Answer: Option A) [tex]729c^6 + 1,458c^5d^2 + 1,215c^4d^4 + 540c^3d^6 + 135c^2d^8 + 18cd^{10} + d^{12}[/tex] is the correct expansion.
Explanation:
on applying binomial theorem, [tex](a+b)^n=\sum_{r=0}^{n} \frac{n!}{r!(n-r)!} a^{n-r} b^r[/tex]
Here a=3c, [tex]b=d^2[/tex] and n=6,
Thus, [tex](3c+d^2)^6=\sum_{r=0}^{6} \frac{6!}{r!(6-r)!} (3c)^{n-r} (d^2)^r[/tex]
⇒ [tex](3c+d^2)^6= \frac{6!}{(6-0)!0!} (3c)^{6-0}.(d^2)^0+\frac{6!}{(6-1)!1!} (3c)^{6-1}.(d^2)^1+\frac{6!}{(6-2)!2!} (3c)^{6-2}.(d^2)^2+\frac{6!}{(6-3)!3!} (3c)^{6-3}.(d^2)^3+\frac{6!}{(6-4)!4!} (3c)^{6-4}.(d^2)^4+\frac{6!}{(6-5)!5!} (3c)^{6-5}.(d^2)^5+\frac{6!}{(6-6)!6!} (3c)^{6-6}.(d^2)^6[/tex]
⇒[tex](3c+d^2)^6= \frac{6!}{(6-)!0!} (3c)^6.d^0+\frac{6!}{(5)!1!} (3c)^5.d^2+\frac{6!}{(4)!2!} (3c)^4.d^4+\frac{6!}{(6-3)!3!} (3c)^3.d^6+\frac{6!}{(2)!4!} (3c)^2.d^8+\frac{6!}{(1)!5!} (3c).d^{10}+\frac{6!}{(0)!6!} (3c)^0.d^{12}[/tex]
⇒[tex](3c+d^2)^6=(3c)^6.d^0+\frac{720}{120} (3c)^5.d^2+\frac{720}{48} (3c)^4.d^4+\frac{720}{36} (3c)^3.d^6+\frac{720}{48} (3c)^2.d^8+\frac{720}{120} (3c).d^{10}+.d^{12}[/tex]
⇒[tex](3c+d^2)^6=729c^6 + 1,458c^5d^2 + 1,215c^4d^4 + 540c^3d^6 + 135c^2d^8 + 18cd^{10} + d^{12}[/tex]
Answer:
Option (a) is correct.
[tex](3c+d^2)^6=729c^6+1458c^5d^2+1215c^4d^4+540c^3d^6+135c^2d^8+18cd^{10}+d^{12}[/tex]
Step-by-step explanation:
Given : [tex]\left(3c+d^2\right)^6[/tex]
We have to expand the given expression and choose the correct from the given options.
Consider the given expression [tex]\left(3c+d^2\right)^6[/tex]
Using binomial theorem ,
[tex]\left(a+b\right)^n=\sum _{i=0}^n\binom{n}{i}a^{\left(n-i\right)}b^i[/tex]
We have [tex]a=3c,\:\:b=d^2[/tex]
[tex]=\sum _{i=0}^6\binom{6}{i}\left(3c\right)^{\left(6-i\right)}\left(d^2\right)^i[/tex]
also, [tex]\binom{n}{i}=\frac{n!}{i!\left(n-i\right)!}[/tex]
for i = 0 , we have,
[tex]\frac{6!}{0!\left(6-0\right)!}\left(3c\right)^6d^2^0=729c^6[/tex]
for i = 1 , we have,
[tex]\frac{6!}{1!\left(6-1\right)!}\left(3c\right)^5d^2^1=1458c^5d^2[/tex]
for i = 2 , we have,
[tex]\frac{6!}{2!\left(6-2\right)!}\left(3c\right)^4d^2^2=1215c^4d^4[/tex]
for i = 3 , we have,
[tex]\frac{6!}{3!\left(6-3\right)!}\left(3c\right)^3d^2^3=540c^3d^6[/tex]
for i = 4 , we have,
[tex]\frac{6!}{4!\left(6-4\right)!}\left(3c\right)^2d^2^4=135c^2d^8[/tex]
for i = 5 , we have,
[tex]\frac{6!}{4!\left(6-4\right)!}\left(3c\right)^2d^2^4=18cd^{10}[/tex]
for i = 6 , we have,
[tex]\frac{6!}{6!\left(6-6\right)!}\left(3c\right)^0d^2^6=d^{12}[/tex]
Thus, adding all term together, we have,
[tex](3c+d^2)^6=729c^6+1458c^5d^2+1215c^4d^4+540c^3d^6+135c^2d^8+18cd^{10}+d^{12}[/tex]
Thus, Option (a) is correct.
