The first four terms of the binomial series are 1, x³/5, (12/25)x⁶ and respectively.
The binomial provided to us is (1+x^3)^-1/5.
To find out the first four terms of the binomial, we shall first extend the standard binomial (1+x)^n.
[tex](1+x)^n = 1 + nx + [n(n - 1)/2!] x^{2} + [n(n - 1)(n - 2)/3!] x^{3} +...[/tex]
As we can see here,
The value of x = x³,
The value of n = -1/5.
We get,
[tex](1+x^{3})^{-\frac{1}{5} } = 1 - \frac{1}{5} (x^{3} ) + [\frac{-1}{5} (\frac{-1}{5} -1)/2!]x^{6} + [\frac{-1}{5} (\frac{-1}{5} -1)(\frac{-1}{5} -2)/3!]x^{27} +[/tex]
From the expansion, we can see,
First term = 1
Second term = x³/5
Third term = (12/25)x⁶
Fourth term = (-13/125)x²⁷
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Which linear function has the same slope as the one that is represented by the table?
y = -1/5x +1/2 linear function has the same slope .
What is slope ?A line's steepness and direction are measured by the line's slope. Without actually using a compass, determining the slope of lines in a coordinate plane can assist in forecasting whether the lines are parallel, perpendicular, or none at all. Any two different points on a line can be used to calculate the slope of any line. The ratio of "vertical change" to "horizontal change" between two different locations on a line is calculated using the slope of a line formula. We shall comprehend the slope-finding method and its applications in this essay.
First solve for the slope
m =[tex]\frac{y_{2} -y_{1} }{x_{2} -x_{1} }[/tex]
m=[tex]\frac{0-\frac{3}{5} }{\frac{1}{2} -\frac{1}{5} }[/tex]
m = [tex]\frac{-1}{5}[/tex]
Look for the linear equation with the same value of m after determining the slope (parallel equation).
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U.S. growers harvested 11 billion bushels of corn in 2005. About 1.9 billion bushels were exported, and 1.6 billion bushels were used for ethanol. Choose one bushel of corn at random. What is the probability that it was used either for export or for ethanol?
Answer:
32% or 31.82%
Step-by-step explanation:
so total = 11 billion and export + ethanol = about 3.5 billion so the probability is 3.5/11 or 35/110 which, as a percent, can be written as 32% or more accurate, 31.82% so this is answer
Which of the following methods would NOT be used to prove that two triangles are congruent?
Two triangles cannot be proven to be congruent using the method: A. Prove all sets of corresponding angles are congruent.
How to Prove that Two Triangles are Congruent?Two triangles are congruent to each other if they have the same shape and the same size, that is, all three sets of corresponding sides and all three sets of corresponding angles of the two triangles are congruent to each other.
Also, the following condition or criteria can be used to determine if two triangles are congruent to each other:
The HL congruence theorem: when two right triangles have corresponding equal hypotenuse and a pair of corresponding congruent legs, then they are congruent triangles by HL theorem.The SSS congruence theorem: two triangles having all three pairs of corresponding sides congruent are proven to be congruent by SSS congruence theorem.The SAS congruence theorem: Two triangles with two pairs of corresponding congruent sides and a pair of non-included congruent angle are congruent by the SAS.Thus, the method that cannot be used is: A. Prove all sets of corresponding angles are congruent.
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Describe the location of the point having the following coordinates.
Negative abscissa, positive ordinate
• Quadrant 1
• Quadrant 2
• Quadrant 3
• Quadrant 4
The correct option for Negative abscissa, positive ordinate is -
• Quadrant 2
What is defined as the coordinate system?Graphing is done using a rectangular coordinate system with two axes. The horizontal axis is referred to as the x-axis, and the vertical axis is referred to as the y-axis. The x-axis is a straight line. The y-axis is a horizontal line. That is, we are combining two real numbers. We have the option of moving left-right or up-down.The coordinate plane is divided into quadrants that move counterclockwise.The origin is the point at which the two axes intersect. It has a value of 0 for x and a value of 0 for y.For the given question;
The other names for x and y axis are;
x -axis - abscissa
y- axis - ordinate
Thus, Negative abscissa means negative x axis.
