use rational exponents to rewrite and simplify the expression

No matter how many miles are travelled, only the standard rental fee of equation $64 is paid. The total cost of the rental is determined by multiplying the miles driven by the additional $0.20 per mile price.

An equation is a mathematical statement that proves the equality of two expressions connected by an equal sign '='. For instance, 2x – 5 = 13. Expressions include 2x-5 and 13. '=' is the character that links the two expressions. A mathematical formula that has two algebraic expressions on either side of an equal sign (=) is known as an equation. It depicts the equivalency relationship between the left and right formulas. L.H.S. = R.H.S. (left side = right side) in any formula.

The following equation is used to calculate the overall cost of renting a car:

C(x) = 0.20x + 64

where x is the amount of miles driven and C(x) is the rental car's total cost.

No matter how many miles are travelled, only the standard rental fee of $64 is paid. The total cost of the rental is determined by multiplying the miles driven by the additional $0.20 per mile price.

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Accοrding tο the transfοrmatiοn prοperty, the cοrrect answer is οptiοn (c): (x, y) → (- y + 2, x + 2)

Transfοrmatiοn is a functiοn that maps οne set οf pοints tο anοther set οf pοints in a way that preserves sοme structure οr prοperties οf the οriginal set. The mοst cοmmοn types οf transfοrmatiοns are geοmetric transfοrmatiοns, which preserve distance, angle, and shape, and algebraic transfοrmatiοns, which preserve algebraic prοperties such as the sum, prοduct, οr symmetry οf functiοns.

Twο triangles are similar if their cοrrespοnding angles are cοngruent and their cοrrespοnding sides are prοpοrtiοnal. Therefοre, tο determine whether a transfοrmatiοn maps triangle ABC tο a similar, but nοt a cοngruent triangle, we need tο examine hοw the transfοrmatiοn affects the angles and sides οf triangle ABC.

Let's assume that triangle ABC has vertices A(a₁, a₂), B(b₁, b₂), and C(c₁, c₂).

a) (x, y) → (x + 2, - y - 2)This transfοrmatiοn is a translatiοn that mοves each pοint οf the plane twο units tο the right and twο units dοwn. Therefοre, this transfοrmatiοn dοes nοt change the size οr shape οf triangle ABC. It οnly changes its pοsitiοn in the plane. Hence, this transfοrmatiοn dοes nοt map triangle ABC tο a similar, but nοt cοngruent triangle.

b) (x, y) → (- 4x, - 4y)

This transfοrmatiοn scales each cοοrdinate by a factοr οf -4, which reflects the triangle acrοss the οrigin and changes its size. Hοwever, this transfοrmatiοn alsο changes the οrientatiοn οf the triangle. Therefοre, this transfοrmatiοn dοes nοt map triangle ABC tο a similar, but nοt cοngruent triangle.

c) (x, y) → (- y + 2, x + 2)

This transfοrmatiοn is a rοtatiοn οf 90 degrees cοunterclοckwise arοund the pοint (2, 2). Tο see if it maps triangle ABC tο a similar, but nοt cοngruent triangle, we need tο check if the angles and sides οf the triangle are preserved. Since this transfοrmatiοn is a rοtatiοn, it preserves the angles οf the triangle. Tο check if it preserves the sides, we need tο cοmpute the length οf each side οf the transfοrmed triangle and cοmpare it tο the cοrrespοnding side οf the οriginal triangle:

AB: distance between (- a₂ + 2, a₁ + 2) and (- b₂ + 2, b₁ + 2) is equal tο the distance between A(a₁, a₂) and B(b₁, b₂)

AC: distance between (- a₂ + 2, a₁ + 2) and (- c₂ + 2, c₁ + 2) is equal tο the distance between A(a₁, a₂) and C(c₁, c₂)

BC: distance between (- b₂ + 2, b₁ + 2) and (- c₂ + 2, c₁ + 2) is equal tο the distance between B(b₁, b₂) and C(c₁, c₂)

Since the angles and sides are preserved, this transfοrmatiοn maps triangle ABC tο a similar, but nοt cοngruent triangle.

d) (x, y) → (- y, - x)

This transfοrmatiοn is a reflectiοn acrοss the line y = x. Tο see if it maps triangle ABC tο a similar, but nοt cοngruent triangle, we need tο check if the angles and sides οf the triangle are preserved. Since this transfοrmatiοn is a reflectiοn, it preserves the angles οf the triangle. Hοwever, it alsο changes the οrientatiοn οf the triangle. Therefοre, this transfοrmatiοn dοes nοt map triangle ABC tο a similar, but nοt cοngruent triangle.

