Answers
Answer:
In counting to a base, when we get to the point where the base integer comes up, we replace by 10, on the first time and by 10 at every other time the integer is about to come up
We now this in base 10 as when we get to 10 we input 10 and so and so till we get to 99 where since we are in base 10 the next number is 100 because that is the highest we can go while in base 11 100 in base is equivalent to 91
So we have for base 5
1 , 2, 3, 4, 10, 11, 12, 13, 14, 20, 21, 22, 23, 24, 30, 31, 32, 33, 34, 40, 41, 42, 43, 44, 100, 101, 102, 103, 104, 110, 111, 112, 113, 114, 120, 121, 122, 123, 124, 130, 131, 132, 133, 134, 140, 141, 142, 143, 144, 200
We see that on getting to 4 instead of the next digit to be 5 we replace it with 10 because 5 cannot be displayed in base 5
Similarly after 14 is 20 because 5 cannot be displayed at 24 the next number is 30 because 5 cannot be displayed
at 34 the next 40, while at 44 the next is (we could have written 50 but 5 is 10 in base 5 so we write 100) 100
At 144, we note that 44 = 100 + 44, therefore, the next number will be 100 + 44+ 1 = 100 + (45 but 45 = 100 in base 5) 100 so that the next number is 200
Base 2
In base 2 we have;
1, 10, 11, 100, 101, 110, 111, 1000, 1001, 1010, 1011, 1100, 1101, 1110, 1111, 10000, 10001, 10010, 10011, 10100, 10101, 10110, 10111, 11000, 11001, 11010, 11011, 11100, 11101, 11110, 11111, 100000, 100001, 100010, 100011, 100100, 100101, 100110, 100111, 101000, 101001, 101010, 101011, 101100, 101101, 101110, 101111, 110000, 110001, 110010
We see that there are no 2s and after 11 we get 100 just like after 99 we get 100 in base 10 and after 44 we get 100 in base 5
So we need to still have an idea of the number system just as we did for base 10
Step-by-step explanation:
Related Questions
can somebody help me plz
Answers
Answer:
126 people
Step-by-step explanation:
9/5 is the ratio tea/coffee
let x be the one who preferred coffee and x+36 preferred tea
9/5=x+36/x
9x=5x+5(36)
9x-5x=180
4x=180
x=180/4=45
x=45 coffee
x+36=45+36=81
45+81=126
check : 81/45= 9/5
There are 6 women and 9 men eligible to be in a committee of 5. Find the expected number of women on the committee given that at least one woman must be on the committee. Round the probabilities of the distribution to four decimal places or keep them as fractions. Round the answer to two decimal places.
Answers
Answer:
P = 0.2517
Step-by-step explanation:
In this case we must calculate the probability of event, which would be the number of specific events (that is, at least one woman and the rest men, 4), then it would be to choose 1 of 6 women by 4 of 9 men divided by the number of total events, which would be to choose 5 (committee size) out of 15 (9 men + 6 women, total number of people)
P (at least one woman) = 6C1 * 9C4 / 15C5
we know that nCr = n! / (r! * (n-r)!)
replacing we have:
6C1 = 6! / (1! * (6-1)!) = 6
9C4 = 9! / (4! * (9-4)!) = 126
15C5 = 15! / (5! * (15-5)!) = 3003
Therefore it would be:
P (at least one woman) = 6 * 126/3003
P = 0.2517
That is, approximately 1 out of 4 women.
10. 80 machines can produce 4800 identical pens in 5 hours. At this rate
a) how many pens would one machine produce in one hour?
b) how many pens would 25 machines produce in 7 hours?
Answers
Answer:
a) 12 (Simply divide 4800/5 to get 960. Then divide 960/80 to get 12)
b) 2100 (Simply multiply 12 by 25 by 7)
Hope it helps <3
The summer has ended and it’s time to drain the swimming pool. 20 minutes after pulling the plug, there is still 45 000L of water in the pool. The pool is empty after 70 minutes.
Calculate the rate that the water is draining out of the pool. (Hint: remember this line is sloping down to the right)
Answers
Answer:
900L per minute
Step-by-step explanation:
1- 70 - 20 = 50
2- in this 50 min the 45000L has been drawned
3- 45000L / 50 = 900L
.. ..