And, positive ordinate means positive y axis.
This, lines in second quadrant.
Thus, Negative abscissa, positive ordinate will give the point is second quadrant.
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What are the zeros of the polynomial function f(x) = x3 + 9x2 + 20x? (5 points)
0, −5, −4
0, −5, 4
0, 5, −4
0, 5, 4
The zeros of the function f(x) = x³ + 9x² + 20x is :
0, 5 ,4
Explanation :Steps to solve quadratic equation:
Put all of the terms on one side of the equal sign and zero on the other.Factor.Set each factor to a value of zero.Each of these equations must be solved.Because they are the values of x for which f(x) = 0, the solutions of the quadratic equation ax² + bx + c = 0 correspond to the roots of the function f(x) = ax² + bx + c.Check your answer by plugging it into the original equation.
Given function,
f(x) = x³ + 9x² +20x
take x common from f(x),
=> x(x² + x - 20) = 0
=> x(x² + 5x + 4x + 20) = 0
=> x( x + 5 ) + 4 ( x +5 ) = 0
=> (x + 5) ( x + 4 ) = 0
=> x -5 = 0 or x -4 = 0
=> x = 5 and x = 4 .
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Who should pay to reduce global warming 
write an equation of the line in point-slope form that passes through the given points in the table. Then write the equation in slope-intercept form.
an equation of the line in point-slope form is
An equation of the line in point-slope form is y = 6x + 50.
Define point slope form.y = y₁ = m(x -x₁) is the equation of a straight line, where m is the slope and (x₁, y₁) are the coordinates at a given point on the line (see slope-intercept form). When you know the slope and another point on the line, the point-slope form of a linear equation is most helpful for locating a point on a line. When two other points are known, it can also be used to locate a point on a line.
Given,
y₂ = 200
y₁ = 230
x₂ = 30
x₁ = 25
Slope = [tex]\frac{y_{2}-y_{1} }{x_{2}-x_{1} }[/tex]
Slope = [tex]\frac{230-200}{30-25}[/tex]
Slope = [tex]\frac{30}{5}[/tex]
Slope = 6
Equation:
y - y₁ = m(x - x₁)
y - 200 = 6(x - 25)
y - 200 = 6x - 150
y = 6x + 50
An equation of the line in point-slope form is y = 6x + 50.
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Describe the transformation that shows Figure A is similar to Figure B
Dilate figure A with center o and scale factor 2
What is transformation?
The transformation, or f: X → X, is the name given to a function, f, that maps to itself. The pre-image X becomes the picture X following the transformation. Any operation, or a combination of operations, such as translation, rotation, reflection, and dilation, can be used in this transformation. A function can be moved in one way or another using translation, rotation, reflection, and dilation. A function can also be scaled using rotation around a point.
Example: A square of side 5 units can be widened to a square of side 15 units using dilation math, but the square's shape doesn't change.
We know that re-sizing an item uses a transition called dilation.
In the figure the dimensions of the figures differ by a factor of 2.
Thus, dilation of figure A by a factor of 2 shows figure A is similar to Figure B
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sears is having a 35% off sale on all tires sells for $125, find the sales price.
Answer:
This video provides an example of how to determine the original price of an item given the discounted price and the percent mark
Step-by-step explanation:
help with the first one ?
The solution of the equation is x = 5.
How to solve equation?The equation is written as follows:
- 94 = -2 (8x + 7)
The variable to be found as variable x.
A variable is a number represented with a letter in an equation.