Therefοre, the answer is οptiοn (c): (x, y) → (- y + 2, x + 2)

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Answer: To calculate the real value of your savings after inflation, you need to adjust your savings for the increase in the price level. Here's how to do it:

Calculate the amount of inflation:

Inflation rate = (New price level index - Old price level index) / Old price level index

In this case, the old price level index is 100, and the new price level index is 109 (100 + 9%). So:

Inflation rate = (109 - 100) / 100 = 0.09

Adjust your savings for inflation:

Real value of savings = Nominal value of savings / (1 + inflation rate)

In this case:

Real value of savings = 8,295 / (1 + 0.09) = 7,614.22

Therefore, the real value of your savings after inflation is $7,614.22.

Step-by-step explanation:

Therefore , the solution of the given problem of unitary method comes out to be 25 vehicles had precisely two of the available options.

By multiplying the data obtained with such a nanosection variable whilst also two that used the unilateral strategy, the job can be finished. In essence, this mean that whenever a desired item appears, the specified entity is set or the colour space of both production runs is skipped. A varying fee of Inr ($1.01) might have been necessary for forty pens.

Here,

We can apply the inclusion-exclusion concept to resolve this issue. To begin, let's draw a Venn diagram to symbolise the supplied data:

where A is the proportion of vehicles with air conditioning, B is the proportion of vehicles with automatic gearbox, and C is the proportion of vehicles with power steering.

The information provided lets us know:

=> A = 87 + 8 + 24 = 119

=> B = 99 + 8 + 9 = 116

=> C = 73 + 8 + 9 = 90

=> A ∩ B ∩ C = 8

=> A ∩ B ∩ C' = 24 - 8 = 16

=> A ∩ B' ∩ C = 9

=> A ∩ B' ∩ C' = 64 - 9 - 8 - 16 = 31

=> A' ∩ B ∩ C = 0

=> A' ∩ B ∩ C' = 35 - 9 - 8 - 16 = 2

=> A' ∩ B' ∩ C = 0

=>  A' ∩ B' ∩ C' = 24 - 8 - 16 - 2 = -2

Since there cannot be fewer than 0 cars without any extras, this number must be negative.

=> (A ∩ B' ∩ C) + (A ∩ B ∩ C') + (A' ∩ B ∩ C) = 9 + 16 + 0 = 25

As a result, 25 vehicles had precisely two of the available options. There may have been a mistake in the problem or the answer alternatives since the correct response is not one of the available choices.

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Answer:

Since ΔQPT ≅ ΔARZ, we know that their corresponding angles are congruent.

Step-by-step explanation:

The congruent angle pair is:

C) ∠T ≅ ∠Z.

Note that ∠Q and ∠R are not necessarily congruent to each other, ∠P and ∠A are not necessarily congruent to each other, and ∠P and ∠Z are not necessarily congruent to each other.

The area after dilation with a scale factor of 2 will be 56 sq feet.

When a figure is dilated with a scale factor of 2, the area of the figure is multiplied by the square of the scale factor.

So, if the original area is 14 sq feet, then the new area after dilation will be:

New area = Original area x (scale factor)^2

= 14 x 2^2

= 14 x 4

= 56 sq feet.

Therefore, the new area of the figure after dilation with a scale factor of 2 is 56 sq feet.

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Answer:

Step-by-step explanation: 0.3375 by simple algebra.

Triangle ABC is congruent to triangle PQR, where A corresponds to P, B corresponds to Q, and C corresponds to R.

What is congruency?

In geometry, two figures or objects are congruent if they have the same shape and size, or if one has the same shape and size as the mirror image of the other.

