Expand
(2x - 3)4
Can you help me expand this by using the binomial theorem?
Answers
The value of expanding (2x -3)^4 is 16x^4 + 96x^3 +216x^2 -216x + 81
How to expand the expression?The expression is given as:
(2x -3)^4
Using the binomial expansion, we have:
\((2x -3)^4 = ^4C_0 * (2x)^4 * (-3)^0 +^4C_1 * (2x)^3 * (-3)^1 + ^4C_2 * (2x)^2 * (-3)^2 + ^4C_3 * (2x)^1 * (-3)^3 + ^4C_4 * (2x)^0 * (-3)^4\)
Evaluate the combination factors.
So, we have:
\((2x -3)^4 = 1 * (2x)^4 * (-3)^0 + 4 * (2x)^3 * (-3)^1 + 6 * (2x)^2 * (-3)^2 + 4 * (2x)^1 * (-3)^3 + 1 * (2x)^0 * (-3)^4\)
Evaluate the exponents and the products
\((2x -3)^4 = 16x^4 + 96x^3 +216x^2 -216x + 81\)
Hence, the value of expanding (2x -3)^4 is 16x^4 + 96x^3 +216x^2 -216x + 81
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Related Questions
There are 46 children attending the Hefner field trip. Each van will hold 8 children. How many van's will be needed to transport each child to and from the field trip?
Answers
Answer:
6
Step-by-step explanation:
There are 46 children attending the field trip.
Each van needs to transport 8 children.
To find the number of vans they need, we will divide the number of children by the number of children per van:
46 / 8 = 5.75 ≅ 6 vans
They need 6 vans.
Find the value of the expression (look at the image)
Answers
Answer:
1
Step-by-step explanation:
Anything raised to the power of zero is one. And 1^-2 is is 1.
Point K is rotated 90°. The coordinate of the pre-image point K was (2, –6) and its image K’ is at the coordinate (−6, −2). Find the direction of the rotation. The direction of rotation was .
Answers
Answer:
Hello! The answer to your question will be below.
Step-by-step explanation:
The answer would be clockwise.
So point k was rotated 90 degrees clockwise.....
Review.....
Question:
Point K is rotated 90 degrees. The coordinate of the pre-image point K was (2,-6) and it’s image K’ is at the coordinate (-6,-2).Find the direction of the rotation.
THE DIRECTION OF THE ROTATION WAS CLOCKWISE.
Hope this helps! :)
⭐️Have a wonderful day!⭐️
Answer:
clockwise.
So point k was rotated 90 degrees clockwise.....
Review.....
Question:
Point K is rotated 90 degrees. The coordinate of the pre-image point K was (2,-6) and it’s image K’ is at the coordinate (-6,-2).Find the direction of the rotation.
Step-by-step explanation:
an 8 sided regular polygon (regular octagon) is inscribed in a circle whose radius is 16 feet. find the area of the polygon.
Answers
The area of the regular octagon inscribed in a circle with a radius of 16 feet can be found using the formula A = (2 + 2sqrt(2))r^2, where r is the radius of the circle. Plugging in the value for r, we get:
A = (2 + 2sqrt(2))(16)^2
A = (2 + 2sqrt(2))(256)
A = 660.254 ft^2
Therefore, the area of the regular octagon is approximately 660.254 square feet.
To derive the formula for the area of a regular octagon inscribed in a circle, we can divide the octagon into eight congruent isosceles triangles, each with a base of length r and two congruent angles of 22.5 degrees. The height of each triangle can be found using the sine function, which gives us h = r * sin(22.5). Since there are eight of these triangles, the area of the octagon can be found by multiplying the area of one of the triangles by 8, which gives us:
A = 8 * (1/2)bh
A = 8 * (1/2)(r)(r*sin(22.5))
A = 4r^2sin(22.5)
We can simplify this expression using the double angle formula for sine, which gives us:
A = 4r^2sin(45)/2
A = (2 + 2sqrt(2))r^2
Therefore, the formula for the area of a regular octagon inscribed in a circle is A = (2 + 2sqrt(2))r^2.