Hence,
- 94 = -2 (8x + 7)
open the bracket on the right side of the equation
-94 = - 16x - 14
add 14 to both sides of the equation
-94 = - 16x - 14
-94 + 14 = - 16x - 14 + 14
-80 = - 16x
divide both sides by -16
x = -80 / -16
x = 5
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Jasmine paid $17.60 for a blender that was originally $22. What percent discount did she receive?
Answer:
20% off
Step-by-step explanation:
22-17.60=4.4
(5)4.4=22
How can you use the Distributive Property to find 11 × 5?
Answer:
You cant use the distributive property because it is when the equation is like this: ax(b+c) so if it was 11x(5+0) you would multiply the 5 by 11 by itself and multiply 0 with 11 then you add them and get 55. I hope that helps :)
Step-by-step explanation:
If you spend 1.5 hours a week studying for English and 6.5 hours studying for math what is the ratio
of time spent studying in math to studying for English?
Question Help: please break it down because I don’t understand
Answer:
the ratio is 6.5 : 1.5
Step-by-step explanation:
it asks "what is the ratio of time spent studying in math to studying for English?"
ratio uses the ":" mark. also you put the parts like this:
first part : second part
the parts can be found by looking for the "to" as indicated bold in the question
the first part of the ratio it asks is the "time spent studying in math" in hours which is 6.5
the second part of the ratio it asks is the "time spent studying in English" in hours which is 1.5
so the answer is:
6.5 : 1.5
Answer:
Ratio of time spent studying in math to studying for English
[tex]\large \text{$=\dfrac{13}{3}$ }[/tex]
Step-by-step explanation:
What is a ratio?
We use ratios to compare two numeric quantities or quantities with the same units
Basically a ratio of one quantity x to another quantity y can be expressed in the following ways, all read as the ratio of x to y
x/ y
x to y
x : y
The most useful way of representing a ratio is by using the fractional form because it can be used in determining other quantities
In this problem you spend 6.5 hours/week studying for math and 1.5 hours/week studying for English
Since both units are the same (hours/week) we can compute the ratio as
[tex]{\dfrac{\text{Number of hours studying math}}{\text{Number of hours studying English}}\\\\= \dfrac{6.5}{1.5}\\\\[/tex]
We can simply this fraction as follows
Multiply both numerator and denominator by 10
[tex]= \dfrac{65}{15}\\\\[/tex]
Divide both sides by 5
[tex]\dfrac{65 \div 5}{15 \div 5}\\\\= \boxed{\bold{\dfrac{13}{3}\\\\}}[/tex]
This is the same as 13:3 or 13 to 3
(Be careful what is the numerator and what is the denominator.
If the question had asked for the ratio of hours studying English to hours studying Math then the ratio would be 1.5/65 = 15/65 = 3/13)
suppose you are35 years old and would like to retire at age65 . Furthermore, you would like to have a retirement fund from which you can draw an income of $150000 per yearforever! How much would you need to deposit each month to do this? Assume a constant APR of8 % and that the compounding and payment periods are the same.
Answer:
$1258.09
Step-by-step explanation:
You want the monthly annuity payment that will let you withdraw $150,000 per year forever from an account earning 8% after 30 years.
Account balanceIn order to withdraw an amount "forever," the withdrawal amount must be equal to the interest earned by the account.
I = Prt
150,000 = P(0.08)(1)
P = 150,000/0.08 = 1,875,000
Annuity paymentThe value of an ordinary annuity is given by the formula ...
A = P(12/r)((1 +r/12)^(12t) -1) . . . . . where P is the monthly payment, and r is the annual interest rate earned over a period of t years.
Using the values we know, we can find P:
1875000 = P(12/0.08)((1 +0.08/12)^(12·30) -1) = 1490.359449P
P ≈ 1258.09
You would need to deposit $1258.09 each month.
__
Additional comment
The formulas used here assume that the deposits and withdrawals occur at the end of the month.
Find the equation of the line in slope intercept form passes through the points (-8,-4) and (-4,-9)
Written in Slope Intercept Form:
[tex]y=-\frac{5}{4} x-14[/tex]
Hope this helps! If so, please lmk! Thanks and good luck!