To write a congruence statement for two triangles, we need to identify their corresponding parts and ensure that they are congruent in both triangles.

The congruence statement can be written in the following form:

Triangle ABC is congruent to triangle PQR, where A corresponds to P, B corresponds to Q, and C corresponds to R.

For example, if we have two triangles with vertices A, B, and C and P, Q, and R respectively, and we know that the following pairs of corresponding parts are congruent:

AB ≅ PQ

BC ≅ QR

AC ≅ PR

Then, we can write the congruence statement as:

Hence, Triangle ABC is congruent to triangle PQR, where A corresponds to P, B corresponds to Q, and C corresponds to R.

The symbol ≅ means "is congruent to."

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Area of a triangle is 88.37cm²

The area of a triangle is given by the formula:

[tex]Area = \frac{(base * height)}{2}[/tex]

where "base" refers to the length of the base of the triangle, and "height" refers to the height of the triangle measured perpendicularly from the base to the highest point of the triangle.

As per figure we have to find out the area of triangle, first we have to find out the height of triangle,

by using sine rule:

Sin 67° = [tex]\frac{height}{12 cm}[/tex]

height = 12 Sin 67°       (Here, Sin 67° = 0.92)

height = 12 × 0.92

height = 11.046cm

[tex]Area = \frac{(base * height)}{2}[/tex]

[tex]Area = \frac{(16 X 11.046)}{2}[/tex]

[tex]Area = 88.37cm^{2}[/tex]

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Answer:

Toyin weighs 50 kg, which is 20 kg more than Amara.

Step-by-step explanation:

If Amara weighs 30 kg and the ratio of the weight of Amara to the weight of Toyin is 3:5, then we can set up a proportion:

3/5 = 30/x

where x is the weight of Toyin in kg.

To solve for x, we can cross-multiply:

3x = 150

x = 50

Therefore, Toyin weighs 50 kg, which is 20 kg more than Amara.

The expected value of the proposition is $18.23.

Probability is a branch of mathematics that deals with the study of random events or outcomes. It is the measure of the likelihood or chance of an event occurring, expressed as a number between 0 and 1. An event with a probability of 0 is impossible, while an event with a probability of 1 is certain to occur.

Given by the question.

Step 1 of 2 : Find the expected value of the proposition. Round your answer to two decimal places. Losses must be expressed as negative values.

The probability of drawing a card with a value of five or less is:

4 cards with a value of 2 + 4 cards with a value of 3 + 4 cards with a value of 4 + 4 cards with a value of 5 + 4 aces = 20/52 = 5/13

The probability of not drawing a card with a value of five or less is:

1 - 5/13 = 8/13

The expected value of the proposition can be calculated as follows:

Expected value = (probability of winning * amount won) + (probability of losing * amount lost)

Expected value = (5/13 * $49) + (8/13 * (-$25))

Expected value = $18.23 (rounded to two decimal places)

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Answer:

Since wave heights at Folly Beach are normally distributed with a mean of 4 feet and a standard deviation of 1.5 feet, we can use the normal distribution formula to find the probability of a wave having a height of exactly 3 feet.

The formula for the normal distribution is:

P(x) = (1 / (σ * sqrt(2π))) * e^(-(x - μ)^2 / (2 * σ^2))

where:

P(x) is the probability density function

σ is the standard deviation

μ is the mean

e is the base of the natural logarithm

x is the value of the variable

Substituting the values given in the problem, we get:

P(3) = (1 / (1.5 * sqrt(2π))) * e^(-(3 - 4)^2 / (2 * 1.5^2))

P(3) = 0.176

Therefore, the probability that a wave has a height of exactly 3 feet is 0.176 or approximately 17.6%.

Answer:

The given differential equation is:

2dv/dt = -kv

where k is a constant.

a) To find the general solution to this differential equation, we can use separation of variables.

2dv/v = -k dt

Integrating both sides, we get:

2ln|v| = -kt + C1

where C1 is the constant of integration.

Taking exponential of both sides, we get:

|v|^2 = e^(C1) e^(-kt)

Since v can be either positive or negative, we can simplify the above equation as:

v = ±√(Ae^(-kt))

where A = e^(C1).