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The combined perimeter of an equilateral triangle and square is 13.
Find the dimensions of the triangle and square that produce a minimum total area.
The measurement of square on each side
The measurement of triangle on each side
Find the dimensions of the triangle and square that produce a maximum total area.
The measusrement of square on each side
The measurement of triangle on each side
Answers
To minimize the total area of an equilateral triangle and square, the side length of the square should be 2.167 and the side length of the triangle should be 3.833.
To find the dimensions that minimize the total area, we can set up equations based on the given information. Let's denote the side length of the square as 's' and the side length of the equilateral triangle as 't'. The perimeter of the square is 4s, and the perimeter of the equilateral triangle is 3t. Given that the combined perimeter is 13, we have the equation 4s + 3t = 13.
To minimize the total area, we need to consider the formulas for the areas of the square and equilateral triangle. The area of the square is given by A_square = \(s^2\), and the area of the equilateral triangle is given by A_triangle = (\(\sqrt{(3)}\)/4) *\(t^2\).
To find the values that minimize the total area, we can substitute s = (13 - 3t)/4 into the equation for A_square and solve for t. By finding the derivative of the total area with respect to t and setting it equal to zero, we can find the value of t that minimizes the area.
Similarly, to find the dimensions that maximize the total area, we follow the same process but this time maximize the total area by finding the value of t that maximizes the area.
Performing the calculations, we find that to minimize the total area, the side length of the square is approximately 2.167 and the side length of the triangle is approximately 3.833. To maximize the total area, the side length of the square is approximately 4.333 and the side length of the triangle is approximately 1.667.
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1
What is the radius of a circle with circumference 17 mm? Round to the nearest tenth. Use 3.14 for TT.
Answers
Answer: 2.7mm
Step-by-step explanation:
The formula for the circumference of a circle is given as 2πr. In thkw case, thw circumference is 17mm.
Therefore, 2πr = 17
2 × 3.14 × r = 17
6.28r = 17
r = 17/6.28
r = 2.7mm
Therefore the radius of the circle is 2.7mm
Use an algebraic equation to find the measures of the two angles described below. Begin by letting x represent the degree measure of the angle's complement.
The measure of the angle is 64 degrees greater than its complement.
What is the measure of the complement?
Answers
Answer:
13 degrees
Step-by-step explanation:
Let x be the degree measure of the angle's complement
Then we have the other angle is x + 64
Because the 2 angles complement each other so the sum of the 2 angles is 90
So we have the equation: 2x + 64 = 90
<=> 2x = 26
<=> x = 13
The value of k is 4 times as many as 21.
Answers
If we multiply 21 by 4, we get 84 .the value of k is 84, as it is 4 times as many as 21.
What is multiplication ?Multiplication is a mathematical operation that combines two or more numbers to find their product. It is an operation of repeated addition or scaling up of a number. In multiplication, the numbers being multiplied are called factors, and the result is called the product. Multiplication is denoted by the "×" symbol or by placing the numbers adjacent to each other.
According to the given information:The value of k is 4 times as many as 21," we can break it down as follows:
"4 times" means multiplication, where we are multiplying a number by 4.
"As many as" implies equality, meaning the value on one side of the comparison is equal to the value on the other side.
"21" is a specific number mentioned in the statement.
Therefore, if we multiply 21 by 4, we get 84 .the value of k is 84, as it is 4 times as many as 21.
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Suppose that you are interested in estimating the average number of miles per gallon of gasoline your car can get. You calculate the miles per gallon for each of the next twelve times you fill the tank. Suppose that in truth, the values for your car are bell-shaped, with a mean of 25 miles per gallon and a standard deviation of 1. Find the possible sample means you are likely to get based on your sample of twelve observations. Consider the intervals into which 68%, 95%, and almost all of the potential sample means will fall, using the Empirical Rule. (Round all answers to the nearest thousandth.)
About 68% of possible sample means will be in the range between ____ and ____ .
About 95% of possible sample means will be in the range between ____ and ____ .
About 99.7% of possible sample means will be in the range between ____ and ____ .