If x - 10 is a factor of x2 - 8x - 20, what is the other
factor?
X +
Answers
Answer:
(x + 2)
Step-by-step explanation:
When we factor the expression x² - 8x - 20, we should get (x + 2)(x - 10).
Alternatively, we can use synthetic division or long division to get our answer.
Answer:
x + 2
Step-by-step explanation:
got it right edg '22
Which statement explains how the lines x+y=2 and y=x+4 are related?
(1) They are parallel.
(2) They are perpendicular.
(3) They are the same line.
4) They are not related.
Answers
Answer:
(2)They are perpendicular.
Step-by-step explanation:
which linear inequality is represented by the graph?
Answers
y < or = 1/2x + 3
Explanation
We know when x = 0, y has to be 3 (y intercept).
Therefore y>= or <= 2x + 4 is wrong since substituting x = 0 gives y=4.
Since you need the shaded part of the graph that’s under the line you need all the y values that are smaller than and equal to the inequality.
If it was greater than or smaller to the area above the line would be shaded.
(1) 10x’y' + 15xy? :
Answers
Answer:
factor: 5(2x'y'+3xy)
Step-by-step explanation:
thats for factoring, i didnt know what you needed
Answer:
25xy
Step-by-step explanation:
collect like terms
A car travels the first 50 km of its journey at an average speed of 25 m/s and the next 120 km at an On his outward journey, Ali travelled at a speed of s km/h for 2.5 hours. On his return journey, he increased his speed by 4 km/h and saved 15 minutes. Find Ali's average speed for the whole journey. Speed of 80 km/h. The car completes the last part of its journey at an average speed of 90 km/h in 35 minutes. Find the average speed for its entire journey, giving your answer in km/h.
Answers
Answer:
The car's average speed for the entire journey = 84.315 km/h
Step-by-step explanation:
Correct Question
A car travels the first 50km of its journey at an average speed of 25m/s and the next 120 km at an average speed of 80km/h. the car completes the last part of its journey at an average speed of 90km/h in 35 minutes. Find the average speed for its entire journey, giving your answer in km/h.
Solution
Average speed is given as total distance travelled divided by total time taken.
So, we will compute the distance covered for each part of the journey and the corresponding time it takes to cover each of these distances.
- The car travels the 50 km first part of the journey at a speed of 25 m/s.
25 m/s = 90 km/h
We have the distance covered in the first part of the journey, now, we need the time taken to cover the distance.
Speed = (Distance/Time)
Time = (Distance/Speed)
Distance = 50 km, Speed = 90 km/h
Time = (50/90) = 0.5556 hr
- The next part, the car covers 120 km at a speed of 80 km/h
Time = (Distance/Speed) = (120/80) = 1.5 hr
- For the last part of the journey, the car travels with an average speed of 90 km/h for 35 minutes.
35 minutes = (35/60) hr = 0.5833 hr
Here, we need to calculate the distance covered for the last part.
Speed = (Distance/Time)
Distance = (Speed) × (Time) = 90 × 0.5833 = 52.5 km
Total distance covered = 50 + 120 + 52.5 = 222.5 km
Total time taken = 0.5556 + 1.5 + 0.5833 = 2.6389 hr
Average Speed = (222.5/2.6389) = 84.315 km/h
Hope this Helps!!!
I need help with this problem ASAP please i have until tmrw to finish my course entirely so if anyone can help it would be greatly appreciated
Answers
Answer:
Equation: f(x) = -x² + 3
Step-by-step explanation:
If we want to find the roots/solve for x, we can either graph the problem or use quadratic formula to find when f(x) = 0:
To get the equation of the graph, our parent function is: f(x) = a(x - h)² + k, or f(x) = x². Since we are reflecting over the x-axis with a vertical stretch and vertically moving up to (0, 3), we are modifying a and k only.
Let R be the relation on the set of ordered pairs of positive integers such that ((a, b), (c, d)) ∈ R if and only if ad = bc. Arrange the proof of the given statement in correct order to show that R is an equivalence relation. (Prove the given relation is reflexive first, and then symmetric and transitive.)