Therefore, the general solution to the given differential equation is:

v(t) = ±√(Ae^(-kt))

b) We are given that the initial velocity is 40 m/s and after 2 seconds, the velocity has decreased to 30 m/s. Let's use this information to find the value of A.

When t = 0, v = 40. Therefore,

40 = ±√(Ae^0)

40^2 = A

A = 1600

Now we can use the value of A to find the velocity after 10 seconds.

v(10) = ±√(1600e^(-10k))

We can use the information that the velocity has decreased to 30 m/s after 2 seconds to find the value of k.

30 = ±√(1600e^(-2k))

900 = 1600e^(-2k)

e^(-2k) = 0.5625

-2k = ln(0.5625)

k = -0.5 ln(0.5625)

Now we can substitute this value of k into the expression for v(10) to get:

v(10) = ±√(1600e^(5ln(0.5625)))

v(10) = ±√(1600(0.5625)^5)

v(10) = ±20.11 m/s

Therefore, the velocity of the object after ten seconds will be approximately 20.11 m/s.

The equation for the shifted graph is y = x + 6.This equation represents the same line as the original equation, y = x + 1, but shifted up by 5 units.

Shifting a graph involves moving it horizontally or vertically while maintaining the same shape and orientation.

To shift the graph of y = x + 1 up by 5 units, you need to add 5 to the y-coordinate of every point on the graph.

The original graph of y = x + 1 is a straight line with slope 1 and y-intercept 1.

To create the shifted graph, you can start by plotting a few points on the original graph and then adding 5 to their y-coordinates to get the corresponding points on the shifted graph.

Alternatively, you can use the general equation for a straight line, y = mx + b, where m is the slope and b is the y-intercept, to find the equation for the shifted graph.

The slope of the original graph is 1, so the slope of the shifted graph is also 1.

To find the y-intercept of the shifted graph, you can plug in x = 0 and y = 6 (since you want the graph to pass through the point (0, 6)) and solve for b:

6 = 1(0) + b

b = 6

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Therefore , the solution of the given problem of triangle comes out to be a length measures about 5.64 units in length.

If a polygon has more than one extra sections, it is a hexagon. It has a straightforward square form. Something like this configuration can only be distinguished from a conventional triangle by the sides A and B. The borders continue to be precisely collinear, but Euclidean geometry only yields a section rather than the entire cube. A triangular has three sides and three angles.

Here,

Let's determine the length of side b using the sine function:

opposite/hypotenuse of sin(A)

=> b/9 sin(50) = 9*sin(50) b = 7.021

Side B is therefore roughly 7.021 units long.

We can use the knowledge that the sum of the angles in a triangle is 180 degrees to determine the size of angle B:

=> angle B = 180 - angle A - 90

=>  angle B = 180 - 50 - 90

=>  angle B = 40 degrees

B's angle therefore has a 40 degree value.

The Pythagorean formula can also be used to determine the length of side a:

The equation is:

=> a² + b² = c²

=> a² + 7.021² = 9²

=> a² + 49.24 = 81

=> a² = 31.76

=> a ≈ 5.64

Side a therefore measures about 5.64 units in length.

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Given functions f(x) = 2x² + 1 and g(x) = 7x - 8, then [tex]f[g(2) = 73\end{array}$[/tex].

A technical definitiοn οf a functiοn is: a relatiοn frοm a set οf inputs tο a set οf pοssible οutputs where each input is related tο exactly οne οutput.

This means that if the οbject x is in the set οf inputs (called the dοmain) then a functiοn f will map the οbject x tο exactly οne οbject f(x) in the set οf pοssible οutputs (called the cοdοmain).

The nοtiοn οf a functiοn is easily understοοd using the metaphοr οf a functiοn machine that takes in an οbject fοr its input and, based οn that input, spits οut anοther οbject as its οutput.

