Can some help simplify this question, with positive exponents

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Answer:

To solve the equation, we can begin by isolating the expression in the parentheses:

2(x^2 + x - 11)^3/2 = 54

Divide both sides by 2:

(x^2 + x - 11)^3/2 = 27

Take the cube root of both sides:

x^2 + x - 11 = 3∛27

Simplify the cube root:

x^2 + x - 11 = 3∛(27) = 3*3 = 9

Rearrange the equation:

x^2 + x - 20 = 0

Factor the quadratic:

(x + 5)(x - 4) = 0

Therefore, the solutions are x = -5 and x = 4.

To check for extraneous solutions, we need to make sure that these values do not make the original equation undefined. Since there are no square roots or denominators, there are no restrictions on x, and both solutions are valid.

So, the solutions to the equation are -5 and 4, in increasing order.

Answer:

A ratio of 3:1 means that there are 4 parts altogether. The fractions from the ratio can therefore be deduced as. 43 and 41. These represent the percentages: 75%:25%

Answer:

Step-by-step explanation:

The ratio can be converted into a fraction

therefore we can divide by the denominator in order to convert 30 Euros to a 3:1 ratio.

The fraction =3/4 and 1/4

so we have to divide 30 by 4

= 7.5 Euros

so;

3:1

(22.5):(7.5)

(45/2):(15/2)

45:15

divide by 15

3:1

95% of samples will produce a confidence interval that captures the true proportion. The correct answer is A.

When we say we are 95% confident in a particular interval estimate, it means that if we were to take many different samples from the same population and compute a confidence interval for each sample using the same method, about 95% of those intervals would contain the true population parameter.

So, Option A correctly states that 95% of samples will produce a confidence interval that captures the true proportion. Option B is incorrect because it refers to the sample proportion, rather than the true population proportion. Option C is also incorrect because it refers to the sample, rather than the confidence interval.

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Answer:24,000 pounds

Step-by-step explanation: a ton is 2,000 pounds so 12 times 2,000 is 24,000

The proper way to determine the seat width of a wheelchair for an individual is to measure the widest aspect of their hips or thighs, and then add approximately 2 inches to that measurement to determine the minimum seat width.

The extra 2 inches are required to allow for comfortable placement, movement, and sitting in the wheelchair. It's crucial to remember that this serves only as a basic guideline and that the actual seat width needed may change based on the demands and preferences of the individual.

For instance, people with particular medical issues or disabilities could need seats that are larger or narrower than average. Therefore, it is advised that a healthcare professional be consulted to help identify the best seat width for a particular person, such as an occupational therapist or physical therapist.

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Answer:

  [tex]\sqrt[6]{2}[/tex]

Step-by-step explanation:

You want the expression 2^(2/3)/2^(1/2) in simplest radical form.

The quotient property tells you ...

  (a^b)/(a^c) = a^(b-c)

The relation between fractional exponents and roots is ...

  [tex]a^\frac{b}{c}=\sqrt[c]{a^b}[/tex]

  [tex]\dfrac{2^\frac{2}{3}}{2^\frac{1}{2}}=2^{\frac{2}{3}-\frac{1}{2}}=2^\frac{1}{6}=\boxed{\sqrt[6]{2}}[/tex]

Answer:

First, we need to move the constant term to the other side of the equation:

y + x^2 - 16x = -100

Next, we need to "complete the square" by adding and subtracting a constant term that will allow us to write the left-hand side of the equation as a perfect square. To do this, we need to take half of the coefficient of the x-term (-16), square it, and add that to both sides of the equation:

y + x^2 - 16x + 64 = -36 + 64

Now, we can simplify the left-hand side by factoring it as a perfect square:

(y + (x - 8)^2) = 28

So the equation, in completed square form, is:

y + (x - 8)^2 = 28

Step-by-step explanation:

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If students earned an average of 13.8 credits per hour, then best point estimate for average number of credit hours per semester is 13.8.