Answers
Answer:
The given relation R is equivalence relation.
Step-by-step explanation:
Given that:
[tex]((a, b), (c, d))\in R[/tex]
Where [tex]R[/tex] is the relation on the set of ordered pairs of positive integers.
To prove, a relation R to be equivalence relation we need to prove that the relation is reflexive, symmetric and transitive.
1. First of all, let us check reflexive property:
Reflexive property means:
[tex]\forall a \in A \Rightarrow (a,a) \in R[/tex]
Here we need to prove:
[tex]\forall (a, b) \in A \Rightarrow ((a,b), (a,b)) \in R[/tex]
As per the given relation:
[tex]((a,b), (a,b) ) \Rightarrow ab =ab[/tex] which is true.
[tex]\therefore[/tex] R is reflexive.
2. Now, let us check symmetric property:
Symmetric property means:
[tex]\forall \{a,b\} \in A\ if\ (a,b) \in R \Rightarrow (b,a) \in R[/tex]
Here we need to prove:
[tex]\forall {(a, b),(c,d)} \in A \ if\ ((a,b),(c,d)) \in R \Rightarrow ((c,d),(a,b)) \in R[/tex]
As per the given relation:
[tex]((a,b),(c,d)) \in R[/tex] means [tex]ad = bc[/tex]
[tex]((c,d),(a,b)) \in R[/tex] means [tex]cb = da\ or\ ad =bc[/tex]
Hence true.
[tex]\therefore[/tex] R is symmetric.
3. R to be transitive, we need to prove:
[tex]if ((a,b),(c,d)),((c,d),(e,f)) \in R \Rightarrow ((a,b),(e,f)) \in R[/tex]
[tex]((a,b),(c,d)) \in R[/tex] means [tex]ad = cb[/tex].... (1)
[tex]((c,d), (e,f)) \in R[/tex] means [tex]fc = ed[/tex] ...... (2)
To prove:
To be [tex]((a,b), (e,f)) \in R[/tex] we need to prove: [tex]fa = be[/tex]
Multiply (1) with (2):
[tex]adcf = bcde\\\Rightarrow fa = be[/tex]
So, R is transitive as well.
Hence proved that R is an equivalence relation.
The relation R is an equivalence if it is reflexive, symmetric and transitive.
The order to options required to show that R is an equivalence relation are;
((a, b), (a, b)) ∈ R since a·b = b·aTherefore, R is reflexiveIf ((a, b), (c, d)) ∈ R then a·d = b·c, which gives c·b = d·a, then ((c, d), (a, b)) ∈ RTherefore, R is symmetricIf ((c, d), (e, f)) ∈ R, and ((a, b), (c, d)) ∈ R therefore, c·f = d·e, and a·d = b·cMultiplying gives, a·f·c·d = b·e·c·d, which gives, a·f = b·e, then ((a, b), (e, f)) ∈RTherefore R is transitiveFrom the above proofs, the relation R is reflexive, symmetric, and transitive, therefore, R is an equivalent relation.Reasons:
Prove that the relation R is reflexive
Reflexive property is a property is the property that a number has a value that it posses (it is equal to itself)
The given relation is ((a, b), (c, d)) ∈ R if and only if a·d = b·c
By multiplication property of equality; a·b = b·a
Therefore;
((a, b), (a, b)) ∈ R
The relation, R, is reflexive.Prove that the relation, R, is symmetric
Given that if ((a, b), (c, d)) ∈ R then we have, a·d = b·c
Therefore, c·b = d·a implies ((c, d), (a, b)) ∈ R
((a, b), (c, d)) and ((c, d), (a, b)) are symmetric.
Therefore, the relation, R, is symmetric.Prove that R is transitive
Symbolically, transitive property is as follows; If x = y, and y = z, then x = z
From the given relation, ((a, b), (c, d)) ∈ R, then a·d = b·c
Therefore, ((c, d), (e, f)) ∈ R, then c·f = d·e
By multiplication, a·d × c·f = b·c × d·e
a·d·c·f = b·c·d·e
Therefore;
a·f·c·d = b·e·c·d
a·f = b·e
Which gives;
((a, b), (e, f)) ∈ R, therefore, the relation, R, is transitive.Therefore;
R is an equivalence relation, since R is reflexive, symmetric, and transitive.