Answers
For estimating the average number of miles per gallon of gasoline a car,
a)About 68% of possible sample means will be in the range between 24.711 and 25.288.
b) About 95% of possible sample means will be in the range between 24.422 and 25.578.
c) About 99.7% of possible sample means will be in the range between 24.133 and 25.867.
Let's we are interested in estimating average number of miles per gallon of gasoline a car. Firstly, sample size, n = 12
mean = 25 miles per gallon
standard deviation = 1
Shape of distribution is bell-shaped. Using Empirical rule,
68% of observed data points will lie inside one standard deviation of the mean.95% will fall within two standard deviations, and 99.7% will occur within three standard deviations.Standard error =\( \frac{ \sigma}{\sqrt{n}}\)
\( = \frac{ 1}{\sqrt{12}}\)
= 0.289
a) The 68% of observed data points will lie inside one standard deviation of the mean, i.e \(\mu ± \sigma \)
= 25 ± 0.289
= ( 24.711, 25.288)
b) 95% will fall within two standard deviations,i.e., \(\mu ± 2 \sigma \)
= 25 ± 2×0.289
= (24.422, 25.578)
c) 99.7% will occur within three standard deviations.i.e., \(\mu ± 3 \sigma \)
= 25 ± 3×0.289
= (24.133, 25.867)
Hence, required value is (24.133, 25.867)
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Complete question:
Suppose that you are interested in estimating the average number of miles per gallon of gasoline your car can get. You calculate the miles per gallon for each of the next twelve times you fill the tank. Suppose that in truth, the values for your car are bell-shaped, with a mean of 25 miles per gallon and a standard deviation of 1. Find the possible sample means you are likely to get based on your sample of twelve observations. Consider the intervals into which 68%, 95%, and almost all of the potential sample means will fall, using the Empirical Rule. (Round all answers to the nearest thousandth.)
a) About 68% of possible sample means will be in the range between ____ and ____ .
b) About 95% of possible sample means will be in the range between ____ and ____ .
c) About 99.7% of possible sample means will be in the range between ____ and ____ .
A number x plus 36 is no more than 40. Write this sentence as an inequality
Answers
Answer:
When we say 'as many as' or 'no more than', we mean 'less than or equal to' which means that a could be less than b or equal to b. But, when we say 'at least', we mean 'greater than or equal to'. Here a could be greater than b or equal to b.
50 PTS Determine if you can use the hypotenuse-leg congruency property to prove the triangles are congruent.
A. No, the hypotenuse-leg congruence property cannot be applied, because we don't know if a leg of one triangle is congruent to a leg of the other triangle.
B. No, the hypotenuse-leg congruence property cannot be applied, because we don't know if the hypotenuses are congruent.
C. Yes, the hypotenuse-leg congruence property can be applied.
D. No, the hypotenuse-leg congruence property cannot be applied, because we don't know if the triangles are right triangles.
Answers
Let R be the region in the fourth quadrant enclosed by the x-axisand the curve Y = X^2 - 2KX, where k > 0. if the area of the region R is 36, thenthe value of k is
(A) 2
(B) 3
(C) 4
(D) 6
(E) 9
Answers
We want to find the value of k given that the area of the region R is 36. To do this, we need to find the limits of integration and set up the integral for finding the area of the region R. The answer is (B) 3.
Since R is in the fourth quadrant and is enclosed by the x-axis and the curve Y = X^2 - 2KX, we can find the limits of integration by setting Y = 0 and solving for X:
0 = X^2 - 2KX
X(X - 2K) = 0
X = 0 or X = 2K
So the limits of integration are from X = 0 to X = 2K.
Now we can set up the integral for finding the area of the region R:
∫[0,2K] (X^2 - 2KX) dX
Integrating this expression gives:
[(1/3)X^3 - KX^2] evaluated at 0 and 2K
Plugging in the limits of integration and simplifying, we get:
(8/3)K^3 - 4K^3
Simplifying this expression, we get:
(2/3)K^3 = 36
Solving for K, we get:
K^3 = 54
Taking the cube root of both sides, we get:
K = 3
Therefore, the answer is (B) 3.
