Order the simplification steps of the expression below using the properties of rational exponents. \(\sqrt[3]{7}=7^{\frac{1}{3}}\)

A.) \(\left(7^{\frac{1}{3}}\right)^3=7^{\frac{1}{3}}\cdot 7^{\frac{1}{3}}\cdot 7^{\frac{1}{3}}=7^{\frac{1}{3}}\cdot ^{\frac{1}{3}}\cdot ^{\frac{1}{3}}=7^{\frac{3}{3}}=7^1=7\\\)

B.) \(\left(7^{\frac{1}{3}}\right)^3=7^{\frac{1}{3}}\cdot 7^{\frac{1}{3}}\cdot 7^{\frac{1}{3}}=7\cdot 7^{\frac{1}{3}}=3\cdot \frac{1}{3}\cdot 7=1\cdot 7=7\)

C.) \(\left(7^{\frac{1}{3}}\right)^3=7^{\frac{1}{3}}\cdot \:7^{\frac{1}{3}}\cdot \:7^{\frac{1}{3}}=7^{\frac{1}{3}}+\:^{\frac{1}{3}}+\:^{\frac{1}{3}}=7^{\frac{3}{3}}=7^1=7\)

D.) \(\left(7^{\frac{1}{3}}\right)^3=7^{\frac{1}{3}}\cdot \:7^{\frac{1}{3}}\cdot \:7^{\frac{1}{3}}=7\cdot \left(\frac{1}{3}+\frac{1}{3}+\frac{1}{3}\right)=7\cdot \frac{3}{3}=7\cdot 1=7\)

Aproximate speed of the Earth's rotation around the Sun is 18.5 miles per second (30 km per second). 110,000 kilometers per hour is equal to 30 kilometers per second.

"A method to find a single unit value from a multiple unit value and to find a multiple unit value from a single unit value."

We always count the unit or amount value first and then calculate the more or less amount value.

For this reason, this procedure is called a unified procedure.

Many set values ​​are found by multiplying the set value by the number of sets.

A set value is obtained by dividing many set values ​​by the number of sets.

Hence, Aproximate speed of the Earth's rotation around the Sun is 18.5 miles per second (30 km per second). 110,000 kilometers per hour is equal to 30 kilometers per second.

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the arc-length of the curve for a single arch of a cycloid of radius R is 8R

To find the arc-length of a single arch of a cycloid of radius R, we can use the formula:

L = ∫√(1 + (dy/dx)^2)dx

First, we need to find an expression for y in terms of x. A cycloid of radius R is given by the parametric equations:

x = R(θ - sinθ)
y = R(1 - cosθ)

To express y in terms of x, we can eliminate θ by solving for sinθ and cosθ in terms of x/R:

sinθ = (x/R) - sin((x/R))
cosθ = 1 - cos((x/R))

Substituting these expressions into the equation for y, we get:

y = R(1 - (1 - cos((x/R))) = Rcos((x/R))

Now, we can find dy/dx by differentiating with respect to x:

dy/dx = -Rsin((x/R))

Substituting this expression and the expression for y into the formula for arc-length, we get:

L = ∫√(1 + (-Rsin((x/R)))^2)dx
 = ∫√(1 + R^2sin^2((x/R)))dx

To evaluate this integral, we can use the trigonometric identity:

1 - cosθ = 2sin^2(θ/2)

Letting u = (x/2R), we can rewrite the integral as:

L = 2R∫√(1 + 4sin^2(u))du
 = 2R∫√(4cos^2(u/2))du
 = 4R∫cos(u/2)du

Integrating, we get:

L = 8Rsin(u/2) + C
 = 8Rsin(x/4R) + C

So the arc-length of a single arch of a cycloid of radius R is given by:

L = 8Rsin(x/4R) + C
Hi! To find the arc-length of a single arch of a cycloid of radius R, we'll use the given hint and the formula for arc-length in parametric form. The parametric equations for a cycloid are:

x = R(θ - sinθ)
y = R(1 - cosθ)

To find the arc-length (L), we need the derivatives dx/dθ and dy/dθ:

dx/dθ = R(1 - cosθ)
dy/dθ = R(sinθ)

Now, we can use the formula for arc-length:

L = ∫√[(dx/dθ)^2 + (dy/dθ)^2]dθ

Substitute the derivatives:

L = ∫√[R^2(1 - cosθ)^2 + R^2(sinθ)^2]dθ

Use the hint: 1 - cos(θ) = 2 sin^2(θ/2), which gives us (1 - cosθ) = 2R^2sin^2(θ/2)

L = ∫√[R^2(2R^2sin^2(θ/2)) + R^2(sinθ)^2]dθ

Factor out R^2:

L = R∫√[2sin^2(θ/2) + sin^2θ]dθ

Now, evaluate the integral over the range of a single arch, which is [0, 2π]:

L = R∫[0 to 2π]√[2sin^2(θ/2) + sin^2θ]dθ

The exact evaluation of this integral is not straightforward, but it's known that the arc-length of a single arch of a cycloid of radius R is:

L = 8R

So, the arc-length of the curve for a single arch of a cycloid of radius R is 8R.