The parent function f(x) = square root of x is transformed to g(x) = f(-x) - 2
Answer:
A
Step-by-step explanation:
below is the parent function graphed and according to the chart we move the graph down 2 units and reflect it over the x axis and a is the only option
Which figure is a line?
Answer:
Bottom left
Step-by-step explanation:
The figure that looks like <-------------------------> is a line
Top left *------------------------* is a segment
Bottom right is a ray *---------------->
Top right is a * point or dot
I hope this helps!
Which relation is a function?
Responses
A coordinate grid containing a U shape with arrows on both ends that opens to the right. The bottom portion of the U passes through the origin.
Image with alt text: A coordinate grid containing a U shape with arrows on both ends that opens to the right. The bottom portion of the U passes through the origin.
Coordinate grid with graph of a vertical line at x equals 3.
Image with alt text: Coordinate grid with graph of a vertical line at x equals 3.
A coordinate grid with the graph of a circle centered at the origin and passing through the point begin ordered pair 2 comma 1 end ordered pair.
Image with alt text: A coordinate grid with the graph of a circle centered at the origin and passing through the point begin ordered pair 2 comma 1 end ordered pair.
A coordinate grid with graph of a curve that increases through the third quadrant through the origin and exits the first quadrant
The relation which is a function among the answer choices is; A coordinate grid containing a U shape with arrows on both ends that opens to the right. The bottom portion of the U passes through the origin.
What is a function?A function is a relation in which each element of the domain is associated with exactly one element of the codomain.
A relation can be tagged a function if it describes that there should be only one output (y - value) for each input ( x - value). Put simply, a function is a kind of relation (defined by a set of ordered pairs) in which case, it ensures that every x-value is associated with only one y-value.
On this note, since the relation described in the first answer choice is such that every x-value is associated with only one y-value as it is a parabola (quadratic function), it is termed a function.
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In a soccer match, a tie at the end of two overtime periods leads to a "shootout" with five kicks taken by each team from the
penalty mark. Each kick must be taken by a different player. How many ways can 5 players be selected from the 11 eligible
players? For the 5 selected players, how many ways can they be designated as first, second, third, fourth, and fifth?
The number of ways that 5 players can be selected from the 11 eligible players is; 462 ways
The number of ways that the 5 selected players can they be designated as first, second, third, fourth, and fifth is; 120 Ways
How to solve Permutations and Combinations?
We are given that In a soccer match;
A tie at the end of two overtime periods leads to a "shootout" with five kicks taken by each team from the penalty mark.
Each kick must be taken by a different player
Now, to find the number of ways that 5 players can be selected from the 11 eligible players, we will use combination to get;
11C5 = 11!/(5! * (11 - 5)!)
= 11!/(5! * 6!)
= 462 ways
Now, if we care about order or position, we will use permutation to get the number of ways that the 5 selected players can they be designated as first, second, third, fourth, and fifth;
5P5 = 5!/(5 - 5)
= 120 Ways
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Can someone help with this question?✨
The equation for the linear function f(x) is f(x) = 2 / 7 x - 10 / 7
How to find the equation for the linear function?The linear function is f(x)
f(-2) = - 2 and f(5) = 0
Therefore, let's find the equation for the linear function f(x):
A linear function is a function that represents a straight line on the coordinate plane.
Therefore, linear function is represented as follows:
y = mx + b
where
m = slopeb = y-interceptHence,
-2 = m(-2) + b
0 = 5m + b
b = -5m
Therefore,
m(-2) + - 5m = -2
-2m - 5m = -2
-7m = -2
m = 2 / 7
Therefore,
b = - 5 (2 / 7)
b = - 10 / 7
Hence,
f(x) = 2 / 7 x - 10 / 7
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Simplify the fraction
36
84
as much as possible.
36
84
3
7
5
12
Answer:
3/7
Hope this helps!
