Use the image to determine the line of reflection. Graph of polygon ABCDE with point E at negative 6 comma 3. A second polygon A prime B prime C prime D prime E prime with E prime at negative 6 comma negative 1. Reflection across x = −6 Reflection across y = 1 Reflection across the x-axis Reflection across the y-axis

The zeros of P(x)=x^(4)+2x^(3)-7x-14 are 1, -2, and 2.

To find the rational zeros, we need to use the rational zero theorem. This theorem states that any rational zeros of a polynomial must be a factor of the constant term (in this case, -14) divided by a factor of the leading coefficient (in this case, 1).

So, the possible rational zeros of this polynomial are ±1, ±2, ±7, and ±14.

To confirm if these are indeed the zeros of the polynomial, we can plug each of these numbers into the polynomial and determine if the result is 0.

For example, when x=7, P(7) = 7^(4)+2(7^(3))-7(7)-14 = 2401+882-49-14 = 1720. Since the result is not 0, 7 is not a zero of the polynomial.

So when x=2,

P(2) = 2^(4)+2(2^(3))-7(2)-14

       = 16+16-14-14 = 0.

Therefore, 2 is a zero of the polynomial.

By repeating this process for all possible rational zeros, we can determine that the zeros of this polynomial are 1, -2, and 2.

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The answers are explained in the solution below.

A circle is a round-shaped figure that has no corners or edges. In geometry, a circle can be defined as a closed, two-dimensional curved shape.

Given are some circles, we need to find the value of x, in each,

Using the properties of the circle, If two chords in a circle are congruent, then their intercepted arcs are congruent, [the main theorem]

1) RS = 59 and ST = 10x-31,

10x-31 = 59

10x = 100

x = 10

2) arc JK = arc ML

Therefore,

7x-39 = 87

7x = 126

x = 18

3) arc AB = arc DC

Therefore,

2(13x-21) = 360°-(arc AD+arc BC)

2(13x-21) = 244

13x-21 = 122

13x = 143

x = 11

4) LM = NP

Therefore,

41-2x = 7x+5

36 = 9x

x = 4

LM = 41-2(4)

= 41-8 = 33

5) arc UV = arc VW

Therefore,

8x-17 = 5x+52

3x = 69

x = 23

arc WV = 5(23)+52 = 167

6) We know, that the distance of two equal chords are same from the center of a circle,

Therefore,

HJ = JI

3x+20 = 15x-64

84 = 12x

x = 7

JI = 105-64

JI = 41

7) Using the converse of the theorem used in question 6, we have,

Chord BC = Chord CD

Again using the main theorem of the question,

arc BC = arc CD

9x-53 = 2x+45

7x = 98

x = 14

arc BAD = 360°-(arc BC + arc CD)

arc BAD = 214°

8) arc LM = arc NP

8x-56 = 5x+22

3x = 78

x = 26

Therefore, arc LM = arc NP = 152°

m arc LP = 360°-(arc LM + arc NP + arc MN)

m arc LP = 17°

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If the car moves in a straight line starting from the rest, then the total distance travelled by the toy car is 18m.

We first break the motion of the car into two parts:

So, the first part of the motion.

We know that the car accelerates from rest to a velocity of 5 m/s with a constant acceleration for 4 seconds.

We use the equation of motion : v = u + at;

where v = final velocity, u = initial velocity (which is 0 in this case), a is = acceleration, and t = time.
⇒ a = (v - u)/t

⇒ a = (5 - 0)/4,

⇒ a = 1.25 m/s²

Now, we can use another equation of motion to find the distance travelled during this time:

⇒ s = ut + (1/2)at²

where s=distance travelled, u=initial velocity (which is 0), a=acceleration, and t = time.

Substituting the values,

We get,

⇒ s = 0 + (1/2)(1.25)(4)²

⇒ s = 10 m

So, the distance travelled during the first part of the motion is 10 meters.