The "best-point" estimate is defined as a single value that is used to approximate an unknown population parameter, based on a sample of data. It is called the "best" point estimate because it is the one that is closest to the true value of the population parameter.

The "best-point" estimate for average number of "credit-hours" per semester for all students at "local -college" is "sample-mean", which is written as :

⇒ sample mean = (sum of credit hours in sample)/(sample size),

Since study found that students earned an average of 13.8 credit hours per semester, we use this value as an estimate for population mean.

⇒ sample mean = (118 ) × (13.8)/(118),

Simplifying,

We get,

⇒ sample mean = 13.8 credit hours per semester,

Therefore, the best point estimate is 13.8 credit hours per semester.

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Answer:

The total area of the shaded region is 16 square units

Step-by-step explanation:

Main Steps:

Step 1. Find length of Square edge & Equilateral triangle leg

Step 2. Find the length from top to bottom of the full diamond

Step 3. Find the Area of the full diamond

Step 4. Find the Area that isn't shaded

Step 5. Find the Area that is shaded

Step 1. Find length of Square edge & Equilateral triangle edge

Since the Square has an area of 16 square units, using the formula for area of a square, the edge length must be 4 units

[tex]A_{square}=edge^2[/tex]

[tex](16~\text{units}^2)=edge^2[/tex]

[tex]4~\text{units}=edge[/tex] of square

Since one edge of the equilateral triangle shares one edge of the square, they are the same length.  Thus, the equilateral triangle also has edge length of 4 units.

Step 2. Find the length from top to bottom of the full diamond

Equilateral triangles have a height that is exactly [tex]\dfrac{\sqrt{3}}{2}[/tex] the length of their base.  So the height of the equilateral triangle is [tex]\dfrac{4\sqrt{3}}{2}[/tex], or more simply [tex]2\sqrt{3}~\text{units}[/tex].

The full diamond, from top to bottom, is the height of two of those equilateral triangles, plus the height of the central square:

[tex]2\sqrt{3}~\text{units} + 4~\text{units} + 2\sqrt{3}~\text{units}[/tex]

Combining like terms, the length of the Diamond diagonal from top to bottom is [tex](4 + 4\sqrt{3})~\text{units}[/tex].

Note, that due to symmetry of construction, this is also the length of the diamond's diagonal from left to right.

Step 3. Find the Area of the full diamond

One fact from geometry is that the area of a rhombus (all side lengths congruent), is given by the following formula:

[tex]A_{\text{rhombus}}=\frac{1}{2}d_{1}d_{2}[/tex], where d1 and d2 are the lengths of the two diagonals.

The "diamond" that was constructed, has all sides congruent (all sides the same length), so it is a rhombus (with matching diagonal lengths that we just calculated in Step 2)

Thus, the Area of the rhombus is:

[tex]A_{\text{rhombus}}=\frac{1}{2}((4 + 4\sqrt{3})~\text{units})((4 + 4\sqrt{3})~\text{units})[/tex]

Using FOIL,

[tex]A_{\text{rhombus}}=\frac{1}{2}(4^2 + 4*4\sqrt{3}+4*4\sqrt{3}+ (4\sqrt{3})^2)~\text{units}^2[/tex]

[tex]A_{\text{rhombus}}=\frac{1}{2}(16 + 32\sqrt{3}+ (16*3))~\text{units}^2[/tex]

[tex]A_{\text{rhombus}}=\frac{1}{2}(64 + 32\sqrt{3})~\text{units}^2[/tex]

Distributing...

[tex]A_{\text{rhombus}}=(32 + 16\sqrt{3})~\text{units}^2[/tex]

Step 4. Find the Area that isn't shaded

The area that isn't shaded is the area of the square, plus 4 equilateral triangles.

