Rewrite in simplest rational exponent form. Show each step of your process

The value of the legs of the triangle would be hypotenuse/√2.

You can use basic trigonometry to solve this problem. We know that the sine of the angle made by the hypotenuse and the base is equal to the ratio of perpendicular height and the hypotenuse, given that the angle is 45° the perpendicular will be equal to ⇒ hypotenuse × sin(45°).

That is perpendicular=hypotenuse/√2

Similarly, the cosine of the angle made by the hypotenuse and the base is equal to the ratio of base length and the hypotenuse, given that the angle is 45° the base will be equal to ⇒ hypotenuse × cos(45°)

That is base=hypotenuse/√2

Therefore, the value of the legs of the 45°-45°-90° triangle would be hypotenuse/√2.

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In random assignment design the subjects are not randomly assigned to an experimental group and control group.

The complete question:

"A researcher wishes to examine the effects of a new medical treatment on patients with Alzheimer's disease. The researcher plans to examine the effects on a group of randomly selected subjects. What is wrong with this design?

Select all that apply.

A. Subjects are not randomly assigned to an experimental group and

control group.

B. A control group is not used.

C. Only one researcher is used.

D. An entire population is not used."

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The function [tex]$$f(x)= \begin{cases}1-x^2 & x \leqslant 1 \\ \ln (x) & x \geqslant 1\end{cases}$$[/tex]  is continuous on (-∞, ∞).

As per the given data the function f(x) is given by:

[tex]$$f(x)= \begin{cases}1-x^2 & x \leqslant 1 \\ \ln (x) & x \geqslant 1\end{cases}$$[/tex]

Here we have to determine that the function f(x) is continuous on (-∞, ∞)

If we show that f(x) is continuous at x = 1 then f(x) is continuous on (-∞, ∞)

What are continuous function?

A continuous function in mathematics is one where changes in the parameter cause constant changes in the function's value (i.e., a change without a leap). This shows that there are no abrupt changes in value or discontinuities.

To show f(x) is continuous at x = 1

[tex]\lim _{x \rightarrow 1^{-}} f(x)=\lim _{x \rightarrow 1^{+}}[/tex] f(x)

[tex]\rightarrow \lim _{x \rightarrow 1^{-}} f(x) & =\lim _{x \rightarrow 1^{-}}\left(1-x^2\right) \\[/tex]

= 1 - 1

= 0

[tex]\lim _{x \rightarrow 1^{+}} f(x) & =\lim _{x \rightarrow 1^{+}} \ln (x) \\[/tex]

= ln 1

= 0

Therefore  [tex]\lim _{x \rightarrow 1^{+}} f(x) & =\lim _{x \rightarrow 1^{-}} f(x)-0[/tex].

Hence f(x) is continuous on (-∞, ∞)

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According to triangle inequality theorem a triangle with two sides 7 units and 17 units, has a third side which measures 10 units.

What is Triangle Inequality Theorem?

According to triangle inequality theorem, the sum of the lengths of any two sides of a triangle is greater than the length of the third side.

The two sides of a triangle are -

a=7, b=x and c=17, where a, b and c are the sides of the triangle.

As, 17 is a bigger number so let it be the biggest side.

Now, according to triangle inequality theorem -

a + b = c

7 + x = 17

x = 17 -7

x = 10

Therefore, the third side of the triangle measures 10 units.

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Answer:

Yes

Step-by-step explanation:

I think its yes.I think It's Yes.

The perimeter expression of a rectangle has a perimeter represented by the expression 4x−8 is 2(2x - 4)

From the question, we have the following parameters that can be used in our computation:

Perimeter = 4x - 8

Factor out 2 from the expression

So, we have the following representation

Perimeter = 2 * 2x - 2 * 4

So, we have

Perimeter = 2(2x - 4)

Hence, the perimeter is 2(2x - 4)

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Answer:

33 ounces

Explanation:

20% of 27.5 ounces is 5.5 ounces.

To find how much more juice the container has, we add 5.5 to 27.5, which equals to 33 ounces

The fraction expression as a product expression is 1/2 * 2/5

From the question, we have the following parameters that can be used in our computation:

A mathematical expression

The mathematical expression is given as

1/5

In this case, the expression is a fraction

Using the above as a guide, we have the following:

Fraction = 1/5

Multiply the fraction by 1

So, we have the following representation

Fraction = 1/5 * 1

Express 1 as 2/2

This gives

Fraction = 1/5 * 2/2

Swap the factors of the expression

This gives

Fraction = 1/2 * 2/5

Hence, the product expression is 1/2 * 2/5

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Earth's rotation is what causes day and night. As Earth rotates on its axis, half of the planet is facing the Sun, while the other half is facing away from the Sun.

