what is the fourth term in the binomial expansion (a+b)^6)

Answers

Answer 1

Answer:

[tex]20a^3b^3[/tex]

Step-by-step explanation:

Binomial Series

[tex](a+b)^n=a^n+\dfrac{n!}{1!(n-1)!}a^{n-1}b+\dfrac{n!}{2!(n-2)!}a^{n-2}b^2+...+\dfrac{n!}{r!(n-r)!}a^{n-r}b^r+...+b^n[/tex]

Factorial is denoted by an exclamation mark "!" placed after the number. It means to multiply all whole numbers from the given number down to 1.

Example:  4! = 4 × 3 × 2 × 1

Therefore, the fourth term in the binomial expansion (a + b)⁶ is:

[tex]\implies \dfrac{n!}{3!(n-3)!}a^{n-3}b^3[/tex]

[tex]\implies \dfrac{6!}{3!(6-3)!}a^{6-3}b^3[/tex]

[tex]\implies \dfrac{6!}{3!3!}a^{3}b^3[/tex]

[tex]\implies \left(\dfrac{6 \times 5 \times 4 \times \diagup\!\!\!\!3 \times \diagup\!\!\!\!2 \times \diagup\!\!\!\!1}{3 \times 2 \times 1 \times \diagup\!\!\!\!3 \times \diagup\!\!\!\!2 \times \diagup\!\!\!\!1}\right)a^{3}b^3[/tex]

[tex]\implies \left(\dfrac{120}{6}\right)a^{3}b^3[/tex]

[tex]\implies 20a^3b^3[/tex]


Related Questions

Simplify the root below
\frac{a}{b} \sqrt[]{ \frac{405}{324} }

Answers

Answer:

  (√5)/2

Step-by-step explanation:

To simplify the root, we remove perfect squares.

__

Factors that are the same in numerator and denominator cancel. A square under the radical can have its root brought out of the radical.

  [tex]\sqrt{\dfrac{405}{324}}=\sqrt{\dfrac{9^2\cdot5}{9^2\cdot2^2}}=\boxed{\dfrac{\sqrt{5}}{2}}[/tex]

Show me 2 line up and down are they parallelside by side on a graph

Answers

Two (2) lines are considered to be parallel if their slopes are the same (equal) and they do not intersect.

The condition for two parallel lines.

In Geometry, two (2) lines are considered to be parallel if their slopes are the same (equal) and they've different y-intercepts.

This ultimately implies that, two (2) lines are parallel under the following conditions:

m₁ = m₂

Note: m is the slope.

In conclusion, two (2) parallel lines are shown in the image attached below for more understanding,

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Which of the following is the conjugate of the expression below when x>-2?
6+√x+2
OA. 6+√x+2
B. 6-√x-2
C. 6-√√x+2
D. 6+√x-2

Answers

C

WHY? Because it is c I took this

Math Topic: Sum, Difference, and Product of two functions

Help with this!

Answers

The expression are; [tex](h - g) (x) = 3x^{3} -x^{2}[/tex], [tex](h + g) (x) = 3x^{3} +x^{2}[/tex] , [tex](h.g) (-1) = -3[/tex].

What is a function?

The function is a type of relation, or rule, that maps one input to specific single output.

Given function;

g(x) = [tex]x^{2}[/tex]

h(x) = [tex]3x^{3}[/tex]

The expression;

[tex](h - g) (x) = 3x^{3} -x^{2}[/tex]

[tex](h + g) (x) = 3x^{3} +x^{2}[/tex]

[tex](h.g) (-1) = x^{2} .3x^{3} \\\\(h.g) (-1) = 3x^5\\\\(h.g) (-1) = 3(-1)^5\\\\(h.g) (-1) = -3[/tex]

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A count of patients needing a blood transfusion in the emergency room at a city hospital would best be modelled by what
distribution?
A. geometric
B. continuous
C. normal
D. Poisson

Answers

The count of patients needing a blood transfusion can be modelled as a continuous distribution.