Based on a similar question posted online, it is required to rank the given options in the order to show that R is an equivalence relation.
Learn more about equivalent relations here:
https://brainly.com/question/1503196
Mrs. Sing invests $12,876 for her business at an annual interest rate of 7 percent for 3 years. Which number will Mrs. Sing substitute for r in the simple interest formula? I = p r t
Answers
Answer:
she wills substitute r with 7%.
Step-by-step explanation:
Just did it.
Answer:
R = 7%
Principal (P)
Rate (R)
Time (t)
Step-by-step explanation:
The full intrest rate formula is i=prt
You are looking for r
R is the interest rate 7%
So i = 12,876*7%*3years will be the full equation.
3. In the polygon below, what kind of
angle is P?
A Acute
B Obtuse
C Right
D Straight
Answers
Answer:a
Step-by-step explanation:
Find the difference in area between the large circle and the small circle. Click on the answer until the correct answer is showing.
Answers
A=4[tex]\pi -8[/tex]
that is your answer :-)
Answer:
[tex]A = 4\pi - 8[/tex]
Step-by-step explanation:
ody
What is the volume of the cylinder to the nearest whole number? a) 942 cm3 b) 3,534 cm3 c)471 cm3 d) 9,420 cm3
Answers
Answer:
V = 3534 cm^3
Step-by-step explanation:
The volume of a cylinder is given by
V = pi r^2 h
the radius is 7.5 and the height is 20
V = pi ( 7.5)^2 * 20
V =1125 pi cm^3
Using the pi button for pi
V =3534.291735 cm^3
Rounding to the nearest whole number
V = 3534 cm^3
Answer:
b)3534 cm^3
Step-by-step explanation:
To find the area of a cylinder, you first have to find the area of the base which is in the shape of a circle. The area of a circle is given by the equation πr^2. In this case r, the radius, is 7.5 cm. So plugging in 7.5 for r you get 7.5^2 × π. Plugging in 3.1415 for π you get ~176.671. Now all you do is multiply this by the height, 20cm, and get the answer of ~3534 cm^3.
Find the measure of angle b.
your answer_____
Answers
Answer:
b = 131
Step-by-step explanation:
The two angles form a straight line so they add to 180 degrees
b+49 = 180
Subtract 49 from each side
b = 180-49
b =131
Answer:
Angle b is 131°
Step-by-step explanation:
Angles on a straight line add up to 180°
To find b add 49 and b and equate it to 180°
That's
b + 49 = 180
b= 180 - 49
b = 131°
Hope this helps you
Can someone help me out with this please
Answers
Answer:
143.81
Step-by-step explanation:
Trapezoid Area
A = 2b/2 * h
A = 9 + 23/2 * 7
A = 32/2 * 7
A = 16 * 7
A = 112
Semi-circle Area
A = πr²/2
A = π4.5²/2
A = π20.25/2
A = 63.62/2
A = 38.81
Total Area
112 + 38.81
143.81
A SQUARE CARPET IS LAID IN ONE CORNER OF A RECTANGULAR ROOM, LEAVING STRIPS OF UNCOVERED FLOOR 2M WIDE ALONG ONE SIDE AND 1M ALONG OTHER . THE AREA OF THE ROOM IS 56m SQUARED .FIND THE DIMENSIONS OF THE CARPET
Answers
Answer:
Step-by-step explanation:
A square has equal sides. Let x represent the length of each side of the square carpet. The diagram representing the room and the carpet is shown in the attached photo. Therefore, the length of the room would be (x + 2)m while the width of the room would be (x + 1)m
Since the area of the room is 56m², it means that
(x + 2)(x + 1) = 56
x² + x + 2x + 2 = 56
x² + 3x + 2 - 56 = 0
x² + 3x - 54 = 0
x² + 9x - 6x - 54 = 0
x(x + 9) - 6(x + 9) = 0
x - 6 = 0 or x + 9 = 0
x = 6 or x = - 9
Since the dimension of the carpet cannot be negative, then x = 6
The dimension of the carpet is 6m × 6m
PLEASEEEEEE HELP MEEEEE 100 points!!!!!!!!!!!! The map shows the location of a mall, library, and school in a city: Coordinate grid shown from negative 12 to positive 12 on x axis at intervals of 2, and negative 12 to positive 12 on y axis at intervals of 2. A triangle is shown with vertices labeled Library, Mall, and School. Library is the ordered pair negative 10, 10 , Mall is the ordered pair 10, 10, and School is the ordered pair 10 and negative 11. Sarah traveled from the school to the mall and then from the mall to the library. Bret traveled from the school to the library. How many miles did Sarah and Bret travel altogether? (1 point) Select one: a. 12 miles b. 29 miles c. 41 miles d. 70 miles
Answers
Sarah: School to mall = 21
mall to library = 20
Total distance for Sarah = 20 + 21 = 41 miles.