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Answer:

  x = 5

Step-by-step explanation:

You want the equation of a line with undefined slope and an x-intercept of 5.

Slope is the ratio of "rise" to "run" for a line. If the line is vertical, the "run" is zero, making the denominator of the ratio is zero. Division by zero gives an "undefined" result.

If the slope of a line is "undefined", it is a vertical line. It has the same x-value everywhere. Its equation is ...

  x = c . . . . . for some constant

Here, the vertical line crosses the x-axis at x=5. That is the equation of the line:

  x = 5

The Laplace transform is a mathematical technique used to convert a function of time, f(t), into a function of a complex variable, s. The transform is defined by an integral that takes the function f(t) and transforms it into the function F(s) defined by:

We can use the definition of Laplace transform to find L{f(t)}:

L{f(t)} = ∫₀^∞ e^(-st) * f(t) dt

For 0 ≤ t < 1, f(t) = t, so we have:

L{f(t)} = ∫₀¹ e^(-st) * t dt

Integrating by parts with u = t and dv/dt = e^(-st), we get:

L{f(t)} = [-te^(-st)/s]₀¹ + ∫₀¹ e^(-st)/s dt

= [-te^(-st)/s]₀¹ + [-e^(-st)/(s^2)]₀¹

= [e^(-s) - 1 + s]/(s^2)

For t ≥ 1, f(t) = 2 - t, so we have:

L{f(t)} = ∫₁^∞ e^(-st) * (2 - t) dt

Integrating by parts with u = 2 - t and dv/dt = e^(-st), we get:

L{f(t)} = [(2 - t)*e^(-st)/s]₁^∞ - ∫₁^∞ (-e^(-st)/s) dt

= [(2 - e^(-s))/s] - [e^(-s)/s^2]

Therefore, the Laplace transform of f(t) is:

L{f(t)} = [e^(-s) - 1 + s]/(s^2) for 0 ≤ t < 1

= [(2 - e^(-s))/s] - [e^(-s)/s^2] for t ≥ 1

Note: The square brackets [] indicate the limits of integration.

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Answer:14-14

Step-by-step explanation:

By using percentage, it can be calculated that

Percentage decrease in the screen time for Lora  = 40 %

What is percentage?

Suppose there is a number and the number has to be expressed as a fraction of 100. The fraction is called percentage.

For example 9% means \(\frac{9}{100}\). Here 9 is expressed as a fraction of 100

Here, Percentage decrease will be calculated

Screen time for Lora last week = 840 minutes

Screen time for Lora this week = 504 minutes

Decrease in screen time for Lora = (840 - 504) minutes

                                                        = 336 minutes

Percentage decrease in the screen time for Lora = \(\frac{336}{840}\times 100\) = 40 %

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Answer:

x ≈ 14.4 cm

Step-by-step explanation:

the third angle of the triangle = 180° - 40° - 46° = 94°

Using the Sine rule in the triangle

\(\frac{x}{sin46}\) = \(\frac{20}{sin94}\) ( cross- multiply )

x × sin94° = 20 × sin46° ( divide both sides by sin94° )

x = \(\frac{20sin46}{sin94}\) ≈ 14.4 cm ( to the nearest tenth )

Answer:

14 cm (rounded to nearest whole number)

Step-by-step explanation:

Law of Sine:

\(\displaystyle \large{\dfrac{\sin A}{a} = \dfrac{\sin B}{b} = \dfrac{\sin C}{c} = 2R}\)

Given angles are:

Definition of Euclidean Triangle:

Find another angle:

So another angle is 94°.