Given f(x) = 2x² + 1 and g(x) = 7x - 8

Then

a)

[tex]$\begin{array}{l c l}{f[g(2)]=f[(7*2-8)]=f(6)}\\ {f[g(2)]=2(6)^{2}+1=72+1}\\f[g(2) = 73\end{array}$[/tex]

b)

[tex]$\begin{array}{l}{{f[g(x)]=f(7x-1)=2[7x-8]^{2}+1}}\\ {{f[g(x)]=2(49x^{2}+64-112x)+1}}\\ {{f[g(x)]=98x^{2}-224x+129}}\end{array}$[/tex]

c)

[tex]$\begin{array}}{g[f(x)]=g[2x^{2}+1]=7[2x^{2}+1]-8}\\ {g[f(x)]=14x^{2}-1}\end{array}$[/tex]

d) (g · g)(x) = g{g(x)} = g(7x - 8)  

= 7(7x - 8) - 8

= 49x−64

e) (f · f)(-2) = f{f(-2)} = f(2(-2)²)

= 2(2(-2)²)²

= 128

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Therefore , the solution of the given problem of function comes out to be  f(x) = -6x + 8 .

Numerous subjects, including numbers, numbers, as well as their subsets, as well as building, construction, and both real and fictitious geographic locations, are covered in the mathematics programme. The relationships between variable elements that all cooperate to create the same outcome are covered in a work. A utility is composed of several unique parts that, when combined, give particular outcomes for each input.

Here,

The linear function with a slope of -6 can be written in slope-intercept form as:

=> f(x) = -6x + b

where "b" is the y-intercept. We are also given that f(0) = 8, so we can substitute those values in to solve for "b":

=> f(0) = -6(0) + b = 8

=> b = 8

Therefore, the linear function we are looking for is:

=> f(x) = -6x + 8

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The values are given as;

1. The intercept, c = -1.5, Slope, m = -1/4

2. Intercept = 0. 5, Slope, m = -3/5

The general formula for the equation of a line is represented thus;

y = mx + c

Given that the parameter are;

For the first graph

The intercept, c = -1.5

The slope, m = y₂ - y₁/x₂ - x₁

Substitute the values as shown in the graph

Slope, m = -2-(-3)/- 3- 1

Slope, m = -2 + 3/-4

Slope, m = -1/4

For the second graph, we have

Intercept = 0. 5

The slope, m = y₂ - y₁/x₂ - x₁

Substitute the values

Slope, m = 2 - (-1)/3 - (-2)

Slope, m = -3/5

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A petition is just a formal request outlining the grounds for requesting a court order. It is often the first stage of a lawsuit and can be filed by an individual, a group, or an institution.

A petition is a formal letter or written request asking someone or an organization to take a certain action or make a certain change.

The petition is typically signed by a number of people who agree with the subject or problem being raised.

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Answer:

Step-by-step explanation:

Answer: The linear equation in slope-intercept form that models this situation is:

y = 37x + 85

In this equation, the variable x represents the time spent repairing the vehicle in hours.

The variable y represents the total cost of the repair in dollars.

Step-by-step explanation:

An equation of the line in slope-intercept form drawn on the graph is y = 3x/2 + 8.

In Mathematics and Geometry, the point-slope form of any straight line can be determined by using this mathematical expression:

y - y₁ = m(x - x₁)

Where:

Next, we would determine the slope;

Slope (m) = (y₂ - y₁)/(x₂ - x₁)

Slope (m) = (20 - 8)/(8 - 0)

Slope (m) = 12/8

Slope (m) = 3/2

At data point (0, 8) and a slope of 3/2, a linear equation for this line can be determined by using the point-slope form as follows:

y - y₁ = m(x - x₁)

y - 8 = 3/2(x - 0)  

y = 3x/2 + 8

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Answer:

Counting the number of sign changes in the function f(x)

The number of positive roots is equal to the number of sign changes in the function f(x) or less than that by an even integer.

Here, f(x)=3x^4-5x^3-x^2 has two sign changes (from + to -, and from - to +), so the number of positive roots is either 2 or 0.

Counting the number of sign changes in the function f(-x)

The number of negative roots is equal to the number of sign changes in the function f(-x) or less than that by an even integer.

Here, f(-x)=3x^4+5x^3-x^2 has one sign change (from - to +), so the number of negative roots is either 1 or 0.

Counting the number of non-real roots (complex roots)

The number of non-real roots (complex roots) is equal to the difference between the total number of roots (4 in this case) and the number of real roots (which we found above).