The half of the planet facing the Sun experiences daylight, while the half facing away from the Sun experiences darkness. This cycle repeats itself every 24 hours, resulting in the day and night cycle that we experience.

Earth's rotation causes day and night by rotating on its axis, with half of the planet facing the Sun and the other half facing away from the Sun. This cycle of daylight and darkness lasts for  creating the day and night cycle that we observe.Earth's rotation is what causes day and night. As Earth rotates on its axis, half of the planet is facing the Sun, while the other half is facing away from the Sun.

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The inverse of ∼P→∼Q is P->Q.

Given ∼P→∼Q

Its inverse, ∼(∼P)→∼(∼Q), means that P→Q.

If the hypothesis is correct but the conclusion is untrue, a conditional statement is false. If the statement in the example above read, "If you achieve good marks then you will not get into a good college," it would be untrue. A conditional statement that has been modified or rearranged is referred to as a related conditional. An if-then statement, also known as a conditional statement, is a set of hypotheses followed by a conclusion. It is noted that p→q. Read this: if p, then q.

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The seven different types of triangles are Equiangular Triangle , Acute Angled Triangle , Right Angled Triangle , Obtuse Angled Triangle , Equilateral Triangle , Isosceles Triangle and Scalene Triangle .

The Triangle is defined as a polygon with three sides having three vertices .

The Seven different types of Triangles are :

(i) Equiangular Triangle :

It is a type of triangle in which the measure of all the three sides of triangle are equal .

(ii) Acute Angled Triangle :

It is a type of triangle in which all the angles of triangle are acute , that means angle is less than 90 degree .

(iii) Right Angled Triangle :

It is the type of triangle in which the measure of one angle is 90 degree .

(iv) Obtuse Angled Triangle :

It is defined as a triangle in which one angle of the triangle is obtuse , that means the angle is greater than 90 and less than 180 degree .

(v) Equilateral Triangle :

It is a triangle in which the measure of all the sides of the triangle are equal .

(vi) Isosceles Triangle :

It is a type of triangle in which the measure of the two sides of the triangle are equal .

(vii) Scalene Triangle :

It is defined as a triangle in which the measure of all the sides of triangle are different ,

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P (X)= [tex]x^3+4 x^2-3x+2[/tex]  moreover, [tex]\frac{P(x)}{x - 1} = x^{2} + 5x +2[/tex] as a result, the actual claim is  [tex]$p(x)=(x-1)\left(x^2+5 x+2\right)+\frac{4}{x-1}$[/tex]

The likelihood of an event occurring is determined by probability:       P(A) = f / N. Probability and odds are related, but probability is necessary for odds to exist. Prior to calculating the likelihood that an event will occur, probability is required.

The steps listed below can be used to determine an event's likelihood: Step 1: Identify an event that has only one outcome. Step 2: Compile a list of all potential outcomes, including any positive ones. Step 3: Subtract the number of favourable outcomes from the total number of outcomes that are feasible. The probability of A or B is equal to p(A or B) = p(A) + p(B) if occurrences A and B are mutually exclusive (B). A probability is a number that expresses the possibility or likelihood that a specific event will take place.

Therefore the correct answer is option A ) [tex]$p(x)=(x-1)\left(x^2+5 x+2\right)+\frac{4}{x-1}$[/tex]

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Answer:

5 kg.

Step-by-step explanation:

2 kg for $0.96 is more expensive than 5 kg for $1.95

The ratio for cos (cosine) of angle θ is: cos(θ) =  adjacent side of θ / hypotenuse

We know that in right triangle the cosine of an angle is nothing but the ratio of the side adjacent of  given angle to the hypotenuse.

In right tirnagle for an angle θ, the ratio for cosine of angle θ would be:

cos(θ) =  adjacent side of θ / hypotenuse

Consider following right triangle PQR with angle PQR = 90 degrees

Here, PR is the hypotenuse.

Assume that angle QPR measure α degrees and angle PRQ measures θ degrees.

By definition of cosine of angle θ,

cos(θ) =  adjacent side of θ / hypotenuse

cos(θ) =  QR / PR

And by definition of cosine of angle α,

cos(α) =  adjacent side of α / hypotenuse

cos(α) = PQ / PR

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Elizabeth has a box that can hold exactly 24 unit cubes. It could also be said that her box has 24 cubic units. So, option D. is correct.

