Is differentiating 8c to the fourth power equal to four times 8c to the 4-1 power?

[tex]8x^3y^9\\[/tex] is equivalent to [tex]8x^4y^12[/tex] when expressed in terms of factors.

An expression is a combination of numbers, variables, and mathematical symbols that represent a mathematical relationship or formula. Expressions can be simple or complex and can include operations such as addition, subtraction, multiplication, and division, as well as exponents, radicals, and functions.

In the given question,

One possible expression that is equivalent to [tex]8x^3y^9[/tex] is:

[tex](2x^2y^3)^2 * (2y^6)\\[/tex]

When we expand this expression, we get:

[tex](2x^2y^3)^2 = (2^2)(x^2)^2(y^3)^2 = 4x^4y^6(2y^6) = 2y^6[/tex]

Multiplying these two expressions together, we get:

[tex](4x^4y^6) * (2y^6) = 8x^4y^12[/tex]

So [tex]8x^3y^9[/tex] is equivalent to [tex]8x^4y^12[/tex] when expressed in terms of factors.

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Answer:

4.8 litres

Step-by-step explanation:

Based on the given information, Elizabeth needs 5 boxes of blueberries for each litre of smoothie. She has 24 boxes of blueberries available.

So, the total amount of smoothie that Elizabeth can make in litres would be:

24 boxes of blueberries / 5 boxes of blueberries per litre

This can be calculated as:

24 boxes ÷ 5 boxes per litre = 4.8 litres

So, Elizabeth can make 4.8 litres of smoothie using the available 24 boxes of blueberries.

Under the HL criteria, the triangles are congruent. The proof is so finished.

Since M is the midpoint of BC, we have BM = CM.

Therefore, by Hypotenuse-Leg (HL) congruence criterion, we have:

Therefore, ∆ABM ≅ ∆DCM; the triangles are congruent by HL criterion.

Hence, the proof is complete.

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The antiderivative of cos(sin x) is 1/2 (sin⁻¹(sin x) + 1/2 sin(2 sin⁻¹(sin x))) + C.

Integration is a mathematical technique that involves finding the integral of a function. The integral of a function is the area under the curve of the function, and it is represented by a symbol ∫. Integration is the inverse of differentiation, which is a technique for finding the derivative of a function.

There are two main types of integrals: definite integrals and indefinite integrals. A definite integral is the integral of a function over a specific interval, while an indefinite integral is the integral of a function without specifying the interval. Definite integrals are used to find the area under the curve of a function between two points, while indefinite integrals are used to find a family of functions that differ by a constant.

Let u = sin x, then du/dx = cos x and dx = du/cos x. Substituting these expressions into the integral, we get:

∫ cos(sin x) dx = ∫ cos(u) du/cos x

= ∫ cos(u) du / √(1 - u²)

We can now use a trigonometric substitution to evaluate this integral. Let u = sin θ, then du/dθ = cos θ and:

cos² θ + sin² θ = 1

cos² θ = 1 - sin² θ = 1 - u²

Substituting these expressions into the integral, we get:

∫ cos(u) du / √(1 - u²) = ∫ cos(θ) cos θ dθ

= ∫ cos² θ dθ

= ∫ (1 + cos(2θ))/2 dθ

= 1/2 ∫ dθ + 1/2 ∫ cos(2θ) dθ

= 1/2 (θ + 1/2 sin(2θ)) + C

Substituting back u = sin x and θ = sin⁻¹(u), we get:

∫ cos(sin x) dx = 1/2 (sin⁻¹(sin x) + 1/2 sin(2 sin⁻¹(sin x))) + C

Therefore, this is the antiderivative of cos(sin x).

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Answer:

$18.28

Step-by-step explanation:

To complete this problem is a straightforward answer:

First, you add 14.75 + 1.03 which equals $15.78 then, you'll want to add 2.50 to 15.78 and there you have your answer!!

Answer:

x^2 - 4x + 1 = 0

x^2 - 4x + 4 = 3

(x - 2)^2 = 3

x - 2 = +√3

x = 2 + √3

Answer:

x = 2 ± √3

Step-by-step explanation:

     x² - 4x + 1 = 0

          x² - 4x = -1

Divide the co-efficient of x by 2. Co-efficient of x is 4, 4÷ 2 = 2.