What is a continuous distribution?

A continuous distribution is a distribution that can take on an infinite number of possible values. A continuous distribution is not defined at specific values. The number of patients can be 1 or 10 or 50 or even 0.

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PLEASE HELP
Write a rational function that meets the following criteria. Leave your answer in factored form.
• Vertical asymptote at x = -2
• Horizontal asymptote at y = 3 and a hole at x = -3

Answers

Answer:

Step-by-step explanation:

Determine the number of crawlers and rovers to make,that will maximize the company profit in a given week

Answers

Answer:

jjjijijijiiijijijijijijij no pliss inglés. ;D.

28.6% of city employees ride the bus to work.Last year,29.7% of city employees rode the bus to work.What is the relative change from last year to this year

Answers

The number of employees taking the bus to an office is reduced by 1.1% from the last year. The relative change will be 1.1%.

What is an expression?

Expression in maths is defined as the collection of the numbers variables and functions by using signs like addition, subtraction, multiplication, and division.

Given that:-

28.6% of city employees ride the bus to work. Last year,29.7% of city employees rode the bus to work.

The relative change will be calculated as follows:-

Relative change =  Present year data - Last year's data

Relative change = 28.6 - 29.7

Relative change  = -1.1%

Therefore the number of employees taking the bus to an office is reduced by 1.1% from the last year. The relative change will be 1.1%.

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. Solve for the missing value (x).
Include the correct unit of measure in your answer.

12.5mg : 5mL = 24mg : x mL

Answers

The missing value of x will be 9.6 mL.

What are ratio and proportion?

A ratio is a collection of ordered integers a and b represented as a/b, with b never equaling zero. A proportionate expression is one in which two items are equal.

The equation is given below.

12.5 mg : 5 mL = 24 mg : x mL

Then the value of x will be

12.5 / 5 = 24 / x

          x = (24 × 5) / 12.5

          x = 9.6 mL

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The inverse of the function f(x) = x + 10 is shown.
==
h(x) = 2x-0
What is the missing value?
01
05
O 10
O 20

Answers

Answer:

So, the answer is 20

Step-by-step explanation:

[tex]f(x) = \frac{1}{2} x + 10 = = = > \\ f(x) - 10 = \frac{1}{2} x = = = > \\ 2f(x) - 20 = x = = = > \\ {f}^{ - 1} (y) = 2y - 20 = = = > {f}^{ - 1} (x) = 2x - 20[/tex]

The missing value of the inverse function is 20.

Option D is the correct answer.

What is a function?

A function has an input and an output.

A function can be one-to-one or onto-one.

It simply indicated the relationships between the input and the output.

Example:

f(x) = 2x + 1

f(1) = 2 + 1 = 3

f(2) = 2 x 2 + 1 = 4 + 1 = 5

The outputs of the functions are 3 and 5

The inputs of the function are 1 and 2.

We have,

To find the inverse of a function, we usually replace f(x) with y, then solve for x in terms of y, and finally replace y with the inverse function notation h(x).

So let's do that with the given function:

f(x) = (1/2)x + 10

y = (1/2)x + 10

2y = x + 20 (multiply both sides by 2 to isolate x)

x = 2y - 20 (subtract 20 from both sides)

Now we can replace x with h(x) and y with x to get the inverse function:

h(x) = 2x - 20

Thus,

The missing value is 20.