A cubic unit is a unit of measurement for volume. It represents the amount of space an object occupies in three dimensions: Length, Width and Height. It is often used to measure the volume of three-dimensional shapes such as cubes, square prisms, cylinders, and spheres. Cubic units are most commonly measured in cubic meters or cubic feet, but they can also be expressed in smaller units such as cubic centimeters or cubic inches.

The total volume of the box is 24 cubic units. A side of the box may measure 2 units x 3 units x 4 units = 24 cubic units.

Choice A is incorrect. This is because the area of ​​a shape is a measure of the two-dimensional space it occupies, and the volume of a shape is a measure of the three-dimensional space it occupies.

Choice B is incorrect. This is because the area of ​​a shape is a measurement of the two-dimensional space it occupies, not its volume.

Option C is incorrect because it states that the box has a volume of 12 cubic units, but it is not because the volume he has is 24 cubic units.

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The equation for new function is,

⇒ f (x) = ∛( 1/4x - 6) - 5

A relation between a set of inputs having one output each is called a function.

Given that;

The parent function is,

⇒ g (x) = ∛x

Now, After  reflected and stretched by 4, we get;

⇒ g (x) = f (1/4x) = ∛ 1/4x

And, After translated 6 units to the right, we get;

⇒ g (x) = f (x - 6) = ∛ 1/4x - 6

And, After 5 unit down, we get;

⇒ g (x) = h (x) - 5 = ∛( 1/4x - 6) - 5

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The value of x in this problem is 2.

Our given algebraic equation is 2.5 ( 6x - 4 ) = 10 + 4 ( 1.5 + 0.5x) and in this, we have to find the final value of x.

Use the transposing method to isolate the variable x and then solve for it.

distribute brackets on both sides of the equation.

= 2.5*6x - 2.5*4 = 10 + 4*1.5 + 4*0.5x

= 15x - 10 = 10 + 6 + 2x

collect terms in x to the left side and numeric values to the right side.

= 15x - 2x = 10 + 6 + 10

= 13x = 26

= x = 26/13 = 2

Therefore the final value of x in this equation will be 2.

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Therefore, the inequality 1.85 < 15 can be used to find h.

In mathematics, "inequality" refers to a relationship between two expressions or values that is not equal to each other. Therefore, inequality emerges from a lack of balance. For instance, let's say you have 225 and want to purchase a new bicycle that costs $250.

When the expression's variables and constants are given values, the calculation it describes yields the expression's value. In an algebraic expression, numbers can be represented by letters.

The inequality 1.85 < 15 can be used to find h, the number of mini-hamburgers Betsy and her sister bought. This inequality represents the constraint that the total cost of the mini-hamburgers (1.85h) must be less than the amount of money they have to spend (15). To solve for h, you would divide both sides of the inequality by 1.85, giving you h < 8.1, so h can be any number less than 8.1.

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Answer:

Factor the numerator and denominator and then divide out any common factors.

Step-by-step explanation:

Given:

A length of a side of an equilateral triangle = 32 cm

Formula Used:

If in an equilateral triangle with side length 'a' then,

Area of an equilateral triangle = [(√3)/4] × a2 

Calculations:

Area of an equilateral triangle = [(√3)/4] × 322

⇒ (1024 × √3)/4

⇒ 256√3 cm2

Answer:  -100

Step-by-step explanation:   -7(15)+5

It said x=15 so put 15 in and solve from there and that would be -100

HOPE THIS HELPS ANY I JUST completed FUNCTIONS AND MAN IT WAS FUN

To compute the sum of dollar amounts that are not whole numbers and to compute products of dollar amounts that are not whole numbers you can use the standard arithmetic operation and place decimal point correctly.

To compute the sum of dollar amounts that are not whole numbers, such as $5.89 and $1.45, you can use the standard arithmetic operations of addition, just like you would with whole numbers.

$5.89 + $1.45 = $7.34

When adding or subtracting decimal numbers, you align the decimal points and perform the operation column by column, just like you would with whole numbers.

To compute products of dollar amounts that are not whole numbers, such as 4 pounds of beef at $5.89 per pound, you can use the standard arithmetic operation of multiplication.

4 × $5.89 = $23.56

When multiplying decimal numbers, you multiply as you would with whole numbers and place the decimal point in the product to the right of the last digit in the product that corresponds to the number of decimal places in the original factors.