To find:

Determine:

Therefore:

\(\displaystyle \large{\dfrac{\sin 46^{\circ}}{x} = \dfrac{\sin 94^{\circ}}{20}}\)

Multiply both sides by 20x:

\(\displaystyle \large{\dfrac{\sin 46^{\circ}}{x} \cdot 20x = \dfrac{\sin 94^{\circ}}{20} \cdot 20x}\\\displaystyle \large{20\sin 46^{\circ}= x\sin 94^{\circ}}\)

Divide both sides by \(\displaystyle \large{\sin 94^{\circ}}\):

\(\displaystyle \large{\dfrac{20\sin 46^{\circ}}{\sin 94^{\circ}} = \dfrac{x\sin 94^{\circ}}{\sin 94^{\circ}}}\\\displaystyle \large{\dfrac{20\sin 46^{\circ}}{\sin 94^{\circ}} = x}\)

Evaluate the expression, hence:

\(\displaystyle \large{x = 14.42...}\)

Round to nearest whole number:

\(\displaystyle \large{x = 14}\)

Therefore, the value of x is 14 cm.

y - 4x = 4

y - 4x - 4 = 0

or

y = 4 + 4x

As per the figure provided the exact value of CE will be equivalent to 12.

According to the figure given in the question, Angles ABE and DBC are vertical angles and thus have the same measure. Since the given segment AE is parallel to a segment of CD, angles A and D are of the same distance by the alternate interior angle theorem. As a result, according to the angle-angle theorem, triangles ABE and DBC are equivalent, with vertex A corresponding to vertex B and vertex E to vertex D, respectively.

Hence, AB ÷ DB = EB ÷ CB

10 ÷ 5 = 8 ÷ CB

Since, CB=4 and CE= CB+BE

CE = 4 + 8

CE=12.

Therefore CE is equal to 12.

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Explanation:

I'll use the quadratic formula to find the roots or x intercepts. This slight detour allows us to factor without having to use guess-and-check methods.

The equation is of the form ax^2+bx+c = 0

This leads to...

\(x = \frac{-b\pm\sqrt{b^2-4ac}}{2a}\\\\x = \frac{-(-11)\pm\sqrt{(-11)^2-4(12)(-5)}}{2(12)}\\\\x = \frac{11\pm\sqrt{361}}{24}\\\\x = \frac{11\pm19}{24}\\\\x = \frac{11+19}{24} \ \text{ or } \ x = \frac{11-19}{24}\\\\x = \frac{30}{24} \ \text{ or } \ x = \frac{-8}{24}\\\\x = \frac{5}{4} \ \text{ or } \ x = -\frac{1}{3}\)

Now use those roots to form these steps

\(x = \frac{5}{4} \ \text{ or } \ x = -\frac{1}{3}\\\\4x = 5 \ \text{ or } \ 3x = -1\\\\4x - 5 =0 \ \text{ or } \ 3x+1 = 0\\\\(4x-5)(3x+1) = 0\)

Refer to the zero product property for more info.

Therefore, the original expression factors fully to (4x-5)(3x+1)

Use the FOIL rule to expand it out and you should get 12x^2-11x-5 again.

----------------------------------------------

We did that factoring so we could find the side lengths of the rectangle.

I'm using the fact that area = length*width

The order of length and width doesn't matter.

From here, we can then compute the perimeter of the rectangle

P = 2(L+W)

P = 2(4x-5+3x+1)

P = 2(7x-4)

P = 14x - 8

Answer:

x=16

Step-by-step explanation:

Answer: 1. A translation takes a line to a parallel line or itself.

Step-by-step explanation:

I took the quiz myself.

When a line is translated, it means the line is moved from one position to another. The true statement is:

(1). A translation takes a line to a parallel line or itself.

In transformation, translation does not change the orientation of a line. This means that:

Point (2) above means that;

Using the above analysis, we can conclude that: option (1) is true

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The value of the additional data point is  \($19.17$\).

Let us first find the mean of the given data:

\(Mean = \frac{\sum_{i=1}^{n} x_i}{n}=\frac{39 + 45 + 43 + 42 + 44}{5}= 42.6\)

Now let's find the value of the additional data point. Let the value of the additional data point be x. Therefore, the new sum of data is

\($(39+45+43+42+44+x)$\).

Total numbers of data are 6 (five given in the set and one additional data point).So, the mean of the resulting data set is given by:

\(32.083 = \frac{(39+45+43+42+44+x)}{6}\)

Multiplying both sides of the equation by 6 we get:

\(6 \times 32.083 = (39+45+43+42+44+x)\)

We have the value of \($39+45+43+42+44$\) which is \($213$\).

Therefore, substituting all the values, we get:

\(193.83 + x = 213\)

On subtracting \($193.83$\) from both sides, we get the value of

\(x. x = 213 - 193.83 = 19.17\)

Therefore, the value of the additional data point is \($19.17$\)

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z is equal to 0.6 cm with an uncertainty of 0.316 cm.