Add the square of 2 to both sides. Square of 2 is 4.

    x² - 4x + 4 = -1 + 4

x² - 2*2x + 2² = 3

{LHS is in the from a² - 2ab + b² = (a -b)²}

          (x - 2)² = 3

Take square root,

          x - 2 = ±√3

Add 2 to both sides,

               x = 2 ± √3

Hence by using Pythagoras the height of tree before it break is 47.02m

Describe Pythagoras.

The Ionian Greek philosopher and mathematician Pythagoras is credited with creating Pythagoreanism. He was born in 570 BCE in Samos, Ionia (Greece), and passed away at Metapontum, Lucania, between 500 and 490 BCE. (Italy). The three sides of a right-angled triangle are the subject of Pythagoras' most famous theorem, which is now referred as as Pythagoras' theorem.

Let A'CB represent the tree before it broke at point C, and let A' be the point at which the tree's top contacts the earth after it broke. Then, at B, ABC is a right triangle.

AB=20 m and BC=15 m

Using Pythagoras theorem,   H²  = P²+ B²

In ΔABC

(AC) ²

=(AB)² +(BC) ²

(AC) ²

=(20) ²+15)²

(AC) ² =400+625

(AC) ²=1025

AC=  32.02

​Hence total height is 32.02+15 =47.02m

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The reasonable prediction through which a green or black marble will be drawn is 35 times.

What about prediction?

In mathematics, prediction refers to the process of using existing data or patterns to make an educated guess about a future event or outcome. It is the process of estimating an unknown value or quantity based on a set of known data or observations.

Prediction is often used in statistics, where data analysis techniques are used to identify patterns and trends in data. Once these patterns are identified, they can be used to make predictions about future events or outcomes. For example, in finance, prediction models are often used to forecast stock prices or market trends.

The accuracy of predictions in mathematics can be improved by using more data, improving the quality of the data, or refining the models and algorithms used to make the predictions. However, even with the best models and data, there is always some degree of uncertainty or error associated with predictions, and this should be taken into account when using predictions to make decisions.

According to the given information:

Given, there are  5 red marbles, 6 blue marbles, 3 green marbles

4 black marbles, 2 yellow marbles .

Here, it is given that marble will be drawn and replace 100 times.

To find reasonable prediction for no of times green or black marble will be drawn

⇒ Total no of times = 100 times

⇒ [tex]\frac{4}{20}[/tex] x 100 +   [tex]\frac{3}{20}[/tex]  x 100

⇒ [tex]\frac{7}{20}[/tex]  x 100

⇒ 35 times

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Answer:

107.7

Step-by-step explanation:

Answer: Use the steps that are easiest to draw a parabola!

Step-by-step explanation:

I used Desmos, here are the key points.

Vertex: (-5,-2)

Extra points to graph:

Hope this helps!

The probability that both Brian and Leo are both assigned window seats on the way to their grandmother's house is the product of the probabilities of each event occurring independently.

Probability is a measure of the likelihood of an event occurring. It is a number between 0 and 1, where 0 represents an impossible event and 1 represents a certain event.

The probability of an event can be determined by dividing the number of favorable outcomes by the total number of possible outcomes. For example, if a coin is tossed, there are two possible outcomes: heads or tails.

In the given question,

The probability that Brian is assigned a window seat on any one flight is 50/150, which simplifies to 1/3. Since there are two flights involved (one to his grandmother's house and one back), we can think of this as two independent events.

The probability that both Brian and Leo are both assigned window seats on the way to their grandmother's house is the product of the probabilities of each event occurring independently.

P(both assigned window seats on the way there) = P(Brian gets window seat) x P(Leo gets window seat) = (1/3) x (1/3) = 1/9.

The probability that Brian is assigned a window seat on the flight to his grandmother's house and the flight home from his grandmother's house is the probability of the intersection of two events: Brian getting a window seat on the flight there and Brian getting a window seat on the flight back.

Since these are two independent events, we can multiply their probabilities:

P(Brian gets window seat on flight there and back) = P(Brian gets window seat on flight there) x P(Brian gets window seat on flight back) = (1/3) x (1/3) = 1/9.

Comparing the two probabilities, we can see that they are the same:

P(both assigned window seats on the way there) = P(Brian gets window seat on flight there and back) = 1/9.

Therefore, the answer to the second part of the question is "the same as".The probability that both Brian and Leo are both assigned window seats on the way to their grandmother's house is the product of the probabilities of each event occurring independently.

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The predictions for the number of times that the following sums would occur are as follows:

To make accurate predictions, we need to follow the binomial distribution rule that assumes the darts to be independent each of other. By this rule, the number of possibilities of the darts being thrown is 2 * 80 which equals 160.

