The fifth term in the binomial expansion of [tex](x+5)^{8}[/tex] is [tex]\boxed{43,750\ x^{4}}[/tex].
Further explanation:
Given:
The binomial term is [tex](x+5)^{8}[/tex].
The expansion of [tex](x+5)^{8}[/tex] is as follows:
[tex]\boxed{{\left({a+b}\right)^n}=\sum\limits_{k=0}^n{{}^n{{\text{C}}_k}{a^{n - k}}{b^k}}}[/tex]
There are [tex]n+1[/tex] terms in the expansion of [tex](a+b)^{n}[/tex].
The sum of indices of [tex]a[/tex] and [tex]b[/tex] is equal to [tex]n[/tex] in every term of the expansion.
The general term [tex]T_{r+1}[/tex] of the binomial term [tex](a+b)^{n}[/tex] is as follows:
[tex]\boxed{{{\text{T}}_{r + 1}}={}^n{{\text{C}}_r}{a^{n - r}}{b^r}}[/tex]
For [tex]5^{th}[/tex] term the value of [tex]r[/tex] is calculated as follows:
[tex]\begin{aligned}r+1&=5\\r&=5-1\\r&=4\end{aligned}[/tex]
Now, the [tex]5^{th}[/tex] term of [tex](x+5)^{8}[/tex] is calculated as follows:
[tex]\begin{aligned}T_{5}&=T_{4+1}\\&=^8C_{4}\cdot x^{8-4}\cdot 5^{4}\\&=\dfrac{8\cdot 7\cdot 6\cdot 5}{4\cdot 3\cdot 2\cdot 1}\cdot x^{4}\cdot 625\\&=625\cdot 70x^{4}\\&=43,750x^{4}\end{aligned}[/tex]
Therefore, the fifth term of the binomial expansion [tex](x+5)^{8}[/tex] is [tex]\boxed{43,750\ x^{4}}[/tex].
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Answer details:
Grade: Senior school
Subject: Mathematics
Chapter: Binomial Theorem
Keywords: Binomial theorem, expansion, (x+5)^8, 175000x3, 43750x4, 3125x5, 7000x5, fifth term, binomial expansion, genral term, binomial, polynomial, indices.
The answer is:
[tex]\[\boxed{43750x^4}\][/tex]
To find the fifth term in the binomial expansion of [tex]\((x + 5)^8\)[/tex], we use the binomial theorem. The binomial theorem states that:
[tex]\[(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\][/tex]
For the expansion of [tex]\((x + 5)^8\)[/tex], we identify:
[tex]\[a = x, \quad b = 5, \quad n = 8\][/tex]
The general term in the expansion is given by:
[tex]\[T_{k+1} = \binom{n}{k} a^{n-k} b^k\][/tex]
We need to find the fifth term, which corresponds to k = 4 (since k starts from 0):
[tex]\[T_5 = \binom{8}{4} x^{8-4} 5^4\][/tex]
First, calculate the binomial coefficient [tex]\(\binom{8}{4}\):[/tex]
[tex]\[\binom{8}{4} = \frac{8!}{4! \cdot 4!} = \frac{8 \times 7 \times 6 \times 5}{4 \times 3 \times 2 \times 1} = 70\][/tex]
Next, calculate [tex]\( x^{8-4} \)[/tex] and [tex]\( 5^4 \):[/tex]
[tex]\[x^{8-4} = x^4\]\[5^4 = 625\][/tex]
Now, combine these results:
[tex]\[T_5 = 70 \cdot x^4 \cdot 625\][/tex]
Finally, multiply the coefficients:
[tex]\[70 \times 625 = 43750\][/tex]
Thus, the fifth term is:
[tex]\[43750x^4\][/tex]
Therefore, the answer is:
[tex]\[\boxed{43750x^4}\][/tex]
You are having a family-barbeque. in attendance are a grandfather, a grandmother, two fathers, two mothers, four children, three grandchildren, one brother, two sisters, two sons, two daughters, one father-in-law, one mother-in-law and one daughter-in-law. if every person is eating two burgers, what’s the minimum amount of burgers you will need?
Final answer:
To calculate the minimum amount of burgers you will need, count the number of individuals who will be eating and multiply it by 2. The Total burgers needed is 50 .