To find the value of k, follow these steps:
1. Determine the x-intercepts of the curve Y = X^2 - 2KX by setting Y = 0.
0 = X^2 - 2KX
X(X - 2K) = 0
The x-intercepts are X = 0 and X = 2K.
2. As we are looking for the area in the fourth quadrant, the integral bounds are from 0 to 2K.
3. Set up the integral for the area:
Area = ∫(X^2 - 2KX) dx, from 0 to 2K
4. Evaluate the integral:
Area = (X^3/3 - KX^2) | from 0 to 2K
= [(8K^3/3 - 4K^3) - (0)]
= 8K^3/3 - 4K^3
= (4K^3/3)
5. Set the area equal to 36 and solve for k:
36 = (4K^3/3)
9 = K^3
k = 3^(1/3) ≈ 2.08
The closest value from the given options is (A) 2.
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A line has a slope of 0 and passes through the point (-13, -11). What is its equation in
slope-intercept form?
Write your answer using integers, proper fractions, and improper fractions in simplest form.
Answers
Answer:
y= - 11
Step-by-step explanation:
A line with a slope of 0 is a horizontal line, parallel to the x- axis with equation
y = c ( c is the value of the y- coordinates the line passes through )
the line passes through (- 13, - 11 ) with y- coordinate - 11, then
y = - 11 ← equation of line
Answer:
y = -11
Step-by-step explanation:
A line with slope as 0, is a horizontal line ( In other words parallel to x-axis).
There is no rise, only run.
Equation of the line is y = c.
y = -11
Evaluate the expression below when x = 6 and y = 2. 6 x 2/y 3
Answers
Answer:
27
Step-by-step explanation:
To evaluate an expression we need to find the numerical value of the expression by substituting appropriate values for the variables and performing the indicated mathematical operations.
Given rational expression:
\(\dfrac{6x^2}{y^3}\)
To evaluate the given expression when x = 6 and y = 2, substitute x = 6 and y = 2 into the expression and solve.
\(\dfrac{6(6)^2}{(2)^3}\)
Following the order of operations, begin by evaluating the exponents first.
To square a number, we multiply it by itself.
\(\implies 6^2 = 6 \times 6 = 36\)
To cube a number, we multiply it by itself twice.
\(\implies 2^3 = 2 \times 2 \times 2 = 8\)
Therefore:
\(\dfrac{6(6)^2}{(2)^3}=\dfrac{6 \cdot 36}{8}\)
Multiply the numbers in the numerator:
\(\dfrac{216}{8}\)
Finally, divide 218 by 8:
\(\dfrac{216}{8}=27\)
Therefore, the evaluation of the given expression when x = 6 and y = 2 is 27.
\(\hrulefill\)
As one calculation:
\(\begin{aligned}x=6, y=2 \implies \dfrac{6x^2}{y^3}&=\dfrac{6(6)^2}{(2)^3}\\\\&=\dfrac{6 \cdot 36}{8}\\\\&=\dfrac{216}{8}\\\\&=27\end{aligned}\)
mini earned $160 during the summer doing chores. she bought 3 dresses worth $12 each using her chore money. how much money was left after she bought the dresses.
a. $112
b. $124
c. $136
d. $148
Answers
Answer:
124
Step-by-step explanation:
Answer:
The answer is $124
Step-by-step explanation:
Look here ez dubs
164-5)-2 -jddjdjej hejdjdjdjdjjd
Answers
Is the pair of equations y 0 and y =- 7 will have?
Answers
No Solution.
Since, the x-axis (equation y=0) does not intersect y=-7 at any point. The given pair of equations has no solution
Three Types of Solutions of a System of Linear EquationsConsider the pair of linear equations in two variables x and y.
a1x + b1y + c1 = 0
a2x + b2y + c2 = 0
Here a1, b1, c1, a2, b2, c2 are all real numbers.