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25. Which of the following expresses 374,274
in scientific notation?
(1)
(2)
(3)
3.74274 x 105
3.74274 x 106
37.4274 X 106
374.274
x 105
× 103
(4)
(5) 3742.74
Ite

Answers

Answer:

3

Step-by-step explanation:

374274 =3.74274 x 100000=3.74272x 10^5

Graph the linear function whose equation is y-2=-(x+1) by following these steps:
y
X
y
4
Step 1: Identify the slope.
slope = =

Answers

Answer:

slope == -1

Step-by-step explanation:

y-2 = -(x+1)

; y-2 = -x-1 expanding the bracket

; y = -x-1+2

; y = 1-x

hence slope equals -1

A tank weighs 5.6kg when it 1/4 filled with water.If it weighs 10.4kg when it is full ,what will be it's weight when it is empty.​

Answers

Answer:

4 kg

Step-by-step explanation:

........................

Answer:

4 kg

Explanation:

Let the mass of tank be 't' and water 'w'

First equation :              (when it is 1/4 filled with water)

[tex]\sf t + \frac{1}{4} w = 5.6[/tex]

Second equation :        (when it is filled fully with water)

[tex]\sf t +w = 10.4[/tex][tex]\sf w = 10.4 - t[/tex]

Insert second equation into first equation :

[tex]\rightarrow \sf t + \dfrac{1}{4} (10.4 - t) = 5.6[/tex]

[tex]\rightarrow \sf t - \dfrac{1}{4} t + 2.6 = 5.6[/tex]

[tex]\rightarrow \sf \dfrac{3}{4} t = 5.6 - 2.6[/tex]

[tex]\rightarrow \sf \dfrac{3}{4} t = 3[/tex]

[tex]\rightarrow \sf t = 4[/tex]

The tank weights 4 kg when it is empty.

supposed an individual has a body fat percentage of 17.5% and weighs 164 lbs. how many pounds of his weight is made up of fat? round your answer to the nearest tenth.

Answers

The amount of pounds of his weight is made up of fat is 28.7lbs

Percentages and values

Given the following parameters

Total weight of body = 164lbs

Percentage fat = 17.5%

Determine the amount made up of fat

Fat amount = 17.5% of 164

Fat amount = 0.175 * 164

Fat amount = 28.7lbs

Hence the amount of pounds of his weight is made up of fat is 28.7lbs

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A rectangular bedroom has an area of 117 ft. the length of the bedroom is 4ft more than the width. find the length and the width of the bedroom

Answers

Answer: Length = 13 ft. , Width = 9 ft.

Step-by-step explanation:

Rectangle area = Length x width

Area = 117 ft.

Length = x + 4

Width = x

To solve

(x)(x + 4) = 117

x^2 + 4x = 117

x^2 + 4x - 117 = 0

(x - 9)(x + 13)

x = 9, -13

x = 9 (measurement cannot be negative)

Length

x + 4

9 + 4 = 13

On the average, five smokers pass acertain street corners every ten minutes, what is the probability that during a given 10 minutes the number of smokers passing will be A. 6 or fewer B. 7 or more C. Exactly 8​

Answers

Answer:

The answer will be A. 6 or fewer.

The probability that during a given 10 minutes the number of smokers passing will be 6 or fewer.

What is probability?

Probability can be defined as the ratio of the number of favorable outcomes to the total number of outcomes of an event. For an experiment having 'n' number of outcomes, the number of favorable outcomes can be denoted by x. The formula to calculate the probability of an event is as follows.

Probability(Event) = Favorable Outcomes/Total Outcomes = x/n

Example

Suppose we have to predict about the happening of rain or not. The answer to this question is either "Yes" or "No". There is a likelihood to rain or not rain. Here we can apply probability. Probability is used to predict the outcomes for the tossing of coins, rolling of dice, or drawing a card from a pack of playing cards.

Terminology of Probability Theory

The following terms in probability help in a better understanding of the concepts of probability.

Experiment: A trial or an operation conducted to produce an outcome is called an experiment.

Sample Space: All the possible outcomes of an experiment together constitute a sample space. For example, the sample space of tossing a coin is head and tail.

Favorable Outcome: An event that has produced the desired result or expected event is called a favorable outcome. For example, when we roll two dice, the possible/favorable outcomes of getting the sum of numbers on the two dice as 4 are (1,3), (2,2), and (3,1).