Explanation:
To calculate the minimum amount of burgers you will need, you need to count the number of individuals who will be eating. Let's count:
1 Grandfather
1 Grandmother
2 Fathers
2 Mothers
4 Children
3 Grandchildren
1 Brother
2 Sisters
2 Sons
2 Daughters
1 Father-in-law
1 Mother-in-law
1 Daughter-in-law
Adding all of these individuals, we get a total of 25 people. Since every person will be eating 2 burgers, we can calculate the minimum amount of burgers we will need by multiplying the total number of people by 2:
Total burgers needed = 25 people * 2 burgers per person = 50 burgers
what are the binomial factors of the expression below? x^2-2x-15 Answers A. (x+3)(x+5) B. (x+3)(x-5) C. (x-3)(x+5) D. (x-3)(x-5)
Answer:
B.[tex](x-5)(x+3)[/tex]
Step-by-step explanation:
We are given that an expression
[tex]a^2-2x-15[/tex]
We have to find the binomial factors of the given expression.
[tex]x^2-2x-15[/tex]
To find the factors of given expression we will use of middle term splitting factor method.
The given can be written as
[tex]x^2-5x+3x-15[/tex]
[tex]x(x-5)+3(x-5)[/tex]
Taking (x-5) common
[tex](x-5)(x+3)[/tex]
Hence, the binomial factors of given expression is given by
[tex](x-5)(x+3)[/tex]
Option B is true.
What is the value of the expression?
Drag and drop the answer into the box to match the expression.
2.54+(−1.66)
Two numbers have prime factorizations of 2 2 · 3 · 5 and 2 · 3 2 · 7.
Which expression can be used to find their least common multiple?
...?
2^2 x 3^2 is the answer... don't be rude and mark the other person bad she gave you a good explanation and fixed your question AND got the answer right... write you question correctly cause you probably just copied it and didn't fix it up!
Just do this for future questions!
Integrate sin^9(x)dx ...?
help please!!!!!!!!!!!!!!!!!!!!!
What are the first 20 digits of pi?
The first 20 digits of pi are 3.1415926535897932384. It's an irrational number used in many mathematical calculations and its decimal representation never ends or repeats.
Explanation:The first 20 digits of the mathematical constant pi (π) are 3.1415926535897932384. Pi is an irrational number, meaning it cannot be expressed exactly as a fraction, and its decimal representation never ends or repeats.It not only appears in geometry and trigonometry, but also in various fields of mathematics and science. Its value is usually approximated as 3.14 or 3.14159 in most mathematical and scientific calculations.Learn more about pi here:https://brainly.com/question/23320385
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X^3+512=0 real or imaginary solution by factoring
What do the sin 25 and cosine 65 have in common. Explain why this is true. ...?
I need to solve this problem in my exam:
Suppose that the dollar cost of producing x radios is c(x) = 400 + 20x - 0.2x^2. Find the average cost per radio of producing the first 35 radios. ...?
The number of cartoons X, watched by Mrs. Kelly's first grade class on Saturday morning is shown below.
X P(X)
0 0.15
1 0.20
2 0.30
3 0.10
4 0.20
5 0.05
Give the standard deviation for the probability distribution above.
In the normal distribution, values in the distribution being measured are close to the mean (high or low values are infrequently found). true or false ?
Answer: True.
Step-by-step explanation:
Hi, the statement "In the normal distribution, values in the distribution being measured are close to the mean (high or low values are infrequently found)." is true.
The normal distribution is a probability function that how the values of a measure are distributed. Most of the values cluster around the mean, it’s a symmetric distribution.
Normal or Gaussian distributions Means that many variables have similar characteristics. So, middle values have higher probability or are more likely to occur, and extremely low or high values are more infrequent.
Ella writes about 7 songs every 5 months. at this rate, how many songs could she write in 50 months?
Which of these shows how the formulas for the area of a circle and the volume of a cylinder are related?
The area of a circle provides the radius of the cylinder’s base. Dividing that by the surface area of the cylinder’s base adds the third dimension of the solid.
The area of a circle provides the height of the cylinder. Multiplying that by the surface area of the cylinder’s base adds the third dimension of the solid.
The area of a circle provides the surface area of a cylinder’s base. Multiplying that by the height of the cylinder adds the third dimension of the solid. ...?