In both examples, it is important to pay attention to the placement of the decimal point and to use proper arithmetic operations. In general, it is important to consider the decimal places when working with decimal numbers and ensure that the final result is rounded to the appropriate number of decimal places, if necessary.

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A particle traveled a distance of 9.0×10¹² meters in 5.0 × 10⁷ seconds.

The division in mathematics is one kind of operation. In this process, we split the expressions or numbers into the same number of parts.

Given:

If a particle traveled a distance of 9.0×10¹² meters at a rate of 1.8×10⁵meters per second.

And the distance,

= rate x time.

Rearranging the formula,

Time = Distance / rate

Time = 9.0×10¹² / 1.8×10⁵

Simplifying by using division operation,

Time = (9.0/1.8) × 10¹² × 10⁻⁵

Time = 5.0 × 10⁷

Therefore, the required specified time is  5.0 × 10⁷ seconds.

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The domain and scope of the identity function are the same. Equation for the identity function is  f(x) = x, or y = x.

An statement, rule, or law in mathematics that specifies the connection between an independent variable and a dependent variable (the dependent variable. A relationship between a group of inputs and one output each is referred to as a function. In plain English, a function is an association between inputs in which each input is connected to precisely one output. Y=X2 is an illustration of this. You only receive one output for y if you enter anything for x. The fact that x is the input value leads us to argue that y is a function of x.

identity function are the same.

Equation for the identity function is

f(x) = x, or y = x.

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The rate exists water being poured into the cup when the water level is 9 cm exists [tex]$$1458 \pi \mathrm{cm}^3 s^{-1}$$[/tex].

To determine the derivative of a composite function in differential calculus, apply the chain rule formula. If y = f(g(x)), then according to the chain rule, the instantaneous rates of change of the functions f and g with respect to x produce the instantaneous rate of change of f with respect to x.

A composition of functions, such as the composition of the functions f and g, f(g(x)), can be used to determine the derivative using the chain rule.

Let us set up the following variables:

R, Radius of conical cup (cm) = 30 cm

H, Height of the conical cup (cm) = 10 cm

Our aim is to find dV/dt when h = 9 and we are given dh/dt = 2

By similar triangles we have

r : h = R : H

r/h = 30/10

⇒ r = 3h

The volume of a cone is 1/3π r²h, So:

V = 1/3π r²h

simplifying the equation, we get

= 1/3π (3h²)h

= 1/3π 9h²h

= 3π h³

Differentiating wrt h we get:

dV/dh = 9π h²

Applying the chain rule we have:

dV/dt = (dV/dh)(dh/dt)

= 9π h² × 2

= 18π h²

If h = 9 then

[tex]$\left[\frac{d V}{d t}\right]_{h=9}=18 \pi\left(9^2\right)=1458 \pi \mathrm{cm}^3 s^{-1}$$[/tex].

Therefore, the rate exists water being poured into the cup when the water level 9 cm exists [tex]$$1458 \pi \mathrm{cm}^3 s^{-1}$$[/tex].

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Answer:

85+7.5v=160

7.5v=160-85

7.5v=75

V=75/7.5

V=10

Ryan would need to take 10 visits to get a free ticket.

For adults, there were 40 tickets sold, and for kids, there were 60 tickets sold.

There are several laws that regulate the sequence in which you should do the operations in math and algebra. The Commutative, Associative, and Distributive Laws are the three that are most frequently discussed.

Let x be the total number of adult tickets sold.

6.50 × x + 2.50 × (100 - x) = 410

6.50 × x + 250 - 2.50x = 410

4x = 410 - 250

x = 40

Hence,

a) 40 tickets were sold by adults, according to item

and

b) 100-40=60 tickets were sold to children.

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Answer:

y = x - 2

Step-by-step explanation:

Here m is the slope and b is the y-intercept.

Choose any two points on the line.            

 (2,0)  ; (0,-2)

Now, find the slope.

        [tex]\sf \boxed{Slope = \dfrac{y_2-y_1}{x_2-x_1}}[/tex]

                   [tex]= \dfrac{-2 - 0}{0 - 2}\\\\=\dfrac{-2}{-2}\\\\= +1[/tex]

m = 1.

Now, y = 1x +  b

The (2 , 0) passes through the line. Substitute x = 2 and y = 0 in the above equation and find the value of 'b'.

       0 = 1*2 + b

       0 =  2 + b

        -2 = b

Equation of line:

              y = x - 2