Note that, a12 + b12 ≠ 0, a22 + b22 ≠ 0
1. If (a1/a2) ≠ (b1/b2), then there will be a unique solution. If we plot the graph, the lines will intersect. This type of equation is called a consistent pair of linear equations.
2. If (a1/a2) = (b1/b2) = (c1/c2), then there will be infinitely many solutions. The lines will coincide. This type of equation is called a dependent pair of linear equations in two variables
3. If (a1/a2) = (b1/b2) ≠ (c1/c2), then there will be no solution. If we plot the graph, the lines will be parallel. This type of equation is called an inconsistent pair of linear equations.
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anova df ss ms f regression 1 12,323.60 12,323.60 90.0481 residual 8 1,154.802 136.8550 total 9 13,478.4 what is the correlation coefficient?
Answers
The correlation coefficient is 0.9541. This can be determined from the information provided in the ANOVA table, specifically the regression sum of squares (SSR) and the total sum of squares (SST).
The correlation coefficient, also known as the Pearson correlation coefficient, is a measure of the strength and direction of the linear relationship between two variables. It ranges from -1 to 1, where values closer to -1 or 1 indicate a stronger relationship, and values closer to 0 indicate a weaker relationship.
To calculate the correlation coefficient, we can use the formula:
r = sqrt(SSR/SST)
From the ANOVA table provided, we can see that SSR = 12,323.64 and SST = 13,538.4. Plugging these values into the formula gives:
r = sqrt(12,323.64/13,538.4) = 0.9541
Therefore, the correlation coefficient for this model is 0.9541, indicating a strong positive linear relationship between the independent variable and the dependent variable
Complete Question:
Using the following information. Coefficients -12.8094 Intercept Independent variable 2.1794 ANOVA df SS MS F 1 90.0481 Regression Residual Total 12,323.64 1,214.762 13,538.4 12,323.64 136.8550 8 1° 9 What is the correlation coefficient? Multiple Choice 0.9103 0.9541 -0.9541
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\(solve \: this\)
.....................
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Answers
Answer:
3 inches
Step-by-step explanation:
If we're gonna have some sort of increase each year, it's gonna pop up in our model attached to a variable, because it represents some kind of change, and variables can change. Looking at our equation, the 28.6 represents our starting point: the boy is 28.6 inches tall when he's 2 years old. The 3a term, then, represents the boy's increasing height each year. When he's 3 years old, a = 1, and so h = 3 + 28.6 = 31.6 inches. When he's 4, a = 2, and h = 3 + 31.6 = 34.6. The increase is always 3 inches.
In a game a player wins if he rolls a 6 on a number cube if the number cube is rolled 18 times then what is an reasonable prediction for the number of unsuccsecful rollls
Answers
Answer:
the reasonable prediction for the number of unsuccessful rollls is 15 ÷ 18
Step-by-step explanation:
The computation of the number of unsuccessful rolls is as follow:
For a number cube, the probability of getting 6 would be 1 ÷ 6 and for not getting is 5 ÷ 6
Now for 18 rolls i.e. there is 3 group of 6 so the chance of not getting would be 15 ÷ 18
Hence, the reasonable prediction for the number of unsuccessful rollls is 15 ÷ 18
The same is to be considered
solve with elimination and substitution method with working please help me out ASAP
2x+3y=5
2x+7y=9
Answers
Answer:
The answer is x=1 and y=1
Step-by-step explanation:
Multiply the second equation by -1, then add the equations together.
(2x+3y=5)
−1(2x+7y=9)
Becomes:
2x+3y=5
−2x−7y=−9
Add these equations to eliminate x:
−4y=−4
Then solve−4y=−4for y:
−4y=−4 Divide -4
\(\frac{-4y}{-4}= \frac{-4}{-4}\)
y=1
Now that we've found y let's plug it back in to solve for x.