Trial: A trial denotes doing a random experiment.

Given:

There are 5 smokers who passes by in every 10 minutes.

So, the probability that during a given 10 minutes the number of smokers passing will be approx 5 or less.

i.e., 6 or fewer.

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Suppose that the total cost of flatiron TVs is given by the function TC(Q) = 500Q + 300 dollars, where Q is the number of TVs produced. Find TC(10).

Answers

The value of TC(10) is 5300 Dollars

Given function is TC(Q) = 500Q +300 dollars

We need to calculate the value for TC (10)

As we know that the equation given is TC(Q) = 500Q + 300 dollars

Q is the number of TVs produced

Therefore, The value of Q is 10 for TC(10)

Substituting the value of Q

in the total cost of flatiron TVs given by function TC(Q) = 500Q + 300 dollars That Q= 100,

TC(10) = 500(10)+300 dollars

∴ TC (10) = 5000+300 dollars

∴ TC (10) = 5300 dollars

Hence the value of TC(10) is 5300 Dollars

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Which is the graph of f(c)=1/4(4)^x?

Answers

Answer: The graph shown

Step-by-step explanation:

Another graph of the function is attached in this answer.

Evaluate the expression.
-2(12 - 2 (-4))²
What is the value of the expression?
Enter your answer in the box.

Answers

Answer:

-2(10(-4))²

-2(-40)²

-2 × 1600

= -3200

Answer: = -3200

Step-by-step explanation:

took the test

Solve for the values of x and y.

Answers

[tex]\quad \huge \quad \quad \boxed{ \tt \:Answer }[/tex]

[tex]\qquad \tt \rightarrow \: x = 6 \:\: units[/tex]

[tex]{\qquad \tt \rightarrow \: y = 20° } [/tex]

____________________________________

[tex] \large \tt Solution \: : [/tex]

Two sides and their opposite angles of a Triangle are given equal.

[tex]\large \textsf{ For x :} [/tex]

[tex]\qquad \tt \rightarrow \: 2x - 1 = x + 5[/tex]

[tex]\qquad \tt \rightarrow \: 2x - x = 5 + 1[/tex]

[tex]\qquad \tt \rightarrow \: x = 6 \: \: units[/tex]

[tex] \large\textsf{For y :} [/tex]

[tex]\qquad \tt \rightarrow \: 3y - 10 = y + 30 [/tex]

[tex]\qquad \tt \rightarrow \: 3y - y = 30 + 10[/tex]

[tex]\qquad \tt \rightarrow \: 2y = 40[/tex]

[tex]\qquad \tt \rightarrow \: y = 20 \degree[/tex]

Answered by : ❝ AǫᴜᴀWɪᴢ ❞

Steps for solving 5(2x-8)= 20 are shown.
5(2x-8)= 20
10x-40-20
10x-40+40-20 + 40
10x = 60
10x 60
10
10
X-6

Which of these is not part of the solution process?
A. Using the distributive property
B. Combining like terms
C. Dividing both sides by 10 to isolate the variable
D. Using the commutative property

Answers

The one which is not part of the solution is option D. Using the commutative property.

What is a system of equations?

A system of equations is two or more equations that can be solved to get a unique solution. the power of the equation must be in one degree.

Given;

5(2x-8)= 20

By Using the distributive property

10x-40-20 = 0

Combining like terms

10x-40+40-20 + 40 = 0

Dividing both sides by 10 to isolate the variable

10x = 60

x = 6

Thus, we can see that all the options are included except option D.

So, the one which is not part of the solution is option D. Using the commutative property.

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Complete the following table using the equation: y = -2x²

Answers

The parabola is a downward parabola which is passing through origin and leading coefficient of the negative 2.

What is the parabola?

It's the locus of a moving point that keeps the same distance between a stationary point and a specified line. The focus is a non-movable point, while the directrix is a non-movable line.