We are given:

x = (3.0 ± 0.2) cm

y = (4.2 ± 0.6) cm

We need to find z = x - (y/2) and its uncertainty.

First, we need to find the central values of x and y:

x_central = 3.0 cm

y_central = 4.2 cm

Next, we need to find the uncertainties of x and y:

x_uncertainty = 0.2 cm

y_uncertainty = 0.6 cm

Now we can use the formula for z = x - (y/2):

z = x_central - (y_central/2) = 3.0 cm - (4.2 cm/2) = 0.6 cm

To find the uncertainty of z, we need to propagate the uncertainties of x and y using the formula:

uncertainty_z = sqrt((uncertainty_x)^2 + ((1/2)*uncertainty_y)^2)

uncertainty_z = sqrt((0.2 cm)^2 + ((1/2)*0.6 cm)^2) = 0.316 cm

Therefore, the final result is:z = (0.6 ± 0.316) cm

Therefore, z is equal to 0.6 cm with an uncertainty of 0.316 cm.

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Answer:

Step-by-step explanation:

The value of z is 0.9 cm and its uncertainty is ±0.36 cm. So we can write the final result as: z = (0.9 ± 0.36) cm

To find z = x - (y/2) and its uncertainty, we first need to calculate the values of x, y, and their uncertainties:

x = (3.0 ± 0.2) cm

y = (4.2 ± 0.6) cm

Using these values, we can find the value of z:

z = x - (y/2)

z = 3.0 cm - (4.2 cm/2)

z = 3.0 cm - 2.1 cm

z = 0.9 cm

Now we need to calculate the uncertainty of z using the formula:

Δz = sqrt( (Δx)^2 + (Δy/2)^2 )

where Δx and Δy are the uncertainties of x and y, respectively.

Δz = sqrt( (0.2)^2 + (0.6/2)^2 )

Δz = sqrt( 0.04 + 0.09 )

Δz = sqrt( 0.13 )

Δz = 0.36

Therefore, the value of z is 0.9 cm and its uncertainty is ±0.36 cm. So we can write the final result as:

z = (0.9 ± 0.36) cm

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Answer:

158 miles

Step-by-step explanation:

D = 59 + 3D  [The distance starts with 59 miles and then adds 3 each day, D]

D =59 + 3*(33)

D = 158 miles

we can conclude that f(0) is not equal to lim(x→0) f(x) for this particular function.

To show that f(0) is not equal to the limit of f(x) as x approaches 0, we need to evaluate these two quantities separately.

First, let's find f(0):

According to the given definition of the function f(x), when x = 0, we have:

f(0) = 1

Now, let's find the limit of f(x) as x approaches 0:

lim(x→0) f(x) = lim(x→0) [x² - 1] = -1

So, we have f(0) = 1 and lim(x→0) f(x) = -1.

Since f(0) is not equal to the limit of f(x) as x approaches 0, we can conclude that f(0) is not equal to lim(x→0) f(x) for this particular function.

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Answer:  Find the Equation of a Line Given That You Know a Point on the Line And Its Slope. The equation of a line is typically written as y=mx+b where m is the slope and b is the y-intercept.

Step-by-step explanation: i hope this help

After 3 hours, the boat is approximately 16.43 miles from the marina, and the bearing from the boat back to the marina is approximately 209.9°

We have,

To solve this problem, we can break down the boat's motion into two components: north-south displacement and east-west displacement.

Given:

The boat travels due south for 1 hour at a constant speed of 15 mph.

The boat then changes course to a bearing of S47°E and travels for 2 hours at the same constant speed of 15 mph.

a.

To find how far the boat is from the marina after 3 hours, we need to calculate the total displacement using the Pythagorean theorem.

First, let's find the north-south displacement:

Distance = Speed x Time = 15 mph x 1 hour = 15 miles

Next, let's find the east-west displacement using the given bearing:

Angle of S47°E = 180° - 47° = 133°

Using trigonometry, we can find the east-west displacement:

East-West Displacement = Distance x cos(Angle) = 15 miles x cos(133°)

Now, let's calculate the total displacement:

Total Displacement = √(North-South Displacement² + East-West Displacement²)

b.

To find the bearing from the boat back to the marina, we can use trigonometry to calculate the angle between the displacement vector and the north direction.

Let's calculate the values:

a. North-South Displacement = 15 miles

b. East-West Displacement = 15 miles x cos(133°)

c. Total Displacement = sqrt(North-South Displacement² + East-West Displacement²)

b. Bearing = atan(East-West Displacement / North-South Displacement) + 180°

Now, let's perform the calculations:

a. North-South Displacement = 15 miles

b. East-West Displacement = 15 miles x cos(133°) ≈ -6.83 miles (rounded to two decimal places)

c. Total Displacement = √(15² + (-6.83)²) ≈ 16.43 miles (rounded to two decimal places)

b.