Use the Pythagorean theorem to find the distance Bret traveled:
Distance = SQRT(21^2 + 20^2)
= sqrt(441 + 400)
= sqrt(841)
= 29 miles
Total distance = 41 + 29 = 70 miles
Answer is D. 70 miles.
Answer:
70
Step-by-step explanation:
i took the test
EXTRA POINTS The amount of people diagnosed is 3,131,953 and the amount of deaths is 132,056 what is the percentage of people who die from the disease?
Answers
2.3%
divide the infected number by the deaths. Then move the decimal points to the left.
Answer:
2.3%
divide the infected number by the deaths. Then move the decimal points to the left.
Step-by-step explanation:
6TH GRAD LE MATH PLEASE HELP ME
Answers
Answer: 6
Step-by-step explanation: I think..... this is a division sign, nvm... so 4 divided by 8/12 is basically 4 x 12/8 which is 48/8 simplified is/equals 6!
Answer:
6
Step-by-step explanation:
4 / 8/12 = 4 * 12/8 = 4 * 3/2 = 6
PLEASE HELP ASAP!:
Solve for a and b
6a-b=-5
4a-3b = -8
Answers
Answer:
a = -1/2
b = 2
Step-by-step explanation:
Step 1: Rewrite 1st equation
-b = -5 - 6a
b = 5 + 6a
Step 2: Substitution
4a - 3(5 + 6a) = -8
Step 3: Solve
4a - 15 - 18a = -8
-14a - 15 = -8
-14a = 7
a = -1/2
Step 4: Plug in a to find b
6(-1/2) - b = -5
-3 - b = -5
-b = -2
b = 2
here are the ingredients for making pineapple sorbet for 6 people. 800 g pineapple 4 egg whites 1/2 lemon 100 g caster sugar Dan makes pineapple sorbet. he uses 2 and a 1/2 lemons How many people does he make pineapple sorbet for?
Answers
Answer:
30 people
Step-by-step explanation:
If you start off with 1/2 a lemon for 6 people, then 1 whole would be 12. 1 1/2 would be 18 people and 2 whole lemons would be 24. so 2 1/2 is 30 people.
Please help with this 3a² = 27. Find a
Answers
Answer:
[tex]a = 3[/tex]
Step-by-step explanation:
[tex]3 {a}^{2} = 27 \\ \frac{3 {a}^{2} }{3} = \frac{27}{3} \\ {a}^{2} = 9 \\ a = \sqrt{9} \\ a = 3[/tex]
Answer: 9
Step-by-step explanation:
First divide both sides by 3
[tex]a^2=9[/tex]
Then root both sides([tex]\sqrt{a^2}=\sqrt{9}[/tex])
a = 9
Hope it helps <3
Edit: :o this is my 250th answer
Which of these sets of side lengths are pythagorean triples!
Answers
Hey there! :)
Answer:
Choices 1, 4 and 5.
Step-by-step explanation:
To solve, we can go through each answer choice and check if they are Pythagorean Triples using the Pythagorean Theorem:
1) 26² = 10² + 24²
676 = 100 + 576
676 = 676. This is correct.