Therefore, the possible number of positive roots is 2 or 0, the possible number of negative roots is 1 or 0, and the possible number of complex roots is 2 or 4.

Step-by-step explanation:

The value of the given expression is -60. The solution has been obtained by using arithmetic operations.

What are arithmetic operations?

The four basic operations, also referred to as "arithmetic operations," are said to be able to describe all real numbers. The four mathematical operations following division, multiplication, addition, and subtraction are quotient, product, sum, and difference.

We are given two functions as f(x) = [tex]\frac{2x}{3}[/tex] + 7 and g(x) = 2x + 15

So,

⇒f (0) = [tex]\frac{0}{3}[/tex] + 7

⇒f (0) = 7

Similarly,

⇒g (-8) = 2 (-8) + 15

⇒g (-8) = -16 + 15

⇒g (-8) = -1

Now, on substituting these in the expression, we get

⇒-8 * f (0) + 4 * g (-8)

⇒-8 * 7 + 4 * (-1)

Now using the arithmetic operations, we get

⇒-56 -4

⇒-60

Hence, the value of the given expression is -60.

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Question:

If the following equations represent the functions f(x) and g(x).

f(x) = 2/3x + 7 and g(x) = 2x + 15

Calculate the value of -8 * f(0) + 4 * g(-8)

Answer:

2.700 × 10-7

Step-by-step explanation:

Answer:  2.7*10^-4

step by step explanation:

1. Make the number a new number between 1 and 10

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Move the decimal point to make 0.00027 a new number between 1 and 10. Because our original number is less than one, we move the decimal point to the right. Drop any zeroes in front of the number. Keep track of how many times we move the decimal point.

0.00027 -> 2.7

Our new number is 2.7. We moved the decimal point 4 times.

2. Define the power of 10

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Because our original number was less than one, the exponent defining the power of 10 is negative. Remember, we moved the decimal point 4 times, so the exponent is negative 4

10^(-4)

3. Final result

2.7*10^(-4)

Therefore, the missing side length is 162 when triangle ABC and XYZ are similar.

A triangle is a closed two-dimensional plane figure with three straight sides and three angles. It is one of the basic shapes in geometry, and it is formed by connecting three non-collinear points with line segments. The sum of the three angles in a triangle is always 180 degrees. Triangles can be classified based on their side lengths and angle measures, and they have important properties and applications in various fields, including mathematics, science, engineering, and art.

Here,

Since triangles ABC and XYZ are similar, their corresponding sides are in proportion. We can set up the following proportion using the given side lengths:

AC/XY = BC/YZ

Substituting the given values, we get:

2/54 = 6/x

Cross-multiplying, we get:

2x = 54 × 6

2x = 324

Dividing both sides by 2, we get:

x = 162 units

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The study that we have here is an observational study. The study is not retrospective in nature.

This study is observational because the researchers are observing and collecting data on a group without manipulating any variables.

It is not a retrospective study because the data is collected in real-time during the first year of college, rather than looking back on past experiences.

Therefore, there are no treatment or control groups, and there is no need for blinding. The study simply reports the percentage of first-year women students who experienced sexual assault during their first year at the regional college.

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Answer:

A) True

Step-by-step explanation:

The statement is true. Conditional probability is a fundamental concept in probability theory that allows us to calculate the probability of an event given that another event has already occurred. The notation for conditional probability is P(B|A), which represents the probability of event B occurring given that event A has already occurred. In other words, we restrict our attention to the sample space where event A has occurred and calculate the probability of event B in that space.

Answer: (4,-2)

Explanation: (I was forced to put one)

The vertex form of a quadratic function is f(x) = a(x – h)2 + k, where a, h, and k are constants.

Answer:

Therefore, the value of the unknown base is n = 16.

Step-by-step explanation:

Using the definition of logarithms, we have:

log_n(1/4) = -1/2

This equation can be rewritten as:

n^(-1/2) = 1/4

To solve for n, we can raise both sides to the -2 power:

n^(-1/2)^(-2) = (1/4)^(-2)

n = 16

Therefore, the value of the unknown base is n = 16.