Bearing = atan(-6.83 / 15) + 180° ≈ 209.9° (rounded to one decimal place)

Therefore,

After 3 hours, the boat is approximately 16.43 miles from the marina, and the bearing from the boat back to the marina is approximately 209.9°

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Answer:

{} , {red}, {blue}, {pink}, {green} , {red, blue}, {red, pink}, {red, green}, {blue, pink} , {blue, green} , {pink, green}, {red, blue, pink}, {red, pink, green}, {blue, pink, green}, {red, blue, green}, {red, blue, pink, green}

Step-by-step explanation:

Given set is:

{red, blue, pink, green}

First of all, we have to calculate the number of subsets

In the given set,

n =  4

So the number of subsets will be: 2^4 = 16

The subsets are:

{} , {red}, {blue}, {pink}, {green} , {red, blue}, {red, pink}, {red, green}, {blue, pink} , {blue, green} , {pink, green}, {red, blue, pink}, {red, pink, green}, {blue, pink, green}, {red, blue, green}, {red, blue, pink, green}

Answer:

Domain is... R

Range is... [-2, +unlimited)

Answer:

d. contour ploughing

hope it helpss

Answer:

9/8

Step-by-step explanation: Done the problem before in my math class.

Answer:

9/8

Step-by-step explanation:

Candance ate 3/4

Serving is 2/3

Divide: 3/4 divided by 2/3

To divide fractions you multiply the inverse.

3/4 x 3/2 = 9/8  

Hope this helps :)

Answer:

\(2.28*10^7\)

Step-by-step explanation:

\((2.4*10^3)(9.5*10^3)=(2.4*9.5)(10^3*10^3)=22.8*10^6=2.28*10^7\)

Answer:

228,000

Step-by-step explanation:

(2.4x10^3) x (9.5x10^3)

(2.4x100) x (9.5x100)

240x950

228,000

From the given price of five books and a DVD , equation which is required to determine the cost of one book is given by 5x + 8 = $26.50.

As given in the question,

Cost of one DVD is equal to $8.

Number of books to be purchased = 5

Total cost of all the five books and a DVD is equal to $26.50

Let 'x' be the cost of each purchased book

Required equation to represent the cost of one book is given by :

5x + 8 = $ 26.50

⇒ 5x = 26.50 -8

⇒ 5x = 18.5

⇒ x = $3.7

Therefore, from the given price of five books and a DVD , equation which is required to determine the cost of one book is given by 5x + 8 = $26.50.

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Answer:

Step-by-step explanation:

10

RNA's capability to self-replicate and catalyze chemical reactions make it the best candidate for the first life-form.

According to the scientists, RNA is considered the best candidate for the first life-form due to the fact that RNA is capable of self-replication and catalysis. This is possible because RNA can act both as a template to produce copies of itself and also as an enzyme to accelerate chemical reactions. These abilities suggest that RNA could have played a role in the emergence of the first living organisms on Earth.

An RNA molecule has the ability to catalyze reactions, which means that it can speed up chemical reactions without itself being altered. Thus, it is capable of serving as an enzyme. Scientists believe that the first life-form must have been capable of self-replication and catalysis, and RNA is capable of both functions.

This capability is significant because it is fundamental for the origin of life. Hence, RNA is believed to have played a significant role in the emergence of the first living organisms on Earth.

To conclude, RNA's capability to self-replicate and catalyze chemical reactions make it the best candidate for the first life-form.

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RNA is considered the best candidate for the first life-form by scientists because RNA is capable of self-replication and catalysis.

The correct option is the first one due to following reasons:

As a result, scientists believe that RNA may have played a crucial role in the origins of life on Earth.

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The statement is true.

An orthogonal set with the same dimension as the initial collection of vectors is created by the Gram-Schmidt process.

Given that,

From a linearly independent collection of {x₁, x₂,..., xp}, the gram-Schmidt process creates an orthogonal set of {v₁, v₂,..., vp} with the feature that for each k, the vectors v₁...vk span the same subspace as that spanned by x₁...xk.

Whether the claim is true or false must be determined.

The statement is true.

An orthogonal set with the same dimension as the initial collection of vectors is created by the Gram-Schmidt process. An orthogonal set is further linearly independent. The orthogonal set produced by the Gram-Schmidt process and the original set will cover the same subspace if their dimensions are the same.

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