The equation of a quadratic function, of vertex (h, k), is given by:

y = a(x – h)² + k

where a is the leading coefficient.

The equation of the parabola will be

y = -x²

The given parabola is a downward parabola which is passing through origin and leading coefficient of the negative 2.

The graph of the function is given below.

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I need this whole page solved please

Answers

Answer:

5. 540°

Step-by-step explanation:

m<a + m<c = 180°

m<h + m<f = 180°

m<k + m<m = 180°

Then m<a + m<c + m<h + m<f + m<k + m<m = 180° x 3 = 540°

One number is 2 more than another number. The product of the numbers is 440. Find the numbers.​

Answers

Answer:

219,221

Step-by-step explanation:

the second number can be represented by the equation n1+2=n2

adding these expressions together you get

(n+2)+n=440

youre going to think of n now as just your first number because the equation solves for n1

with that you will get a number valued at 219

you then solve the first equation n1+2=n2 because we now know the value of n1

219+2=221

so the second numbers value is 221

urgent help algebra 2

Answers

Answer:

y = - 1/4 x + 1

Step-by-step explanation:

Find two convenient integer coordinates like   -4,2 and  4, 0

  use these points to  calculate the slope to be -1/4

   intercept is b = 1

y = -1/4 x + 1

sub to oiVJDF NvlC Nvlkjxdv Nlkj;zsdn

Answers

What does sub to oiVJDF NvlC Nvlkjxdv Nlkj;zsdn mean
no oiVJDF NvlC Nvlkjsjqisksjsiwjbwiqiwjajjdjaiwh

On a coordinate plane, a dashed straight line has a positive slope and goes through (negative 3, 1) and (0, 3). Everything to the left of the line is shaded.

Which linear inequality is represented by the graph?


y <2/3x + 3

y > 3/2x + 3

y > 2/3x + 3

y < 3/2x + 3

Answers

Answer:

The correct linear inequality is y<2/3x+3

Step-by-step explanation:

Linear inequalities are defined as expressions in which two linear expressions are compared using the inequality symbols

In the graph, the grey region corresponds to the region non-allowed by inequality. We see that for x=0, y is allowed to be only less than 3: this means that the correct inequality must be in the form y<mx+3, so only the 1st option or the 4th option.

In order to choose the correct option, we should find the value of m, the slope of the line in the graph. This slope can be found by calculating the variation of y divided by the variation of x:

m=dy/dx

Choosing for example the points x=0 (which corresponds to y=3) and x=3 , we find

So, the equation of the line is y=2/3x+3

and so the correct inequality is

y<2/3x+3

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which choices are equivalent to the expression 3 square root of 5

Answers

Answer:

6.708.

Step-by-step explanation:

hope this helps

Please help


Follow the steps to find the area of the shaded region.


First use the formula below to find the area of the whole sector.


Sector Area= ( angle of sector/360). R2


Sector Area = ? Cm^2
Round to four decimal places.

14 cm

46°

14 cm

Answers

The area of the sector, to 4 decimal places, is: 78.6794 cm².

How to Find the Area of a Sector of a Circle?

Area of sector = ∅/360 × πr².

Given the following:

∅ = 46°Radius (r) = 14 cm

Area of sector = 46/360 × π(14²)

Area of sector ≈ 78.6794 cm²

The area of the sector is approximately 78.6794 cm².

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given: f(x)=2x^2 and g(x)=x-8. Find f(g(x))

Answers

The value of the composite function f(g(x)) is 2(x-8)^2

Composite functions

Composite functions are functions in another function. Given the following functions

f(x)=2x^2 and

g(x)=x-8

Determine f(g(x))

f(g(x)) = f(x-8)

f(x-8) = 2(x-8)^2

f(g(x)) = 2(x-8)^2

Hence the value of the composite function f(g(x)) is 2(x-8)^2

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