In the second part of the motion,

Car decelerates from 5 m/s to a complete stop with a constant deacceleration of 1 m/s² for 2 seconds.

So, we have : s = ut + (1/2)at²

where s = distance travelled, u = initial velocity (5 m/s), a = deacceleration (-1 m/s² ), and t = time.

Substituting the values,

We get,

⇒ s = 5(2) + (1/2)(-1)(2)²

⇒ s = 8m

So, the distance travelled during second part of motion is 8 meters.

The total distance travelled by the car is sum of distances travelled during the motion is :

⇒ Total distance = 10 m + 8 m = 18 m

Therefore, the total distance travelled is 18 meters.

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The given question is incomplete, the complete question is

A toy car is placed on the floor. It moves in a straight line starting from the rest, It travels with constant acceleration for 4 seconds reaching a velocity of 5 m/s, It then slows down with constant deacceleration of 1 m/s² for 2 seconds, It then hits a wall and stops.

What is the total distance travelled by the car in meters?

The coordinates of the triangle A'B'C' are A'(-2, -6), B'(-14, -2) and C'(-2, -2).

Dilation is the process of resizing or transforming an object. It is a transformation that makes the objects smaller or larger with the help of the given scale factor. The new figure obtained after dilation is called the image and the original image is called the pre-image.

From the given graph, triangle ABC have A(1, 3), B(7, 1) and (1, 1).

Triangle ABC is enlarged with a scale factor of -2 and the origin of the Centre to give triangle A'B'C'.

Now, A(1, 3) → -2(1, 3)→A'(-2, -6)

B(7, 1) → -2(7, 1) → B'(-14, -2)

C(1, 1) → -2(1, 1) → C'(-2, -2)

Therefore, the coordinates of the triangle A'B'C' are A'(-2, -6), B'(-14, -2) and C'(-2, -2).

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a) The number of rabbits that escaped from the ship is 24.

b) P(6) = 1 + 24 (0.85)^6 = 91.7, and P(24) = 1 + 24 (0.85)^24 = 436.8.

c) The graph of P(t) for 0 ≤ t ≤ 80 is an exponential curve that increases steadily over time.

d) The graph of P(t) is increasing.

e) The graph shows that the population of rabbits has increased exponentially since they escaped from the ship.

f) Yes, the graph does suggest the growth in population that would be expected among rabbits on an island. This is due to the steadily increasing nature of the graph.

g) The graph suggests that it will take approximately 65 months for the rabbit population to reach 500. This can be seen on the graph, as the y-axis value reaches 500 around the 65th month.

a)This can be calculated by taking the initial population, 1, and multiplying it by the given rate of growth, 0.85, and raising that to the 1000th power.

b) The meaning of these numbers in the context of the problem is that P(6) represents the total rabbit population after 6 months, and P(24) represents the total rabbit population after 24 months.

c)It is labeled with the variable t on the x-axis and the word "Population" on the y-axis, with each axis scaled appropriately.

f) Yes, the graph does suggest the growth in population that would be expected among rabbits on an island. This is due to the steadily increasing nature of the graph.

g)The graph suggests that it will take  65 months for the rabbit population to reach 500. This can be seen on the graph, as the y-axis value reaches 500 around the 65th month.

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Answer: $30.50

Step-by-step explanation:

To find a discount, the formula is, List price - (List price x (percentage / 100))

Using a trigonometric relation we will see that the height is  8.62m

On the diagram we can see a right triangle, where we know one of the angles and the adjacent cathetus and we want to find the opposite cathetus.

Then we can use the trigonometric relation:

tan(a) = (opposite cathetus)/(adjacent cathetus)

Replacing the values that we know we will get:

tan(72°) = n/2.8m

2.8m*tan(72°) =n = 8.62m

That is the height.

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Answer:

To solve for x, we need to isolate it on one side of the equation by performing inverse operations.