Next, there are three possible ways to achieve the sum of 8 which are (1, 7), (2, 6), or (3, 5). The probabilities are: 1/20) * (1/20) = 1/400. Now we multiply these by the number of possibilities to arrive at:

160 * (3/400) = 12

For the sum of 10, there are also three possible ways, namely: (1, 9), (2, 8), or (3, 7). The probability of getting a sum of 10 on any given throw is 3/400. When we express this, we arrive at:

160 * (3/400) = 12

For the sum of 14, there are two possible ways, namely: (5, 9) or (6, 8). Given the probability of 1/400, the probability of getting a sum of 14 on any given throw is 2/400. When we express this with the binomial distribution, we will arrive at 160 * (2/400) = 8

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Answer:

u = -1

Step-by-step explanation:

5(u + 6) - 8u = 33

5u + 30 - 8u = 33

5u - 8u + 30 = 33

-3u + 30 = 33

      -30  = -30

-3u         = 3

u = -1

[tex]3.9ft^{2}[/tex]

Therefore , the solution of the given problem of triangle comes out to be horizontal dimension of 16 inches, and an approximate hypotenuse of 28.84 inches.

A polygon is a hexagon if it has at least one additional segment. Its shape is a straightforward rectangle. Only edges between A and B can separate anything similar from a regular triangle form. The cube is only partially formed by Euclidean geometry, even though the edges are entirely collinear. A circle with four sides and three angles make compose a triangle.

Here,

The backrest is described as having a 24 inch vertical dimension and a 16 inch horizontal dimension. Although the hypotenuse is not named, we may nevertheless determine its length using the Pythagorean theorem:

=> h² = 24² + 16²

=> h² = 576 + 256

=> h² = 832

=> h= 28.84 inches high

As a result, the backrest has the following measurements: a vertical dimension of 24 inches, a horizontal dimension of 16 inches, and an approximate hypotenuse of 28.84 inches.

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Each game should cost us roughly 8 cents.This game's expected value is negative, thus playing it is not a good idea because you will most likely lose money in the long run.

Given problem describes that there can be 4 unique ball colour combinations as :

Provided that each colour contains two balls in the respective boxes, each combination has a probability of happening of 2/4 * 1/3 = 1/6. The likelihood that the payoff will be zero is thus 4 * 1/6 = 2/3.

Red balls make two different combinations as: (red, red) in left box and (red, red) in  right box. Due to the fact that each box contains just one red ball, each combination has a probability of happening of 1/4 * 1/3 = 1/12. The likelihood that the dividend will be $0.50 is thus 2 * 1/12, or 1/6

Blue balls makes two different combinations as: blue and blue in the left box and (blue, blue) in the right box. Due to the fact that each box contains just one blue ball, each combination has a probability of happening of 1/4 * 1/3 = 1/12. Hence, the likelihood that the payoff will be $1 is 2 * 1/12, or 1/6.

In the left box, there is only one conceivable pair of balls that are both green: (green, green). Given that there is only one green ball in the left box, this combination has a probability of happening of 1/4 * 1/3, or 1/12. Hence, there is a 1/12 chance that the payment will be $16.

[tex]Payout\; |\;Probability\\ \ $0 \qquad \qquad 2/3\\$0.50 \quad \qquad 1/6\\$1 \qquad \qquad 1/6\\$16 \qquad \qquad 1/12[/tex]

By multiplying each payout by the associated probability and adding them together, the anticipated value is determined. The predicted outcome in this situation is:

[tex]\$ 0 * 2/3 + \$0.50 * 1/6 + \$1 * 1/6 + \$16 * 1/12 - \$1 = -\$0.0833[/tex]

The anticipated value of playing this game, according to the revised figures above, is [tex]-\$0.0833[/tex], which indicates that you should anticipate losing on average roughly $0.08 every game. Thus, it is unprofitable game to play if played on regular basis.

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During summer vacation, Alexa read, on average, 10 pages per night. The number of pages per night Alexa averaging now is 2 pages.

A % is a quantity or ratio that, in mathematics, represents a portion of one hundred. A dimensionless relationship between two numbers can be represented in a variety of ways, such as through ratios, fractions, and decimals.

By calculating the percentage of the original Wesley is now reading, we can make the inquiry a little simpler. He is reading 20% of what he used to read rather than 80% less, for example.

Now we need to set up a proportion to solve the problem.