2) 49² = 14² + 48²
2401 = 196 + 2304
2401 ≠ 2500. This is incorrect.
3)
16² = 12² + 9²
256 = 144 + 81²
256 ≠ 225. This is incorrect.
4)
41² = 40² + 9²
1681 = 1600 + 81
1681 = 1681. This is correct.
5)
25² = 15² + 20²
625 = 225 + 400
625 = 625. This is correct.
Therefore, choices 1, 4 and 5 are correct.
Answer:
A, D, and E.
Step-by-step explanation:
use
tiles
Algebra
factor for X^2-2x-3
Answers
Answer:
I hope it's correct
How many times would you have to cross the Golden Gate Bridge in order to travel the exact distance between Wolverhampton and Birmingham?
Answers
Answer:
Hey!
Well...the distance from Birmingham to Wolverhampton in about 12 miles (19km)
The length of the Golden Gate bridge is 2,737m OR 2.737 km...in miles it's 1.701
So you will have to cross it about 11.2 times (3 sf)
Step-by-step explanation:
19 km / 1.701km = 11.16990005878895
Rounded up to a degree of accuracy = 3 sf
= 11.2
Hope this helps!
Helppp!!!! please!!!
Answers
Answer:
A) 52.5 inches²
Step-by-step explanation:
The equation for the area of a trapezoid is a=1/2h(b1+b2). This basically means that you take the height of the trapezoid, multiply it by the top base plus the bottom base and divide that by 2. When you do this, you take 8.5 plus 6.5, which equals 15, and multiply that by 7 to get 105. After you get this, you divide it by 2 to get 52.5 inches².
En una fábrica de pinturas cuentan con un tanque de pintura blanca y otro de pintura azul. El litro de pintura blanca cuesta 4 dólares y el litro de pintura azul, 7 dólares. Si se quiere mezclar ambas pinturas para llenar un tanque de 500 litros de capacidad y además se requiere que la mezcla no cueste más de 6 dólares ni menos de 5 dólares el litro. ¿Cuál de las siguientes inecuaciones te ayuda a calcular cuántos litros de pintura blanca, como máximo, debe tener la mezcla? ("x" representa la cantidad de litros de pintura blanca) 1. (4x + 7x)/500 > 5 2. 4x + 7(500 - x) > 5 3. 4x + 7x > 500 4. [4x + 7(500 - x)]/500 > 5
Answers
Answer:
Las inecuaciones que pueden ayudar a calcular cuantos litros de pintura blanca se pueden tener como son [tex]\frac{4\cdot x + 7\cdot y}{500}> 5\,\frac{USD}{L}[/tex] y [tex]\frac{4\cdot x + 7\cdot y}{500}< 6\,\frac{USD}{L}[/tex].
Step-by-step explanation:
Esta situación puede ser descrita mediante una ecuación y una inecuación simultánea. La ecuación es de la capacidad del tanque, mientras que la inecuación es del coste unitario de la mezcla. Sean [tex]x[/tex] y [tex]y[/tex] las capacidades empleadas de pintura blanca y pintura azul en litros, entonces:
Capacidad del tanque (en litros)
[tex]x + y = 500\,L[/tex]
Coste unitario de la mezcla (en dólares por litro)
[tex]5\,\frac{USD}{L} < \frac{4\cdot x + 7\cdot y}{500} < 6\,\frac{USD}{L}[/tex]
Es decir:
[tex]\frac{4\cdot x + 7\cdot y}{500}> 5\,\frac{USD}{L}[/tex] y [tex]\frac{4\cdot x + 7\cdot y}{500}< 6\,\frac{USD}{L}[/tex]
Las inecuaciones que pueden ayudar a calcular cuantos litros de pintura blanca se pueden tener como son [tex]\frac{4\cdot x + 7\cdot y}{500}> 5\,\frac{USD}{L}[/tex] y [tex]\frac{4\cdot x + 7\cdot y}{500}< 6\,\frac{USD}{L}[/tex].