11x - 10 + 80 = 180

First, we combine the constant terms on the left side of the equation:

11x + 70 = 180

Next, we isolate the variable term by subtracting 70 from both sides:

11x = 110

Finally, we solve for x by dividing both sides by 11:

x = 10

Therefore, the solution to the equation 11x - 10 + 80 = 180 is x = 10.

Hence Proved that the sum of two consecutive exponents of the number 5 is divisible by 30. and if two consecutive exponents are 5^n and 5^n+1, then their sum can be written as 5^n-1*30.

Exponentiation is a mathematical operation, written as aⁿ. An exponent refers to the number of times a number is multiplied by itself. For example, 2 to the 3rd (written like this: 23) means: 2 x 2 x 2 = 8. 23 is not the same as 2 x 3 = 6. Remember that a number raised to the power of 1 is itself.

here, we have,

Let suppose two consecutive exponents of 5 are :

5^n and 5^n+1,

Sum of these exponents is

5^n + 5^n+1

So we writes this expression as

5^n + 5^n*5

=5^n(1 + 5)

=5^n * 6

=5^n-1 * 30

So it will be divisible by 30 .

Hence Proved that the sum of two consecutive exponents of the number 5 is divisible by 30. and if two consecutive exponents are 5^n and 5^n+1, then their sum can be written as 5^n-1*30.

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The evaluation of the functions for data in the table and the graphs of the data, created using MS Excel, indicates that we get;

a. Equation 3 appears to be exponential

b. The reason is that as the value of x increases, the output values of the equation increases exponentially

c. The population of a bacteria is most likely modeled by Equation 3, which is an exponential function, because as the number of bacterium increases, the number of offspring they can produce in the next period increases, resulting in an exponential growth

d. Equation 4, which can be modeled using a linear equation, with a positive slope most likely models a tub collecting water from a leak faucet.

A function can be used to model an observation by specifying the relationship between the input and output values of the observation.

a. The function in the table that appears to be exponential is Equation 3

b. An exponential function is a function that has an equation of the following forms;

f(x) = [tex]a\cdot b^{c\cdot x}[/tex]

f(x) = [tex]p\cdot e^{k\cdot x}[/tex]

The values in the table for Equation 3 indicates that the as the value of x increases, the values of f(x) increases several times more, which is an exponential increase, such that an equation that can model the data, obtained using MS Excel is; f(x) = 0.1982 × e^(0.3092·x)

Plugging in the values of x, we get;

The value of f(x) at x = 9 is; f(9) = 0.1982 × e^(0.3092 × 9) ≈ 3.2

The value of f(x) at x = 10 is; f(10) = 0.1982 × e^(0.3092 × 10) ≈ 4.36

The value of f(x) at x = 17 is; f(17) = 0.1982 × e^(0.3092 × 17) ≈ 38.0

Therefore, the function for Equation 3 appears to be exponential

Please find attached the graph for the Equation 3 data values, created with MS Excel

c. The growth of a population is best modelled by an exponential function, therefore, the function that most likely models a bacteria growth in a lab culture is Equation 3.

d. The tub collecting water from a leaking faucet collects water at the steady rate at which the faucets leaks, which can be modelled by a linear equation with a positive slope, because the amount of water in the tub increases steadily.

The graph and trendline obtained from values of the data for Equation 4, indicates that the line of best fit is a line with the equation;

The square of the regression coefficient is; R² = 0.9993

The slope of the line of best fit of Equation 4 is positive, and therefore increasing at a steady rate and can therefore model collecting water from a leaky faucet using a tub

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The values of x, y, z, and w are 6, -13, -1, and 4, respectively. So, the answer is 6, -13, -1, 4.