The correct answer is statement (c) The area of a circle provides the surface area of a cylinder’s base. Multiplying that by the height of the cylinder adds the third dimension of the solid.
The formula for the volume of a cylinder demonstrates how the area of a circle is used to find the volume of the cylinder. Specifically, the volume of a cylinder is given by [tex]\pi r^2h[/tex], where r is the radius of the base (a circle), and h is the height of the cylinder.
This means that the area of a circle [tex](\pi r^2)[/tex] is multiplied by the height (h) of the cylinder to obtain the volume. This clearly shows how the area of a circle is extended into three dimensions to derive the volume of a cylinder.
Complete question :
Which of these shows how the formulas for the area of a circle and the volume of a cylinder are related?
(a) The area of a circle provides the radius of the cylinder’s base. Dividing that by the surface area of the cylinder’s base adds the third dimension of the solid.
(b) The area of a circle provides the height of the cylinder. Multiplying that by the surface area of the cylinder’s base adds the third dimension of the solid.
(c) The area of a circle provides the surface area of a cylinder’s base. Multiplying that by the height of the cylinder adds the third dimension of the solid.
Is there any rational numbers between 5/7 and 6/7 and why? ...?
The reference angle for an angle whose measure is -440 is _____. 60 80 100
The reference angle for an angle whose measure is -440 is 80.
Step-by-step explanation:Reference angle--
It is the angle which is made from the terminal side of the given angle and it is always made by taking x-axis as a reference.
Here we are given a angle whose measure is:
-440.
This means that the angle covers more than one rotation and ends in the fourth quadrant.
Hence, the reference angle is:
-440+360=80
HELP ASAP!!
given the function f (x)=2x-5 over 3 , which of the below expressions is correct ?
Multiply. write your answer in simplest form. 35/70
Which of the following is not a cost of being financially irresponsible?
You are late to work because you forgot to set the alarm.
You have no money to pay the cable bill, but you have a new flat screen television.
You can't afford to go to the movies.
You end up paying more for things because of credit card interest.
Simplify: (4xy^-2) / [ 12x^(-1/3) y^5]
Solve the problem.
When granular materials are allowed to fall freely, they form conical (cone-shaped) piles. The naturally occurring angle of slope, measured from the horizontal, at which the loose material comes to rest is called the angle of repose and varies for different materials. The angle of repose θ is related to the height h and base radius r of the conical pile by the equation θ = cot-1(r/h). A certain granular material forms a cone-shaped pile with a height of 19 feet and a base diameter of 30 feet. What is the height of a pile that has a base diameter of 104 feet?
The angle of repose is calculated using the equation θ = cot-1(r/h). For a pile with a height of 19 feet and a base diameter of 30 feet, the angle of repose is approximately 38.03 degrees. Using this angle and the base diameter of 104 feet, we can find that the height of the new pile is approximately 77.07 feet.
The angle of repose θ is related to the height h and base radius r of the conical pile by the equation θ = cot-1(r/h).
Given a pile with a height of 19 feet and a base diameter of 30 feet, we can find its base radius by dividing the diameter by 2, which gives us a radius of 15 feet.
Now, we can substitute the given values into the equation to find the angle of repose θ.
θ = cot-1(15/19)
Using a calculator, we find that the angle of repose is approximately 38.03 degrees.
Next, we can use the height of the new pile and the known angle of repose to find its base radius.
Given a base diameter of 104 feet, the radius is half of the diameter, so the radius is 52 feet.
Now, we can use the equation θ = cot-1(r/h) and solve for h to find the height of the new pile.
h = r / tan(θ)
Substituting the known values, we get:
h = 52 / tan(38.03)
Using a calculator, we find that the height of the pile with a base diameter of 104 feet is approximately 77.07 feet.
A cannery processed 1,445 pounds of strawberries in 4.5 hours. The cannery processed 2,320 of strawberries in 7 hours. Write a linear equation to model the relationship between the weight of strawberries S and time T. How many pounds of strawberries can be processed in 11 hours?
The linear equation to model the relationship between the weight of strawberries and time can be found using the slope and y-intercept. The equation is S = 690T - 560.5. In 11 hours, the cannery can process 7109.5 pounds of strawberries.
