when there are 4 clowns:

=> monkey = k/clown = 180/4 = 45

Therefore,

The number of monkeys when number of monkeys varies inversely to the number of clowns is 45.  

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