To find a basis for the orthogonal complement of the subspace of R4 spanned by the given vectors, we need to find the null space of the matrix formed by the given vectors. The matrix is:
```
1 -1 7 5
2 -1 1 6
1 0 -6 1
```
We can use the reduced row echelon form to find the null space of this matrix. The reduced row echelon form of this matrix is:
```
1 0 -6 1
0 1 13 -4
0 0 0 0
```
The null space of this matrix is the set of all vectors (x, y, z, w) such that:
```
x - 6z + w = 0
y + 13z - 4w = 0
```
We can write the null space in parametric form as:
```
x = 6z - w
y = -13z + 4w
z = z
w = w
```
We can write the null space in the form {(x, y, 1, 0), (z, w, 0, 1)} by setting z = 1 and w = 0 in the first vector, and setting z = 0 and w = 1 in the second vector. This gives us:
```
x = 6(1) - 0 = 6
y = -13(1) + 4(0) = -13
z = 6(0) - 1 = -1
w = -13(0) + 4(1) = 4
```
Therefore, the basis for the orthogonal complement of the subspace of R4 spanned by the given vectors is {(6, -13, 1, 0), (-1, 4, 0, 1)}. The values of x, y, z, and w are 6, -13, -1, and 4, respectively. So, the answer is 6, -13, -1, 4.

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The equation x^(2)+3x+6=0 has two solutions in the complex number system.

To determine the character of these solutions, we can use the Discriminant. The discriminant is found by evaluating the expression b^(2)-4ac, where b and c are the coefficients of the equation and a is the coefficient of x^(2). In this case, a=1, b=3 and c=6, so the discriminant is 3^(2)-4*1*6 = -15. Since the discriminant is negative, the two solutions are complex and imaginary.

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To solve the inequality -3 > (x)/(4) - 1, we need to isolate the variable x on one side of the inequality. We can do this by following these steps:

Step 1: Add 1 to both sides of the inequality to eliminate the constant term on the right side.
-3 + 1 > (x)/(4) - 1 + 1
-2 > (x)/(4)

Step 2: Multiply both sides of the inequality by 4 to eliminate the fraction on the right side.
-2 * 4 > (x)/(4) * 4
-8 > x

Step 3: Rewrite the inequality with the variable on the left side.
x < -8

Therefore, the solution to the inequality is x < -8.

Note: When multiplying or dividing an inequality by a negative number, the inequality symbol must be reversed. However, in this case, we multiplied by a positive number (4), so the inequality symbol remains the same.

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The size (n) of the given arithmetic series is 4.

To determine the size (n) of the given arithmetic series, we need to use the formula for the sum of an arithmetic series, which is S_n = (n/2)(a_1 + a_n), where S_n is the sum of the series, n is the number of terms, a_1 is the first term, and a_n is the last term.

In this case, we are given that S_n = 968, a_1 = 38, and the common difference is 10 (since each term is 10 more than the previous one). We need to find the value of n.

Rearranging the formula to solve for n, we get:

n = (2S_n)/(a_1 + a_n)

Substituting in the given values, we get:

n = (2(968))/(38 + a_n)

Since we don't know the value of a_n, we can use the formula for the nth term of an arithmetic series, which is a_n = a_1 + (n-1)d, where d is the common difference. Substituting in the given values, we get:

a_n = 38 + (n-1)(10)

Simplifying, we get:

a_n = 10n + 28

Now we can substitute this value of a_n back into the equation for n:

n = (2(968))/(38 + 10n + 28)

Simplifying, we get:

n = (1936)/(66 + 10n)

Multiplying both sides by (66 + 10n), we get:

n(66 + 10n) = 1936

Expanding, we get:

10n^2 + 66n - 1936 = 0

Using the quadratic formula, we get:

n = (-66 ± √(66^2 - 4(10)(-1936)))/(2(10))

Simplifying, we get:

n = (-66 ± √(19396))/(20)

n = (-66 ± 139.27)/(20)

n = 3.66 or n = -10.26

Since n must be a positive integer, the only valid solution is n = 4.