Explanation:To write a linear equation to model the relationship between the weight of strawberries S and time T, we need to determine the slope and the y-intercept. The slope represents the rate at which strawberries are processed per hour, while the y-intercept represents the initial amount of strawberries processed.
Using the given information, we can use the formula for calculating the slope (m) of a line: m = (y2 - y1) / (x2 - x1). Plugging in the values, we have: m = (2320 - 1445) / (7 - 4.5) = 690 pounds/hour.
Next, we can calculate the y-intercept (b) using the formula: b = y - mx. Plugging in the values for one of the points (1445 pounds in 4.5 hours), we have: 1445 = 690(4.5) + b. Solving for b, we get: b = -560.5 pounds.
Therefore, the linear equation to model the relationship between the weight of strawberries S and time T is: S = 690T - 560.5. To find the number of pounds of strawberries that can be processed in 11 hours, we can substitute T = 11 into the equation: S = 690(11) - 560.5 = 7109.5 pounds.
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Working alone, it would take Shandra 45 minutes to mow the lawn. To mow the same lawn, also working alone, it would take Kelly 1 hour and 15 minutes. If they work together, the girls believe it will take them 1 hour to mow the lawn.
Which best describes the reasonableness of their solution? Their estimate is longer than the time it will actually take: Their estimate is accurate: Their estimate is shorter than the amount of time it will actually take: Their estimate is longer than the time it will actually take:
The correct answer is:
Their estimate is longer than the time it will actually take.
( Since, Estimated time-- 1 hour=60 minutes
Time actually taken= 28.125 minutes )
Step-by-step explanation:Time taken by Shandra alone to mow the field is: 45 minutes.
Time taken by Kelly alone to mow the field is: 1 hour 15 minutes.
We know that:
1 hour=60 minutes
Hence, 1 hour 15 minutes=60 minutes +15 minutes=75 minutes.
By looking at the time we consider that the total work be of 225 units.
This means that the Efficiency of Shandra is: 225/45=5 units/minutes.
Similarly,
Efficiency of Kelly is: 225/75=3 units/minutes.
Now, when they will work together there efficiency will be added.Hence, 5+3=8
Hence, the time taken by both of them when they work together is:
225/8
= 28.125 minutes.
But it was estimated that they took 1 hour=60 minutes.
Hence, there estimation of time taken was longer than what they actually took.
In the diagram, m∠COE = 55°. If m∠2 = 2x and m∠3 = x+10, what is the measure of angle 2?
15
30
45
60
Answer:
The answer is 30
Step-by-step explanation
You gather what you knowCOE = 55°, m∠2 = 2x, m∠3 = x + 10
Then you set up an equation using that information2x + x + 10 = 55
Combine like terms3x + 10 = 55
Find xX= 15
Plug in "x = 15" to "2x" to find out what the measure of angle 2 is2 x 15 = 30
Therefore m∠2 = 30
Is 61 composite or prime
Hello There!
61 is a prime number.
Its only factors are: 1 and 61. It has no more factors.
#HaveAnAmazingDay
Hope this helps!
[tex]GraceRosalia[/tex]
In what position would you find the 'leading' runner in a race? In what position do you suppose you would find the 'leading coefficient' in a polynomial?
Amanda has a job transporting soft drinks by truck. Her truck is filled with cans that weigh 14 ounces each and bottles that weigh 70 ounces each. There is a combined total of 930 cans and bottles in her truck. Let x be the number of 14 -ounce cans in her truck. Write an expression for the combined total weight (in ounces) of the cans and bottles in her truck.
The combined total weight of the cans and bottles in the truck, if x represents the number of 14-ounce cans, can be represented by the expression 14x + 70(930 - x). This combines the weight of the 14-ounce cans (14x) with the weight of the 70-ounce bottles (70(930-x)).
Explanation:In order to find the expression for the combined total weight (in ounces) of the cans and bottles in the truck, we have to first determine the number of bottles. We know that the total number of cans and bottles is 930, so the number of bottles would be 930 - x. Each can weighs 14 ounces and each bottle weighs 70 ounces. Therefore, the total weight in ounces will be the sum of the weight of cans and the weight of bottles. This can be represented as 14x + 70(930 - x). This expression represents the combined total weight in ounces of the items in Amanda's truck.
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