Here's was the original equation:
2x+3y=5
Substitute 1 for y in 2x+3y=5:
2x+(3)(1)=5
2x+3=5
2x+3+−3=5+−3 ( Add -3 on both sides)
2x=2 Divide by 2
\(\frac{2x}{2} =\frac{2}{2}\)
x=1
the perimeter of a geometric figure is the sum of its sides. if the perimeter of the following pentagon is 42 meters, find the length of each side
Answers
Answer: 2.5 divided by 2 equals 1.25 so each side equals 1.25
Step-by-step explanation:
Answer:
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Step-by-step explanation:
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How do I solve for X
Answers
Answer:
I think x equals 7.5
Step-by-step explanation:
Please let me know if this was correct or what you were looking for. Thanks!
there is a 68% chance tha tu the coach will select you and a 26% chance that the coach will select your friend. what is the probability that you or your friend is selected
Answers
The probability that either you or your friend is selected is 94% or 0.94.
Here, we have,
Given that
there is a 68% chance that you are selected and a 26% chance that your friend is selected,
To find the probability that either you or your friend is selected, we need to add the individual probabilities of each event occurring.
The probability that either of you is selected is obtained by summing these probabilities:
P(you or your friend is selected)
= P(you) + P(your friend)
= 68% + 26%
= 0.68 + 0.26
= 0.94
Therefore, the probability that either you or your friend is selected is 94% or 0.94.
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Solve for x
6(x - 1) = 9(x + 2)
O x= -8
O x= -3
0 x = 3
0 x = 8
Answers
Answer:
\(A.x=-8\)
Step-by-step explanation:
6x-9x=18+6
-3x=24
x=-8
Write the integers in order from least to greatest 25,-13,12,-9,2
Answers
Answer:
-13, -9, 2, 12, 25
the weights of oranges growing in an orchard are normally distributed with a mean weight of 8 oz. and a standard deviation of 2 oz. from a batch of 1400 oranges, how many would be expected to weigh more than 4 oz. to the nearest whole number? 1) 970 2) 32 3) 1368 4) 1295
Answers
The number of oranges that are expected to weigh more than 4 oz is:
1400 - (1400 × 0.0228)≈ 1368.
The mean weight of the oranges growing in an orchard is 8 oz and standard deviation is 2 oz, the distribution of the weight of oranges can be represented as normal distribution.
From the batch of 1400 oranges, the number of oranges is expected to weigh more than 4 oz can be found using the formula for the Z-score of a given data point.
\(z = (x - μ) / σ\)
Wherez is the Z-score of the given data point x is the data point
μ is the mean weight of the oranges
σ is the standard deviation
Now, let's plug in the given values.
\(z = (4 - 8) / 2= -2\)
The area under the standard normal distribution curve to the left of a Z-score of -2 can be found using the standard normal distribution table. It is 0.0228. This means that 0.0228 of the oranges in the batch are expected to weigh less than 4 oz.
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HELP PLEASE DUE IN 3 MINUTES
Answers
Answer:
$3 dollars
Step-by-step explanation:
mode is the one that appears the most frequently in a set and three bucks is shown multiple times the most
What is the surface area of a sphere with radius 2?
O A. 27 units2
O B. 41 units2
C. 87 units2
D. 167 units2
SUBMIT
Answers
\(\textit{surface area of a sphere}\\\\ SA=4\pi r^2~~ \begin{cases} r=radius\\[-0.5em] \hrulefill\\ r=2 \end{cases}\implies \begin{array}{llll} SA=4\pi (2)^2\implies SA=16\pi \\\\\\ SA\approx 50.27 \end{array}\)
Wants to buy new boots that cost $68. The sales tax rate in her city is 5-%. What is the total cost for the boots
Answers
Emily new boots cost a total of $71.4 with the tax included as 5% and the original cost of the boots being $68.
The sales tax rate in Emily's city = 5% and the cost of Emily's new boots is $68
we need to find the amount of sales tax Emily will pay for her boots,
The formula we will use =
Sale tax = Percent sales tax x Cost of the product
when we put the values in the formula, we get:
Sales tax = 5% x $68
= $3.4
now we need to also calculate the total cost for Emily's boots,
Therefore, the total cost = Cost of boots + Sales Tax
when we put the values in the formula, we get:
Total cost = $68 + $3.4
= $71.4
Therefore, the total cost Emily will pay for her boots is $71.4
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