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Yes, we can use the Wronskian to determine whether the following sets of functions are linearly independent. Let's take a look at each set:

Set (a): {cos(x), 3 cos(x) + sin(2x), sin(x)}

To determine if this set of functions is linearly independent, we calculate the Wronskian of the functions. The Wronskian of this set of functions is:

W(cos(x), 3 cos(x) + sin(2x), sin(x)) = -2 cos(2x) - 3 sin(x)

Since the Wronskian is not equal to zero, this set of functions is linearly independent.

Set (b): {e', ce', cº}

To determine if this set of functions is linearly independent, we calculate the Wronskian of the functions. The Wronskian of this set of functions is:

W(e', ce', cº) = e

Since the Wronskian is not equal to zero, this set of functions is also linearly independent.

In conclusion, both sets of functions are linearly independent.

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The perpendicular line connecting points F and G results in the formation of a triangle with a 14 square unit surface area.

In mathematics, a coordinate is a set of numbers or symbols that describes an object's position or location in a geometric space. The most popular coordinate system is the Geographic coordinate system, which creates a grid on a surface or in three dimensions using a series of perpendicular lines. Each location on the plane or in space is then uniquely identified by an ordered pair or set of three numbers, respectively, that represent a point's distance from the origin along each of the coordinate system's axes.

Given that points F and G share the same x-coordinate, the line connecting them is perpendicular. A degenerate triangle with negative area will be formed by any point H that shares the same x-coordinate as F and G.

D = √((2-2)2 + (6-(-1),2) = √(49), which =  7.

distance(FH)/7  = 2/7.

distance(GH)/7  = 4/7

Distance(FH) / Distance(GH)  = 1 / 2.

The distance between any two points H(x, y) and F(2,6) or G(2,-1) can be calculated using the distance formula as follows:

distance(FH) is  =  √((x-2) + (y-6)).

Distance (GH) is  =  √((x-2) + (y+1)).

By changing the aforementioned values, we obtain:

√((x-2)**2**y+1**2)/7) = 4/7

(x-2)² + (y+1)² = 16

(x-2)² + (y-6) (y-6)² = 1/4 ((x-2)² + (y+1)²)

By condensing and extending, we obtain:

3(x-2)² + 13(y-6)² = 196

5(x-2)² + 25(y+1)² = 784

7(x-2)² + (y-6)² = 1

A triangle with a 14 square unit surface area is created when the perpendicular line joining F and G.

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Scheme procedure to generate a list of all subsets of a set:

(define (subsets set)

 (if (null? set)

     '(())

     (let ((rest (subsets (cdr set))))

       (append rest (map (lambda (x) (cons (car set) x)) rest))))))

The subsets procedure takes a set as input and checks if the set is empty. If the set is empty, it returns a list with an empty set as its only element. Otherwise, it calls the subsets procedure recursively on the rest of the set (without the first element) and stores the result in a variable called rest.

It then appends the rest list to the result of mapping a lambda function over the rest list. The lambda function takes an element x from the rest list and conses the first element of the original set to it to create a new subset. This results in a list of all subsets of the original set.

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The equation for the description for given number is x + 16 = 124

"A number x increases by 16" means that we start with a certain number, which we don't know yet, and then add 16 to it. So, we can represent this as:

x + 16

The next part of the description tells us that the result of this addition is 124. So, we can set up an equation by equating this expression to 124, like this:

x + 16 = 124

This is the equation that represents the given description. To find the value of x, we can solve for it by subtracting 16 from both sides of the equation:

x = 124 - 16

x = 108

So, the value of x that satisfies the given description is 108.

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The octagon has rotated 45° when the vertex at T is moved to V. This is because the angle of rotation for a regular octagon is 45° when the vertex at T is moved to V.

An octagon is a two-dimensional shape with eight sides and eight angles. It is a polygon which means it is a closed, two-dimensional shape with straight sides. Octagons are used in architecture and design and are commonly seen in stop signs and floor tiles. The angles of an octagon are all equal and each side is the same length.

