To prove that every subgroup of $D_n$ of odd order is cyclic, we will use the following fact:
Fact: If $G$ is a group of odd order, then every subgroup of $G$ is also of odd order.
Proof of the fact: Let $H$ be a subgroup of $G$. By Lagrange's theorem, the order of $H$ divides the order of $G$. But the order of $G$ is odd, so the order of $H$ is odd as well. $\square$
Now, let $H$ be a subgroup of $D_n$ of odd order. We will show that $H$ is cyclic.
If $H$ is the trivial subgroup, then it is clearly cyclic. Otherwise, $H$ contains at least one non-identity element, say $x$. If $x$ is a reflection, then $x^2$ is the identity and $H$ contains the two elements $x$ and $x^2$, which contradicts the assumption that $H$ has odd order. Therefore, $x$ must be a rotation.
Let $k$ be the smallest positive integer such that $x^k$ is a reflection. Note that $k$ must divide $n$, since $x^n$ is the identity and $x^k$ is a reflection. We claim that $H$ is generated by $x^k$.
First, we show that every power of $x^k$ is in $H$. Let $m$ be an arbitrary integer. If $m$ is even, then $(x^k)^m$ is a rotation and is therefore in $H$. If $m$ is odd, then $(x^k)^m=x^{km}$ is a composition of a rotation and a reflection, and is therefore in $H$.
Next, we show that $x^k$ generates $H$. Let $y$ be an arbitrary element of $H$. If $y$ is a rotation, then $y=x^{km}$ for some integer $m$ (since $x^k$ is a rotation). If $y$ is a reflection, then $yx=x^{-1}y$ is a rotation, so $yx=x^{km}$ for some integer $m$ (since $x^k$ is the smallest power of $x$ that is a reflection). Therefore, $y=x^{-1}(x^{km})=(x^k)^{-1}(x^{km+1})$, which is a power of $x^k$.
Thus, we have shown that $H$ is generated by $x^k$, and since $x^k$ is a rotation, it is of infinite order. Therefore, $H$ is cyclic.
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In the diagram AB/BC = AD/DE
Substitute the known values into the proportion and solve for DE
Answer: 9
Step-by-step explanation:
[tex]\frac{2}[3}=\frac{6}{DE}\\\\\frac{3}{2}=\frac{DE}{6}\\\\DE=9[/tex]
Nora works at a retail store. She earns a commission of $5.50 on each item she sells. If she sells 17 items, how much does she earn from commissions?
The total earnings in commission is $93.50
How to determine the total commission?The given parameters are:
Commission = $5.50
Items = 17
The total commission is:
Total = $5.5 0 * 17
Evaluate
Total = $93.50
Hence, the total earnings in commission is $93.50
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Condense log 2 4 + log 2 5
Answer:
[tex]log_{2}[/tex] 20
Step-by-step explanation:
using the rule of logarithms
loga + logb = logab
then
[tex]log_{2}[/tex]4 + [tex]log_{2}[/tex]5
= [tex]log_{2}[/tex] (4 × 5)
= [tex]log_{2}[/tex]20
The length of a rectangle is 6 inches more than its width. Its area is 91 square inches. The length is .
Answer:
13 in
Step-by-step explanation:
let w be width then length is w + 6
the area (A) of a rectangle is calculated as
A = length × width , then
A = w(w + 6) = 91 , that is
w² + 6w = 91 ( subtract 91 from both sides )
w² + 6w - 91 = 0 ← in standard quadratic form
(w + 13)(w - 7) = 0 ← in factored form
equate each factor to zero and solve for w
w + 13 = 0 ⇒ w = - 13
w - 7 = 0 ⇒ w = 7
however, w > 0 then w = 7
and length = w + 6 = 7 + 6 = 13 in
Find the principal needed now to get the given amount; that is, find the present value.
To get $100 after 4 years at 10% compounded monthly.
The present value of $100 is ?
Simplify the expression below. Then classify the resulting expression
as a monomial, binomial, or trinomial.
3x²+6x+5-3x(2+x)
Which of the following represents the simplified expression and its
polynomial classification?
Answer:
Simplified = 5
Classification = Monomial
Step-by-step explanation:
PART I: Simplify the expressionGiven expression:
3x² + 6x + 5 - 3x (2 + x)
Expand parenthesis by distributive property:
= 3x² + 6x + 5 - 3x (2) - 3x (x)
= 3x² +6x + 5 - 6x - 3x²
Put like terms together:
= 3x² - 3x² + 6x - 6x + 5
= 0 + 0 + 5
= [tex]\boxed{5}[/tex]
PART II: Classify polynomialConcept:
Polynomial is classified by the number of terms a polynomial has.
Monomial: a polynomial with only one termBinomial: a polynomial with two terms...Classify the given expression:
Original = 3x² + 6x + 5 - 3x (2 + x)
Simplified = 5
5 is a constant and it has only one term
Therefore, it is a monomial.
Hope this helps!! :)
Please let me know if you have any questions
The answer is 5, which is a monomial.
Let's simplify using the distributive property.
3x² + 6x + 5 - 3x(2) - 3x(x)3x² + 6x + 5 - 6x - 3x²5If the resulting expression has only one term, it is classified as a monomial.
If there are fifty raffle tickets in a jar, and only two will be drawn out, and you own two of them, what chance do you have of owning both winning tickets?
Answer:
1/25
Step-by-step explanation:
Answer:
I think it may be 1/1225
Step-by-step explanation:
first chance is 2/50 and second chance is 1/49 as one card decrease at first. So multiplying it we get 1/1225.