[tex]A_{\text{not shaded}}=4~A_{\text{equilateral triangle}}+A_{\text{square}}[/tex]

Recall that the area of a triangle is 1/2 * b * h

[tex]A_{\text{equilateral triangle}}=\frac{1}{2}(4)(2\sqrt{3})~\text{units}^2[/tex]

[tex]A_{\text{equilateral triangle}}=4\sqrt{3}~\text{units}^2[/tex]

[tex]A_{\text{not shaded}}=4~A_{\text{equilateral triangle}}+A_{\text{square}}[/tex]

[tex]A_{\text{not shaded}}=4~(4\sqrt{3}~\text{units}^2)+(16~\text{units}^2)[/tex]

[tex]A_{\text{not shaded}}=(16\sqrt{3}~\text{units}^2)+(16~\text{units}^2)[/tex]

[tex]A_{\text{not shaded}}=(16+16\sqrt{3})~\text{units}^2[/tex]

Step 5. Find the Area that is shaded

Lastly, the area that is shaded is the area of the full diamond/rhombus, minus the area that is not shaded:

[tex]A_{\text{shaded}}=A_{\text{Rhombus}}-A_{\text{not shaded}}[/tex]

[tex]A_{\text{shaded}}=((32+16\sqrt{3})~\text{units}^2)-((16+16\sqrt{3})~\text{units}^2)[/tex]

Combining like terms and simplifying...

[tex]A_{\text{shaded}}=16~\text{units}^2[/tex]

Answer:

x= (2010/(c+12)) +25

Step-by-step explanation:

2010= (x-25) (c+12)

x-25 = 2010/(c+12)

x= (2010/(c+12)) +25

The apothem is the vocabulary for a. The regular polygon's area is 319.8 cm².

We may use the following formula to calculate the area of the polygon:

Area = (1/2) × Perimeter × Apothem

The perimeter of a polygon is calculated by multiplying the number of sides by the length of each side, as shown below:

Perimeter = the number of sides multiplied by the length of each side

Perimeter = 6 × 14 cm (because to the regular hexagon shape)

Perimeter = 84 cm

Use the fact that it forms a right triangle with half of one side length and the apothem as the legs and the apothem as the hypotenuse to find the apothem. Thus,

a² = s² - (s/2)²

a² = 196 - 49

a² = 147

a = √(147) = 7.65 cm

Now we can enter our values into the area formula:

Area = (1/2) × Perimeter × Apothem

Area = (1/2) × 84 cm × 7.65 cm

Area = 319.8 cm²

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Step-by-step explanation:

Slope intercept form is   y = mx + b  

Re-arrange given equation  to

2y = 4x-6        divide by 2

y = 2x -3          done.

Answer:      B

Step-by-step explanation:

It is speeding up

122° 43’ 12” is written in decimal degree form, to the nearest thousandth, as approximately: 122.720.

In order to convert 122° 43’ 12” to decimal degree form, we can use the formula which is expressed as:

decimal degrees = degrees + (minutes / 60) + (seconds / 3600)

Substituting the given values into the stated formula above, we have the following:

decimal degrees = 122 + (43 / 60) + (12 / 3600)

decimal degrees = 122.720

Rounding to the nearest thousandth, we get:

122.720 ≈ 122.720

Therefore, 122° 43’ 12” is approximately equal to 122.720 degrees to the nearest thousandth.

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The equation of a circle having endpoints of diameter as (4,9) and (4,-3) is (x - 4)² + (y - 3)² = 36.

The "center" of circle is called as midpoint of diameter.

The "x-coordinate" of "mid-point" is = (4+4)/2 = 4, and

The "y-coordinate" of "mid-point" is = (9+(-3))/2 = 3.

So, center of circle is (4,3),

The radius of circle is half the length of the diameter, which is distance between two endpoints of diameter. We find distance using distance formula:

⇒ distance = √((x₂ - x₁)² + (y₂ - y₁)²), where (x₁,y₁) and (x₂,y₂) are "end-points" of diameter.

⇒ distance = √((4 - 4)² + (9 - (-3))²)

⇒ √(144) = 12.

So, radius is 6.