A rectangle has a width of 7.9 ft and a perimeter of 27.2 ft.
Enter the length of the rectangle, in decimal form, in the box.
I just had the same question and the answer was 5.7. Hope this helps somebody
Explain why you can use subtraction to solve a division problem.
there are 5 sodas, 3 grape sodas, 7 root beers, and 8 lemon lime sodas in a cooler. What is the probability of choosing a grape soda? Give your answer as a ration in its lowest terms.
Which completes the syllogism? If a polygon is a square, then it has four congruent sides. If a polygon has four congruent sides, then it is a rhombus. Therefore, …
A. if a polygon is a square, then it is a rhombus.
B.if a polygon is a rhombus, then it is a square.
C.if a polygon is a rhombus, then it is a parallelogram.
D.if a polygon has four congruent sides, then it is a square.
Answer:
Step-by-step explanation:
Jeremiah has a batting average of 0.312 this baseball season. Express his average as a fraction in lowest terms.
Am I able to evaluate sin(cos^1(7/11)) without a calculator? If yes, how? ...?
Yes, you can evaluate sin(cos^-1(7/11)) without a calculator. The result can be simplified to √(72/121).
Explanation:Yes, you can evaluate sin(cos-1(7/11)) without a calculator. Let's break it down step by step:
cos-1(7/11) represents the inverse cosine of 7/11. Inverse cosine is the angle whose cosine is equal to the given value. So, cos-1(7/11) is the angle whose cosine is 7/11.To evaluate sin(cos-1(7/11)), we need to find the sine of the angle we found in the previous step.Since we know that sin2(θ) + cos2(θ) = 1, we can use this relationship to find sin(θ) when we know cos(θ).Let's assume the angle we found in step 1 is θ. We know that cos(θ) = 7/11, so we can substitute this value into the equation sin2(θ) + (7/11)2 = 1 and solve for sin(θ).By solving the equation, we find that sin(θ) = √(1 - (7/11)2).Finally, we can simplify the expression sin(θ) = √(1 - (49/121)) or sin(θ) = √(72/121).Therefore, sin(cos-1(7/11)) can be evaluated as √(72/121).
How many 20 cm by 30 cm rectangular tiles would you need to tile the outdoor are shown below?
I calculated that the area of the shape is 10.95m squared, but the answer in the book (183 tiles) is different then the answer I got.
Where did I go wrong?
The issue was inconsistent units - converting everything to cm squared, the appropriate number of 20 by 30 cm tiles to cover an area of 10.95m squared is 183.
Explanation:The problem here involves area measurement and conversions. The area of the outdoor space is given in square meters (m²), while the area of the tiles is given in square centimeters (cm²). Therefore, the first step here is to convert all measurements to the same unit.
Since 1 meter is equal to 100 centimeters, we can convert the area of the outdoor space to square centimeters by squaring 100 (which equals to 10,000) and then multiplying it by the area in square meters. That gives us 109,500 cm².
Next, to find the area of a single tile we multiply its length by its width, which is 20 cm by 30 cm, giving us 600 cm². The number of tiles needed will be the total area divided by the area of one tile. So, 109,500 cm² divided by 600 cm² equals to about 182.5.
Since we cannot have a half tile, we round up that number to 183 tiles. Therefore, the correct answer is 183 tiles.
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How many pounds does 64 ounces weigh?