20% of the 10

20/100 = x/10

Cross multiply

20 x 10 = 200

200/100 = 2

He is currently reading 2 pages per night

Thus, the number of pages is 2.

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The answer of the given question based on the quadratic expression is Vertex: (3, -1) ,  Axis of symmetry: x = 3 , y-intercept: (0, 8) , x-intercepts: (2, 0) and (4, 0) , Shape: upward-opening parabola.

A quadratic expression is a mathematical expression that can be written in the form ax² + bx + c, where a, b, and c are constants (numbers) and x is a variable. The term "quadratic" comes from the Latin word "quadratus," which means "square."

To graph the quadratic function f(x) = x² - 6x + 8, we can first find the vertex of the parabola by using the formula x = -b/2a. In this case, a = 1 and b = -6, so the x-coordinate of the vertex is x = -(-6)/(2*1) = 3.

To find the y-coordinate of the vertex, we can substitute x = 3 into the expression for f(x):

f(3) = (3)² - 6(3) + 8 = -1

Therefore, the vertex of the parabola is (3, -1).

we can find y-intercept by setting x = 0:

f(0) = (0)² - 6(0) + 8 = 8

So the y-intercept is (0, 8).

To find the x-intercepts, we can set f(x) = 0 and solve for x:

x² - 6x + 8 = 0

(x - 4)(x - 2) = 0

x = 2 or x = 4

So the x-intercepts are (2, 0) and (4, 0).

Finally, we can see that the leading coefficient of the quadratic expression is positive (a = 1), so the parabola opens upward.

Therefore, the key features of the graph of f(x) = x² - 6x + 8 are:

Vertex: (3, -1)

Axis of symmetry: x = 3

y-intercept: (0, 8)

x-intercepts: (2, 0) and (4, 0)

Shape: upward-opening parabola.

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Answer:

The perimeter is 5 + 12 + 13 = 26 units.

Answer

its scale factor is ​4/3 and 0.75

Step-by-step explanation:

Answer:

Part A: $3,415.50

B: 11.34%

Step-by-step explanation:

Part A: $31,000 is within the values 10,276≤x≤41,175, so use f(x)=0.12x-205.50

Part B: For the effective tax rate, divide the amount in part A by the taxable income

Part C: Compare both the taxable income and the effective tax rate to the income domains given and the % multiplier. This one is mostly about how you describe the situation, so I'll leave that up to you.

To calculate the taxes owed on a taxable income of $31,000, we use the appropriate equation for the tax bracket it falls into and substitute the value of x. The effective tax rate is calculated by dividing the amount of taxes owed by the taxable income and multiplying by 100. The piecewise function represents the marginal tax rate chart by using different equations for each tax bracket.

Part A:

To calculate the amount of taxes owed on a taxable income of $31,000, we need to determine which tax bracket it falls into. Since $31,000 is greater than $10,275 but less than $41,175, it falls into tax bracket 2. To find the amount of taxes owed, we use the equation for tax bracket 2: f(x) = 0.12x - 205.50. Plugging in $31,000 for x, we get:

f(x) = 0.12 * 31000 - 205.50 = $3,574.50

Therefore, the amount of taxes owed on a taxable income of $31,000 is $3,574.50.

Part B:

To determine the effective tax rate, we divide the amount of taxes owed by the taxable income. Using the result from Part A (taxes owed = $3,574.50) and the taxable income of $31,000, we have:

Effective tax rate = (taxes owed / taxable income) * 100 = (3,574.50 / 31,000) * 100 ≈ 11.52%

Therefore, the effective tax rate on a taxable income of $31,000 is approximately 11.52%.

Part C:

The piecewise function represents the amount of taxes owed as a function of the taxable income. Each part of the function corresponds to a different tax bracket, with the equation for that tax bracket. The marginal tax rate chart is represented in the piecewise function by the different equations for each tax bracket. For example, the equation in part 1 of the function (f(x) = 0.10x) corresponds to the 10% marginal tax rate for the tax bracket $0-$10,275.

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The length and width of the rectanhle are Length = 7y and Width = 2y^2 - 9

To find the expressions for the greatest common factor (GCF) of the length and width of a rectangle with an area of 14y^3 - 63y, we can factor the area expression and look for common factors.

14y^3 - 63y = 7y(2y^2 - 9)

The GCF of the length and width will be factors of both 7y and 2y^2 - 9.