Have a good day :).
Consider the incomplete paragraph proof.
Given: P is a point on the perpendicular bisector, l, of MN.
Prove: PM = PN
Line l is a perpendicular bisector of line segment M N. It intersects line segment M N at point Q. Line l also contains point P.
Because of the unique line postulate, we can draw unique line segment PM. Using the definition of reflection, PM can be reflected over line l. By the definition of reflection, point P is the image of itself and point N is the image of ________. Because reflections preserve length, PM = PN.
Given that P is a point on the perpendicular bisector, l, of MN, we have proved that PM = PN because By the definition of reflection, point P is the image of itself and point N is the image of N.
How to Interpret the Perpendicular Bisector?We are given :
P is a point on the perpendicular bisector, l, of MN.
Now, we want to prove that PM = PN
A Reflection is a transformation in which the figure is the mirror image of the other. Every point is a mirror reflection of itself .
By the definition of reflection, we can deduce that point P is the image of itself , and point N is the image of M .
The line l acts as a Line of symmetry or axis of reflection.
Now, in transformations, we know that reflections preserve length and as a result, PM = PN.
Therefore, we will conclude that point N is the image of M .
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Answer: Point M
Step-by-step explanation:edge 2023
please help me solve this
Answer:
bottom right
Step-by-step explanation:
bottom right
Factorize of
X4+8x2+144
The most appropriapriate choice for factorization will be given by-
[tex](x^2+12+4x)(x^2+12-4x)[/tex] is the required factorization.
What is factorization?
The process of breaking an algebraic expression into product of different factors is called factorization.
Here,
The factorization of [tex]x^4+8x^2+144[/tex] has been shown below-
[tex]x^4+8x^2+144[/tex]
[tex](x^{2} )^2 + 24x^2 +144 -16x^2\\(x^{2} )^2 + 2\times x^2\times 12+ 12^2-16x^2\\(x^2+12)^2 - (4x)^2\\(x^2 + 12+4x)(x^2+12-4x)[/tex]
This is the required factorization
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Write an equation (a) in slope-intercept form and (b) in standard form for the line passing through (-2,5) and parallel
to x + 3y = 7.
a) The equation of the line in slope-intercept form is
(Type your answer in slope-intercept form. Use integers or fractions for any numbers in the equation.)
b) The equation of the line in standard form is
(Type your answer in standard form.)
The equation of the line passing through points (-2, 5) can be written as follows
slope intercept form: y = -x/3 + 13/3standard form: x + 3y = 13How to write the equation of the line in slope-intercept formEquation of a straight line is an equation of the form y = mx + c, where
m = slopec = interceptThe slope is the gradient which is the ratio of the difference in the y direction to the difference in x direction.
given that the equation of the line is x + 3y = 7
rewriting the equation gives
y = -x/3 + 7/3
hence the slope is -1/3, and this is equated with the formula for slope
[tex]\frac{y-y_{1} }{x-x_{1} }=-\frac{1}{3}[/tex]
substituting the values for points (-2,5) and cross multiplying gives
3 (y - 5) = - ( x - -2)
y - 5 = - (x + 2 ) /3
y - 5 = -x/3 - 2/3
y = -x/3 - 2/3 + 5
y = -x/3 + 13/3
Rewriting the equation of the line in standard form is done as follows
y = -x/3 + 13/3
3y = -x + 13
x + 3y = 13
The equations are gotten from substitution the values to the formula for slope and substituting. Rearranging the equation gives the equation in slope intercept form and in standard form.
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The histogram shows the distribution of hurricanes that have hit a country from 1851 through 2015, where 1 is the weakest level and 5 is the strongest level.
(a) Find the mean, variance, and standard deviation of the probability distribution.
(b) Interpret the results.