Answer:
4 lbs
Step-by-step explanation:
There are 16 ounces in a pound, so divide 64 by 16 to get the number of pounds. 64 ÷ 16 = 4 lbs
When a number is divided by 5, the result is 50 more than if the number had been divided by 6. what is the number?
what is this equation in slope intercept form? 4y+12x=0
The equation 4y + 12x = 0 in slope-intercept form is y = -3x. The slope (m) is -3, and since there is no constant term, the y-intercept (b) is 0.
Explanation:The equation provided, 4y + 12x = 0, can be converted into slope-intercept form, which is represented by y = mx + b. The slope, m, indicates the steepness of the line, while the y-intercept, b, denotes the y-value when x equals zero or where the line crosses the y-axis. Let's rearrange the equation step-by-step:
First, isolate the y variable by subtracting 12x from both sides: 4y = -12x.Next, divide each side by 4 to solve for y: y = -12x / 4.This simplifies to: y = -3x.Here, the slope (m) is -3, indicating a decrease in y by 3 units for every increase of 1 unit in x. The equation does not have a b term, thereby indicating that the y-intercept (b) is 0. Hence, the slope-intercept form of the equation is y = -3x + 0, which can be simplified to y = -3x.
WriteTwo Equivalent Ratios For The Given Ratio 9/10
...?
What is the resulting ordered pair if the value of the independent variable is x=-3?f(x) = –2x-3
Claire can run 100 meters in 17 seconds. what is her speed in meters per second?
1.) 7, 9, 11, 13, . . . is it 2.) 2,1, 1 , 1
2 4
A.) arithmetic A.) arithmetic
B.) geometric B.) geometric
C.) both C.) both
D.) neither D.) neither
Which of the following is a counterexample to that statement “All prime numbers are odd”
a:The statement has no counterexamples
b:What you are trying to prove
c:The theorems you will use
d: The definitions of important vocabulary
...?
Fill in the blank: A concave polygon is ________ that _______ ...?
What is the formula that relates circumference and radius?
A. C = 2r
B. C = 2/r
C. C = 2D
D. C + 2 = r
Answer:
The formula that relates circumference and radius is [tex]C=2\pi r[/tex].
Step-by-step explanation:
The circumference of a circle is calculated by the formula
[tex]C=2\pi r[/tex]
Where,
C is circumference of the circle.
r is radius of the circle.
π is 22/7 or 3.14.
In the given options π is not missing. So, all the given options are incorrect.
Therefore the formula that relates circumference and radius is [tex]C=2\pi r[/tex].
Which of the following algebraic equations could represent the sentence, "The product of a number and five is 11"?
a,x+ 5 = 11
b,x- 5 = 11
c,= 11
d,5x= 11
Marlene rides her bike at a rate of 16 miles per hour. The time in hours that she rides is represented by the variable t, and the distance she rides is represented by the variable d. Which statements are true of the scenario? Check all that apply.
The independent variable, the input, is the variable d, representing distance.
The distance traveled depends on the amount of time Marlene rides her bike.
The initial value of the scenario is 16 miles per hour
.
The equation t = d + 16 represents the scenario
.
The function f(t) = 16t represents the scenario.
there can be more then one answer
Answer:
B and E are the answers
Step-by-step explanation:
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If f(x)=2x^2-x-6 and g(x)=x^2-4, find f(x)/g(x)
2x+3/x-2
2x-3/x+2
2x+3/x+2
2x-3/x-2
To find f(x)/g(x), we factor both f(x) and g(x), and then simplify by canceling common terms. The simplified result is 2x + 3 / x + 2.
To find the quotient of two functions, f(x) and g(x), we divide the first function by the second. In this case, we have f(x) = 2x² - x - 6 and g(x) = x² - 4. The process involves the following steps:
Write the division of the two functions: f(x)/g(x).Simplify the expressions, if possible, by factoring and reducing them.We start by factoring both f(x) and g(x):
f(x) = (2x + 3)(x - 2)
g(x) = (x + 2)(x - 2)
Now we divide f(x) by g(x):
f(x)/g(x) = ((2x + 3)(x - 2)) / ((x + 2)(x - 2))
Notice that the term (x - 2) is common in both the numerator and the denominator, so we can reduce the expression by canceling out the common term, resulting in:
f(x)/g(x) = (2x + 3) / (x + 2)
The correct result is the second option: 2x + 3 / x + 2.