The only common factor is y, so the length and width expressions are:

Length = 7y

Width = 2y^2 - 9

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Answer:

284.96

Step-by-step explanation:

Hope it helps:)

Answer:

48 units squared

Step-by-step explanation:

The area of a trapezoid is 1/2(base1+base2)*height. By counting,  Base 1 is 4 units, base 2 is 8 units, and the height is 8 units. (Just count the squares each side takes and multiply by 2). So we can plug in these values into the formula above, to get 1/2(4+8)*8. Solving this , we get 1/2(12)*8=6*8=48.

Answer:

48 sq units

Your mistake was probably thinking each square unit on the graph was 1 unit, but if you look closely each one is 2, this is proven by the 6 covering 3 boxes.

Step-by-step explanation:

This is the formula for the area of a trap:

A = (a + b)/2 * h

A is the area

a is one of the bases

b is the other base

h is the height

A = (4+8)/2 * 8

A = (12)/2 * 8

A = 6 * 8

A = 48 sq units

The retirement account is worth $424647.577  after 29 years. Rounded to two decimal places, the answer is  $424647.577

Capital is the amount you owe at any one time. This is exactly your loan amount if you  just took the loan. As you pay EMIs, the principal is reduced and when it reaches zero, your loan is closed.

 To calculate the future value of a pension account, we can use the formula:

FV =[tex]FV = P(1 + r/n)^(nt)[/tex]

where:

FV = Future Value

P = capital (initial amount)

r = annual interest  (as a decimal)

n = number of times the interest rate is increased per year

t = time (in years)

In this case, the principal is $100,509, the annual interest  is 5%, and the interest is compounded quarterly (4 times a year). The  period is 29 years.

So we can plug these values ​​into the formula and get:

FV =[tex]100509(1 + 0.05/4)^(4*29)[/tex]

FV = 100509(1.0125)^116

FV = 100509 (4.22497)

FV = $424647.577

Therefore, the value of the retirement account  after 29 years is $ $424647.577 Rounded to two decimal places, the answer is  $424647.577

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13303.492 ×  (0.875)¹⁰ is  an exponential regression equation for this set of data, rounding all coefficients to the nearest hundredth.

What in mathematics is an exponential?

The formula for an exponential function is f (x) = ax, where x is a variable and an is a constant that serves as the function's basis and must be bigger than 0.

                           The transcendental number, or e, which is roughly equivalent to 2.71828, is the most often used exponential function basis. Calculating the exponential growth or decay of a given collection of data is done mathematically using an exponential function.

According to the question,

  x : 0             1            2        3          4           5             6    

 y  :  13100   11124   10572   9365  8067    6507       6892

so the function

 y = 13303.492 ×  (0.875)ˣ

when x = 10 ,

    y = 13303.492 ×  (0.875)¹⁰   ≈  3500

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Simplifying this equation, we get:

8 + 4k + 4k + 6 = 30

8k + 14 = 30

8k = 16

k = 2

Therefore, the value of k is 1.

In mathematics, a polynomial is an expression consisting of variables (usually represented by x), coefficients, and non-negative integer exponents, which are combined using the operations of addition, subtraction, and multiplication. For example,

[tex]3x^2 + 2x - 1[/tex]

is a polynomial with three terms, or a "trinomial," where the variable x is raised to the powers of 2 and 1, and the coefficients are 3, 2, and -1.

The degree of a polynomial is the highest power of the variable in the expression. For example, the polynomial above has a degree of 2, since the highest power of x is 2.

Polynomials are used in many areas of mathematics, including algebra, calculus, and geometry, and are used to model many real-world phenomena.

We can use the remainder theorem, which states that if a polynomial f(x) is divided by (x - a), then the remainder is equal to f(a). In this case, we know that when the polynomial[tex]l x^3 + kx^2 + 2kx[/tex] + 6 is divided by (x - 2), the remainder is 30. So, we can set up the following equation:

[tex]x^3 + kx^2 + 2kx + 6 = (x - 2)q(x) + 30[/tex]

where q(x) is the quotient when. [tex]x^3 + kx^2 + 2kx + 6[/tex] is divided by (x - 2). We don't need to know what q(x) is, since we're only interested in finding k.

Now, let's substitute x = 2 into the equation above:

[tex]2^3 + k(2^2) + 2k(2) + 6 = (2 - 2)q(2) + 30[/tex]

Simplifying the left-hand side, we get:

[tex]8 + 4k + 4k + 6 = 30[/tex]

[tex]16k = 16[/tex]

[tex]k = 1[/tex]

OR

8 + 4k + 4k + 6 = 30

8k + 14 = 30

8k = 16

k = 2

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