The equation of circle with center (h,k) and radius "r" is : ⇒ (x - h)² + (y - k)² = r²,

Substituting the values

We get,

⇒ (x - 4)² + (y - 3)² = 6²,

⇒ (x - 4)² + (y - 3)² = 36,

Therefore, the equation of the circle is (x - 4)² + (y - 3)² = 36.

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The equation of the parabola is:y = -13/36(x + 2)² + 5.

Equation is a mathematical statement that two expressions are equal. It is written as an expression that shows the equality of two values using mathematical symbols. Equations are used to solve problems by finding the unknown value. They are used in almost all branches of mathematics, and help to explain the behavior of physical and chemical systems.

The equation of a parabola with vertex (x_0,y_0) and that passes through (x_1,y_1) can be expressed as
y = a(x - x_0)² + y_0
where a is a constant.

In this case, the vertex of the parabola is (-2, 5) and it passes through (4, -8). We can substitute these values for x_0, y_0 and x_1, y_1 into the equation to get:

y = a(x - (-2))² + 5

We can solve for a by substituting the coordinates of the given point (4, -8) into the equation:

-8 = a(4 - (-2))² + 5

-8 = a(6)^2 + 5

-8 = 36a + 5

-13 = 36a

a = -13/36

Therefore, the equation of the parabola is:

y = -13/36(x + 2)²+ 5

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The required equation of the parabola is y = -13/36(x + 2)² + 5.

A parabola is a type of curve that is defined by a quadratic equation of the form y = ax² + bx + c.

In geometric terms, a parabola is the set of all points in a plane that are equidistant from a fixed point (called the focus) and a fixed line (called the directrix).

The equation of a parabola with vertex (x₁, y₁) that passes through
(x₂, y₂) can be expressed as

                        => y = a(x - x₁)² + y₁

Where a is a constant.

Here we have

The vertex of the parabola is (-2, 5) and it passes through (4, -8).

Substitute (-2, 5) in the formula y = a(x - x₁)² + y₁

=>  y = a(x - (-2))² + 5

Solve for a by substituting the coordinates of the given point (4, -8) into the equation:

=> -8 = a(4 - (-2))² + 5

=> -8 = a(6)² + 5

=> -8 = 36a + 5

=> -13 = 36a

=> a = -13/36

Therefore,

The required equation of the parabola is y = -13/36(x + 2)² + 5.

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The diameter of the circle is d = 124.73 cm

The diameter of a circle is the line that passes through the center of the circle and touches the circumference at two points.

To find the diameter of the circle, we use the formula for the area of the circle given by A = πd²/4 where d = diameter of circle

Making d subject of the formula, we have that

d = √(4A/π)

Given that the area is

substituting the values of the variables into the equation, we have that

d = √(4A/π)

d = √(4 × 98 cm²/22/7)

d = √(4 × 98 cm² × 7/22)

d = √(4 × 98 cm² × 7/22)

d = √(2744 cm²/22)

d = 124.73 cm

So, the diameter d = 124.73 cm

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There is a 0.5477 probability that a randomly selected customer will arrive less than 5 minutes after the previous customer during the 3 pm to 5 pm time period. (option a)

Given that the arrival rate of cars at the bank's drive-through window during the 3 pm to 5 pm time period is 15 customers per hour, we can calculate the average time between arrivals as follows:

Average time between arrivals = 1 / arrival rate

= 1 / 15

= 0.0667 hours (since there are 60 minutes in an hour, this is equivalent to 4 minutes)

Now, we need to find the probability of a customer arriving less than 5 minutes after the previous customer. Since the time between arrivals follows the exponential distribution, we can use the cumulative distribution function (CDF) of the exponential distribution to calculate this probability. The CDF gives us the probability of an event occurring within a certain time frame.

To find the probability of a customer arriving less than 5 minutes after the previous customer, we need to calculate the probability of an arrival occurring within a 5-minute time frame, which is equivalent to 0.0833 hours. So, we can substitute λ = 15 and x = 0.0833 in the formula for the CDF and get:

F(0.0833) = 1 - e¹⁵ˣ⁽⁻⁰°⁰⁸³³⁾

= 0.5477 (rounded to four decimal places)

Therefore, the answer is option (a) 0.5477.