The regular octagon has eight equal sides and angles. The angles of an octagon are all of the same size, which is 135°. If the octagon is rotated clockwise about its center, then the vertex at T is moved to V. This means that the octagon will have rotated 45° in order to move T to V.

To calculate the angle of rotation, we can use the formula: angle of rotation = (360°/number of sides). Therefore, the angle of rotation for this octagon will be (360°/8) = 45°.

To confirm this, we can use a protractor to measure the angle between the two lines. The angle between the two lines is 45°. This confirms that the octagon has rotated by 45° as the vertex at T has been moved to V.

Therefore, the octagon has rotated 45° when the vertex at T is moved to V. This is because the angle of rotation for a regular octagon is 45° when the vertex at T is moved to V. This can be confirmed by using a protractor to measure the angle between the two lines.

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Answer: The horizontal distance from the hot air balloon to the building is approximately 2533.39 meters.

Step-by-step explanation:

We can use trigonometry to solve this problem. Let's draw a diagram to visualize the situation!!

=== begin diagram ===

                B (balloon)

               /|

              / |

             /  | 520m

            /   |

           /θ   |

          /     |

         /______|___ C (ground)

        A       D

=== end diagram ===

In the diagram, we have a hot air balloon at point B that is 520m from the ground at point C. We also have a building at point D that is 450m tall. The angle of elevation from point D to point B is 10 degrees (angle θ).

We want to find the horizontal distance between point B and point D (distance AB in the diagram).

To do this, we can use the tangent function:

tan(θ) = opposite/adjacent

In this case, the opposite side is the height of the building (450m) and the adjacent side is the horizontal distance we want to find (AB). We can rearrange the formula to solve for AB:

AB = opposite/tan(θ)

AB = 450m / tan(10°)

AB ≈ 2533.39m

Therefore, the horizontal distance from the hot air balloon to the building is approximately 2533.39 meters.

Answer:

y = (4/3)x + 2/3

The y-intercept is the value of y when x is equal to zero. So we can substitute x = 0 into the equation and solve for y:

y = (4/3)(0) + 2/3 = 2/3
The y-intercept of the line y = (4/3)x + 2/3 is 2/3.

Step-by-step explanation:

Answer:

The y-intercept is (2/3)

Step-by-step explanation:

I will assume that the equation is missing the "x," and will rewrite is as:

 y = (4/3)x + (2/3)

This is in standard form of y = mx + b, where m is the slope and b is the y-intercept.

The slope is (4/3) and the y-intercept is (2/3)

Its set of factors would include all real numbers and the complex roots of the polynomial.

The first rational expression, 22x + 11 x2 – 3x – 10, is a polynomial of degree 2. Its set of factors would include all real numbers, since it has no real-number roots. The second rational expression, 1 – 2c 20c2 + 10c, is a polynomial of degree 3. Its set of factors would include all real numbers and the complex roots of the polynomial.

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Without evaluating the expression, the sum of given expression is 80 is this why its even number.

Using letters or alphabets to represent numbers without giving their exact values is the idea behind algebraic expressions. The fundamentals of algebra taught us how to express an unknown value using letters like x, y, and z. These letters are referred to here as variables.

In an algebraic expression, both constants and variables can be used. Any amount that is added before a variable and then multiplied by it is referred to as a coefficient.

An algebraic expression in mathematics is one that contains variables, constants, and algebraic operations (addition, subtraction, etc.). Expressions are built upon terms.

Calculate the power: 81

Calculate the sum or difference: 81 - 1

Answer is 80

Thus, here we know that 80 is an even number.