PLEASE ANYONE HELP MEEE!!!!
An equilateral triangle has an apothem of 5 cm. Find the perimeter of the triangle to the nearest centimeter.
A.
26 cm
B.
52 cm
C.
75 cm
D.
100 cm"
How many atoms of oxygen are in one formula unit of Ca(No3)2?
Determine the coordinates of the vertices of the triangle to compute the area of the triangle using the distance formula (round to the nearest integer).
FIRST GRAPH
A) 20 units^2
B) 30 units^2
C) 40 units^2
D) 50 units^2
Explain why the graph shown may be misleading.
As they are lined out, the amounts must correlate horizontally, so the compacts should be longer than the rest, the mid-size behind it, and the trucks half-way between the compacts and the mid-size.
What is graph?It is defined as the data representation in visual way, and it makes the appearance of the data simpler.
As we can see in the graph:
Total of 40 compact cars, 4 of 10 each.
Total of 30 mid-size, 3 of 10 each.
And Total of 20 trucks,
However, they all appear to have the same start and end points in the graph, with the mid-sizes being longer than the compacts and the trucks being considerably larger than the others above it, virtually implying that the amounts for all are identical for that period.
Thus, as they are lined out, the amounts must correlate horizontally, so the compacts should be longer than the rest, the mid-size behind it, and the trucks half-way between the compacts and the mid-size.
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To get to work, matt walks 0.75 miles from his house to the bus stop and rides the bus 3.8 miles to his office. if he walks at a pace of 3.6 miles per hour and the bus drives at an average speed of 15 miles per hour, how long is with his commute?
what is the correct expansion of the binomial (x+y)^5
Answer: The correct expansion is,
[tex]x^5+5x^4y+10x^3y^2+10x^2y^3+5xy^4+y^5[/tex]
Step-by-step explanation:
Since, by the binomial expansion,
[tex](p+q)^n=\sum_{r=0}^{n} ^nC_r (p)^{n-r}(q)^r[/tex]
Where,
[tex]^nC_r=\frac{n!}{r!(n-r)!}[/tex]
Here, p = x and q = y and n = 5,
By substituting values,
[tex](x+y)^5=\sum_{r=0}^{5} ^5C_r (x)^{n-r}(y)^r[/tex]
[tex] =^5C_0(x)^{5-0}(y)^0+^5C_1 (x)^{5-1}(y)^1+^5C_2 (x)^{5-2}(y)^2+^5C_3 (x)^{5-3}(y)^3+^5C_4 (x)^{5-4}(y)^4+^5C_5(x)^{5-5}(y)^{5}[/tex]
[tex]=1(x)^5(y)^0+\frac{5!}{4!(5-4)!}x^4y^1+\frac{5!}{3!(5-3)!}x^3y^2+\frac{5!}{2!(5-2)!}x^2y^3+\frac{5!}{1!(5-1)!}xy^4+\frac{5!}{5!(5-5)!}x^0y^5[/tex]
[tex]=x^5+\frac{5!}{4!1!}x^4y^1+\frac{5!}{3!2!}x^3y^2+\frac{5!}{2!3!}x^2y^3+\frac{5!}{1!4!}x^1y^4+\frac{5!}{5!0!}x^0y^5[/tex]
[tex]=x^5+\frac{5\times 4!}{4!}x^4y^1+\frac{5\times 4\times 3!}{3!2!}x^3y^2+\frac{5\times 4\times 3!}{2!3!}x^2y^3+\frac{5\times 4!}{4!}x^1y^4+\frac{5!}{5!}y^5[/tex]
[tex]=x^5+5x^4y+10x^3y^2+10x^2y^3+5xy^4+y^5[/tex]
Which is the required expansion.
convert .98 to percent ...?
A rectangle has an area of 3 square feet and a width of 1/2 foot. What is the length of the rectangle ?
Find the measure of
PLEASE HELP
Which is a true statement about any two congruent chords in a circle?
A. They form an angle.
B. They are perpendicular.
C. They are parallel.
D. They are equidistant from the center of the circle.