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Answer:

18 - x

Step-by-step explanation:

f(x) = 6 - x

g(x) = x/3 + 8

f(2x) = 6 - (2x) => 6 - 2x

g(3x + 9) = (3x+9)/3 + 9 = x + 3 + 9 => x + 12

(6 - 2x) + (x + 12)

Simplified:

18 - x

The measure of the sides of the triangle is x = 15√3

Given data ,

We know that in the right triangle supplied, the bottom leg (adjacent side to the 30-degree angle) is "15" and the left leg (adjacent side to the 60-degree angle) is "x"

The ratio of the lengths of the side across from the angle and the side next to it is known as the tangent of an angle in a right triangle

From the trigonometric relations , we get

tan 60 = opposite side / adjacent side

tan(60°) = √3 = x/15

Multiply by 15 on both sides , we get

x = 15√3

Hence , the measure of side is x = 15√3

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The value of the car decreases from year to year by 15%.

A kind of function that would explain this type of pattern is an exponential decay function [tex]f(x) = 10000(0.15)^x[/tex].

You need the decay rate to find the function the actuary is using.

In Mathematics and Statistics, a population or physical asset (car) that decreases at a specific period of time represent an exponential decay. This ultimately implies that, a mathematical model for any population or physical asset that decreases by r percent per unit of time is an exponential equation of this form:

P(t) = I(1 - r)^t

Where:

Next, we would determine the constant ratio in each representation as follows;

Constant ratio, r = a₂/a₁ = a₃/a₂ = a₄/a₃ = a₅/₄

Constant ratio, r = 8520/10,000 = 7213/8250 = 6,100/7213 = 768/192

Constant ratio, r = 85.

By substituting the given parameters, we have the following:

P(t) = I(1 - r)^t

f(x) = 10,000(1 - 85/100)^x

f(x) = 9,600(1 - 0.85)^x

f(x) = 9,600(0.15)^x

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The distance AB is 10 units and other coordinates are calculated below

Given that

C = (3, 3) and D = (3, y)

The length is 7 units

So, we have

y - 3 = 7

Evaluate

y = 4

So, one possible value of y is 4

Given that

B = (r, 2) and C = (5, 2)

The length is 10 units

So, we have

r - 5 = 10

Evaluate

r = 15

So, one possible value of r is 15

Finding the coordinates of A

We have

AB = 4 units and B = (15, 2)

So, we have

A = (15, 2 + 4)

Evaluate

A = (15, 6)

Here, we have

A = (a - 7, b) and D = (a + 3, b)

The distance because they have the same y-coordinates is

Distance = |a - 7 - a - 3|

Evaluate

Distance = 10

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Triangle QRT with coordinates Q(-2,-1), R(1,5), and T(-8,-4) is a scalene triangle.

To classify the triangle QRT by its sides, we need to find the lengths of its three sides and compare them. We can use the distance formula to calculate the length of each side

The length of RT = √[(1 - (-8))² + (5 - (-4))²] = √[9² + 9²]

= √[162]

= 9√(2) units

The length of QT = √[(-8 - (-2))² + (-4 - (-1))²]

= √[(-6)² + (-3)²]

= √[45]

= 3√(5) units

The length of QR = √[(1 - (-2))² + (5 - (-1))²]

= √[3² + 6²]

= √[45]

= 3√(5)

Since all three sides have different lengths, triangle QRT is scalene

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The given question is incomplete, the complete question is:

Given the coordinates, classify triangle QRT by its sides. Q(-2, -1), R(1,5), T(-8,-4)

Given that the triangles are congruent, we can conclude that

From the question, we are to determine what can be concluded from the fact that triangle QGL and triangle BRN are congruent

If ΔQGL and ΔBRN are congruent, we can infer that

QG = BR

LG = NR

QL = BN

Also, in terms of angles

∠LGQ = ∠NRB

∠QLG = ∠BNR

∠LQG = ∠NBR

In addition,

If two triangles are congruent, the areas of the triangles will be equal to each other

Thus,

We can conclude that

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Answer:

Subtract 142 from 180 since angle of a straight line adds upto 180 degrees..