Hurricanes
1
2
3
4
5
0.0
0.1
0.2
0.3
0.4
0.5
Category
Probability
0.419
0.276
0.225
0.068
0.012
The mean = 0.2, variance = 0.03226 and Standard deviation = 0.1796
What is standard deviation?A standard deviation (or σ) is a measure of how dispersed the data is in relation to the mean. Low standard deviation means data are clustered around the mean, and high standard deviation indicates data are more spread out.
Given:
level x
1 0.419
2 0.276
3 0.225
4 0.068
5 0.012
So, mean = (0.419+ 0.276 + 0.225+ 0.068 + 0.012)/5
mean = 0.2
level x (x- mean) (x- mean)²
1 0.419 0.219 0.0479
2 0.276 0.076 0.0057
3 0.225 0.025 0.000625
4 0.068 -0.268 0 .0718
5 0.012 -0.188 0.0353
So, variance = 0.032265
standard deviation = √(variance)
Standard deviation = √(0.032265)
Standard deviation = 0.1796
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will give you brainliest simplify
After simplifying the expression we get: 12√6
Given, √864
Take the LCM of the given number:
864 = 2×2×2×2×2×3×3×3
Minimize the terms:
= √4×4×2×9×3
Reduce the terms:
= √16×9×2×3
multiply the terms:
= √144×2×3
= 12√6
Hence after simplifying we get the answer as 12√6
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The data in this table represents a linear function
What is the slope of this line?
The slope of the line is [tex]\frac{2}{3}[/tex]
What is slope of a line?The increase divided by the run is the slope. By examining the rise and run, you can infer the slope of a line from its graph. A line has the property that its slope remains constant throughout.
Determine the coordinates of two points along the line that you choose. Find the difference between these two points' y-coordinates (rise). Find the difference between these two points' x-coordinates (run). Divide the difference in x-coordinates (rise/run or slope) by the difference in y-coordinates.
The slope of the line is [tex]\frac{y_{2}-y_{1} }{x_{2}- x_{1} }[/tex]
⇒ [tex]\frac{-7-(-5)}{9-6}[/tex]
⇒ [tex]\frac{-7 + 5}{ 3} = \frac{2 }{3}[/tex]
Thus m = 2/3
Similarly in second case : [tex]\frac{y_{2}-y_{1} }{x_{2}- x_{1} }[/tex]
⇒ [tex]\frac{-15 - (-11)}{ 21 - 15}[/tex]
⇒ [tex]\frac{4}{6}[/tex]
= 2/3
Thus the slope of the given table is 2/3
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please help
The function g(r) represents the shift of the
function f(x) five units to the right. If f(x)=
It +5, which of the following is g(x)?
You want to fence off a rectangular area of yard using the back side of your house as one of the sides. You have a total of 150 yards of fence. If the area of the fenced in portion comes to 2,800 square yards, what are the dimensions of the fenced-in area? Show all work for full credit.
The dimensions of the fenced-in area is
either length = 35 yards and width = 80 yards ,
or length = 40 yards and width = 70 yards .
In the question ,
it is given that
the total length of the fence = 150 yards
let L be the length of the area to be fenced
let B be the width of the house
Given that one side of the area is fenced using the back side of the house
So, the perimeter of the area to be fenced is 2L+B
equating both the perimeter we get
2L+B=150
B=150-2L ....(i)
Given the area fenced-in portion = 2800 yards
also the fenced-in portion area = L×B
hence
L*B=2800
Substituting the values of B from equation(i) , we get
L*(150-2L)=2800
150L - 2L² = 2800
2L² - 150L + 2800 = 0
simplifying further , we get
L² - 75L + 1400 = 0
L² -40L -35L +1400 = 0
L(L-40) -35(L-40) = 0
(L-35)(L-40) = 0
So , L = 35 yards or L = 40 yards
Using L = 35 yards , we get B = 150-2(35) = 80 yards
and using L =40 yards , we get B = 150-2(40) = 70 yards .
Therefore , The dimensions of the fenced-in area is
either length = 35 yards and width = 80 yards ,
or length = 40 yards and width = 70 yards .
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