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Answer:

Solve for x: x = x +

solve for y: y = x + 6

Coordinate: (0,6)

I didnnt know which answer you wanted cuz you didn't specify

The correct option is (c) $768x^{\frac{1}{3}}$

The given expression is as follows; \[ \left(\frac{3 \sqrt[3]{x y} y^{-3}}{\sqrt[3]{y} x^{-\frac{4}{3}}}\right)^{-1} \]Firstly, simplify the numerator and the denominator of the expression.\[\frac{3\sqrt[3]{xy}y^{-3}}{\sqrt[3]{y}x^{-\frac{4}{3}}}=\frac{3}{\sqrt[3]{x}x^{-\frac{4}{3}}\sqrt[3]{y}y^{-3}}\]Next, simplify the denominator of the expression.\[\frac{3}{\sqrt[3]{x}x^{-\frac{4}{3}}\sqrt[3]{y}y^{-3}}=3x^{\frac{1}{3}}y^{\frac{10}{3}}\]Now, let us substitute $y=16$ into the expression of $3x^{\frac{1}{3}}y^{\frac{10}{3}}$.\[3x^{\frac{1}{3}}y^{\frac{10}{3}}=3x^{\frac{1}{3}}(16)^{\frac{10}{3}}=48x^{\frac{1}{3}}(2)^{\frac{10}{3}}=768x^{\frac{1}{3}}\]As we obtained the value of the expression by substituting $y=16$, therefore the correct option is (c) $768x^{\frac{1}{3}}$.

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[tex]\frac{1}{36}[/tex]

Negative exponents can be manipulated and solved through exponent properties.

Exponent Properties

There are numerous different exponent properties that allow us to simplify expressions. However, for this question, the important property is known as the negative exponent property. This states that [tex]\displaystyle a^{-b}=\frac{1}{a^b}[/tex]. We can apply this same idea to the question we were given.

Solving

Through the negative exponent property, we know that we can make the exponent positive by taking the reciprocal.

Remember that 6^2 = 36, then solve.

So, the final answer is 1/36.

Answer: 119.32 centimeters

Step-by-step explanation:

Circumference = diameter x π

C = 38 (3.14)

C = 119.32

The circumference is 119.32 centimeters

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The height of the building is approximately 27.62 feet. Rounded to the nearest hundredth, the answer is 27.62 feet.

In arithmetic, we use angles and distance to determine an object's height. The distance between the items is measured horizontally, and the height of an object is determined by the angle of the top of the object with respect to the horizontal.

Trigonometry includes heights and distances, and it has numerous uses in practical daily life. It is used to determine the distance between any two objects, including heavenly bodies or other objects, as well as the height of towers, buildings, mountains, etc. Astronauts, surveyors, architects, and navigators are the main users.

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From equation f (x)=7x-9,   f (x+h)= 7h and we can simplify it as 7.

For part (a), we can start by substituting x+h for x in the equation for f(x):

f(x+h) = 7(x+h) - 9

f(x+h) = 7x + 7h - 9

We can then simplify the expression to:

f(x+h) = 7x + 7h - 9

Now, to find f(x+h) - f(x), we can subtract f(x) from both sides of the equation:

f(x+h) - f(x) = 7x + 7h - 9 - (7x - 9)

f(x+h) - f(x) = 7h

Therefore, for part (a) we can conclude that f(x+h) - f(x) = 7h.

Now, we can simplify the difference quotient using f(x+h) and f(x):

f(x+h) - f(x) = [7(x+h) - 9] - [7x - 9]

= 7h

f(x+h) - f(x) / h = 7h / h = 7

We have already simplified the difference quotient  to 7.

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x + y = 35  and 2x + 4y = 102 represents the required system of equations.

Two or more expressions with an Equal sign is called as Equation.

Let x be the number of chickens and y be the number of cows.

Each chicken has 2 legs, so x chickens have 2x legs.

Each cow has 4 legs, so y cows have 4y legs.

The total number of animals is 35 and the total number of legs is 102, so we can set up the following system of equations:

x + y = 35 (the total number of animals)

2x + 4y = 102 (the total number of legs)

Hence, x + y = 35  and 2x + 4y = 102 represents the required system of equations.

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