Ans will be =38

Answer:

MO/KO = NO/LO

5/10 = 3/LO

5/10 = 3/x

Step-by-step explanation:

Since KLO and MNO are similar, that means the corresponding sides share a common proportion:

Note how corresponding sides MO and KO are given measurements, that means we can find the common proportion by dividing MO/KO (which is 5/10). Knowing the proportion between the corresponding sides between MO and KO, we can find LO by finding its corresponding side and setting up a proportion. LO's corresponding side is NO (with a measurement of 3):

5/10 = 3/LO

Other options also work:

MO/KO = NO/LO               since MO = 5 units, KO = 10 units, & NO = 3 units

5/10 = 3/x                           since any value can be set to a variable, LO = x

Every 30th person on a list of county residents, is the most representative sample of the population of interest.  Option A

A representative sample is a small group of individuals who, as closely as possible, reflect a larger group.

The best representative sample will be obtained by taking every 30th person on a list of county residents.

This is due to the fact that it requires choosing people from the entire county's resident population as opposed to only those who frequent a certain region (options B and C) or a particular group (option D, high school students).

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The expected number of trips a customer needs to make to get a complete set of letters is 6 trips.

This problem can be modeled using the concept of a geometric distribution, where the number of trials required to achieve the first success (getting a complete set of letters) follows a geometric distribution.

The probability of getting a specific letter on any given purchase is 1/6, since there are six letters and they are equally likely to be given away. Therefore, the probability of not getting a specific letter on any given purchase is 5/6.

Let X be the random variable representing the number of purchases needed to obtain a complete set of letters. Since each purchase is independent of the others, X follows a geometric distribution with a probability of success p = 1/6.

The expected value of a geometric distribution with probability of success p is given by E(X) = 1/p. Therefore, the expected number of trips to the store that a customer needs to make to get a complete set is:

E(X) = 1/p = 1/(1/6) = 6

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Answer & Step-by-step explanation:

Part A: The explicit equation for f(n) that represents the number of lionfish in the bay after n years can be written as:

f(n) = 9000 * (1 + 0.69)^n

where 9000 is the initial number of lionfish, 0.69 is the growth rate, and n is the number of years.

Part B: To find the number of lionfish in the bay after 6 years, we substitute n = 6 into the explicit formula from Part A and simplify:

f(6) = 9000 * (1 + 0.69)^6 = 9000 * (1.69)^6 = 9000 * 11.34 = 102,060.5

Rounding to the nearest whole number, there will be approximately 102,061 lionfish in the bay after 6 years.

Part C: The recursive equation for f(n) can be found by subtracting 1,400 from the previous year's population and then applying the annual growth rate. Thus, we have:

f(1) = 9000 f(n) = f(n-1) - 1400 + 0.69f(n-1) = 0.69f(n-1) - 1400

where f(1) is the initial number of lionfish and n represents the number of years after the first year.

The probability of iii-2 and iii-3 having 2 colorblind children and 5 normal children is approximately 0.16, or 16%.

To calculate the probability of having 2 colorblind children and 5 normal children, we need to use the binomial probability formula, which is:

P(X=x) = (n choose x) x pˣ x (1-p)ⁿ⁻ˣ

where:

P(X=x) is the probability of getting k successes in n trials

(n choose k) is the binomial coefficient, which is the number of ways to choose k items from a set of n items

p is the probability of success in each trial

(1-p) is the probability of failure in each trial

x is the number of successes we want to have

In this case, we want to find the probability of having 2 colorblind children and 5 normal children, so we can plug in the values:

P(X=2) = (7 choose 2) * 0.25² * 0.75⁵

Using a calculator or computing by hand, we get:

P(X=2) = 0.16 or 16%.

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