use the bionomial theorem to write the binomial expansion
[tex]( \frac{1}{2}x + 3y) ^{4} [/tex]
Answers
Answer:
[tex]$\left(\frac{x}{2} + 3 y\right)^{4}=\frac{x^{4}}{16} + \frac{3}{2} x^{3} y + \frac{27}{2} x^{2} y^{2} + 54 x y^{3} + 81 y^{4}$[/tex]
Step-by-step explanation:
[tex]$\left(\frac{1}{2}x+3y \right)^4=\left(\frac{x}{2}+3y \right)^4\\$[/tex]
Binomial Expansion Formula:
[tex]$(a+b)^n=\sum_{k=0}^n \binom{n}{k} a^{n-k} b^k$[/tex], also [tex]$\binom{n}{k}=\frac{n!}{(n-k)!k!}$[/tex]
We have to solve [tex]$\left(\frac{x}{2} + 3 y\right)^{4}=\sum_{k=0}^{4} \binom{4}{k} \left(3 y\right)^{4-k} \left(\frac{x}{2}\right)^k$[/tex]
Now we should calculate for [tex]k=0, k=1, k=2, k=3 \text{ and } k =4;[/tex]
First, for [tex]k=0[/tex]
[tex]$\binom{4}{0} \left(3 y\right)^{4-0} \left(\frac{x}{2}\right)^{0}=\frac{4!}{(4-0)! 0!}\left(3 y\right)^{4} \left(\frac{x}{2}\right)^{0}=\frac{4!}{4!}(81y^4)\cdot 1 =81 y^{4}$[/tex]
It is the same procedure for the other:
For [tex]k=1[/tex]
[tex]$\binom{4}{1} \left(3 y\right)^{4-1} \left(\frac{x}{2}\right)^{1}=54 x y^{3}$[/tex]
For [tex]k=2[/tex]
[tex]$\binom{4}{2} \left(3 y\right)^{4-2} \left(\frac{x}{2}\right)^{2}=\frac{27}{2} x^{2} y^{2}$[/tex]
For [tex]k=3[/tex]
[tex]$\binom{4}{3} \left(3 y\right)^{4-3} \left(\frac{x}{2}\right)^{3}=\frac{3}{2} x^{3} y$[/tex]
For [tex]k=4[/tex]
[tex]$\binom{4}{4} \left(3 y\right)^{4-4} \left(\frac{x}{2}\right)^{4}=\frac{x^{4}}{16}$[/tex]
You can perform the calculations, I will not type everything.
The answer is the sum of elements calculated.
Just organizing:
[tex]$\left(\frac{x}{2} + 3 y\right)^{4}=\frac{x^{4}}{16} + \frac{3}{2} x^{3} y + \frac{27}{2} x^{2} y^{2} + 54 x y^{3} + 81 y^{4}$[/tex]
Answer: [tex]\bold{\dfrac{1}{16}x^4 + \dfrac{3}{2}x^3y + \dfrac{27}{2}x^2y^2 +54xy^3+81y^4}[/tex]
Step-by-step explanation:
Binomial Tree
n=0 1
n=1 1 1
n=2 1 2 1
n=3 1 3 3 1
n=4 1 4 6 4 1
Using the Binomial Theorem
[tex]\bigg(\dfrac{1}{2}x+3y\bigg)^4\\\\\\=1\bigg(\dfrac{1}{2}x\bigg)^4(3y)^0\quad \rightarrow \quad \dfrac{1}{16}x^4\\\\+4\bigg(\dfrac{1}{2}x\bigg)^3(3y)^1\quad \rightarrow \quad \dfrac{3}{2}x^3y\\\\+6\bigg(\dfrac{1}{2}x\bigg)^2(3y)^2\quad \rightarrow \quad \dfrac{27}{2}x^2y^2\\\\+4\bigg(\dfrac{1}{2}x\bigg)^1(3y)^3\quad \rightarrow \quad 54xy^3\\\\+1\bigg(\dfrac{1}{2}x\bigg)^0(3y)^4\quad \rightarrow \quad 81y^4[/tex]
______________________
[tex]= \dfrac{1}{16}x^4 + \dfrac{3}{2}x^3y + \dfrac{27}{2}x^2y^2 +54xy^3+81y^4[/tex]