During one month there were 7 days of precipitation. what if there had only been 3 days of precipitation that month? how would that change the measures of center? if necessary, round your answers to the nearest tenth.
The Mean will decrease while the Median and the Mode will remain with the same amount.
For Central Measures, we have Median, Mode and Mean. They try to describe the whole distribution by one value, in the center of it.
For this question, let's pick some examples from the Old Faithful, WY Station to the National Oceanic and Atmospheric Administration from 2015, January
1st Case Scenario:
For 7 days of precipitation, (Snow, ice pellets, etc. rain) in inches:
19 19 19 24 23 23 23 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
Mean = 5
Median=0 (the sum of the 15th+16th position over 2)
Mode=0 (Most common observations)
2nd Case Scenario
For 3 days of precipitation, (Snow, ice pellets, etc. rain) in inches:
0 0 0 24 0 0 23 0 19 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
Mean = 2.2
Median =0
Mode=0
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BE is two units longer than a EDE is five units longer than a E and CE is 12 units longer than AE what is BD
Length of BD is 11 units.
The intersecting chords theorem or just the chord theorem is a statement in elementary geometry that describes a relation of the four line segments created by two intersecting chords within a circle. It states that the products of the lengths of the line segments on each chord are equal.
Let length of AE be x
By intersecting chords theorem,
AE(ED) = BC(DE)
x(x + 12) = (x +2)(x +5)
x² + 12x= x² + 7x + 10
5x = 10
x = 2
BD = BE + DE
= x + x +7
= 2x + 7
= 2(2) + 7
= 11 units
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Suppose matrix B is the inverse of matrix A. Use your knowledge of inverses and matrix multiplication to answer the following: AB
AB will be an Identity matrix.
What is an identity matrix?A square matrix with 1s on the main diagonal and 0s everywhere else is an identity matrix. The identity matrices 22 and 33, for example, are presented below. These are known as identity matrices because they produce the identity matrix when multiplied by a compatible matrix.If the answer to a matrix multiplication problem is an identity matrix, then each of the two matrices is an inverse matrix of the other. When the matrix is multiplied by the original matrix, the result is the identity matrix.As it is given in the description itself, if the answer to a matrix multiplication problem is an identity matrix, then each of the two matrices is an inverse matrix of the other.
Therefore, AB will be an Identity matrix.
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Answer:
AB= I
ABA=A
BBA=B
BBAA=I
Step-by-step explanation:
just took the test on edge. 2022
Find the surface area of the composite figure.
8 cm
6 cm
7 cm
8 cm
SA =
6 cm
[ ? ] cm²
9 cm
6 cm
If you'd like,
you can use a
calculator.
Enter
Answer:
The Answer Is 382...
Step-by-step explanation:
Find the value of x in the given figure
Step-by-step explanation:
3x° + 2x° = 180° ( supplementary angle )
5x° = 180°
x = 180° / 5
x = 36°
[tex]...[/tex]
work out the area of a rectangle with base, b = 20mm and perimeter, p = 50mm.
Answer:
100
Step-by-step explanation:
If the base is 20, then 2 of the sides must have a length of 20. This means that the TOTAL length of the remaining sides is 10. (20*2) + (10) = 50. If we divide that in half, we get 5. That is the side length of the remaining 2 sides. This means that 20 * 5 will give us the area (b*h).
Given,
base, b = 20mm &
perimeter, p = 50mm.
the area of the rectangle : (50 – 20 × 2) ÷ 2 = 5. 5 × 20 = 100 [tex]mm^{2}[/tex].
Perimeter and area are two important and basic mathematical subjects. They help quantify the physical space and also provide a more advanced foundation of mathematics in algebra, trigonometry, and calculus. Perimeter is a measure of the distance around a shape, and area indicates how much area the shape covers.
The perimeter of a 2D shape is the distance around the shape. You can imagine wrapping a string around a triangle. The length of this cord is around the triangle. Or, if you're traveling outside the park, take a route that goes around the edge of the park.
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Pls, can someone help me? (whoever helps me gets BRAINLEIST)
Answer:
25 people
Step-by-step explanation:
These are the pieces of information we know.
- 1/6 who came to the restaurant ordered fish tacos
- There were 30 customers in total
- We need to find the number of people who did not order fish tacos
1. The number of people who ordered fish tacos can be expressed as: 30 x 1/6, which is 5. We know that 5 people ordered fish tacos.
2. Since we want to know the number of people who didn't want fish tacos, we take 5 away from 30: 30 - 5, which is 25.
25 people did not order fish tacos on Tuesday.
Quick algebra 1 question for 50 points!
Only answer if you know the answer, quick shout-out to Yeony2202, tysm for the help!
The number of songs that Deante' would have downloaded during month 9 is 58 songs.
What is a regression line?A regression line is a statistical line that best describes the behavior of a data set and such, it is a line that best fits a set of data.
By using Microsoft excel to plot the data on a scatter plot, we can logically deduce that line of best fit for the songs downloaded is given by:
y = 2.03x + 39.7
Thus, the number of songs that Deante' would have downloaded during month 9 is:
y = 2.03(9) + 39.7
y = 18.27 + 39.7
y = 57.97 ≈ 58 songs.
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Answer:
A) ŷ = 2.03X + 39.73
B) 58
Step-by-step explanation:
A line of best fit is a line drawn on a scatter plot that shows the trend and the best estimation of the relationship between x and y.
After graphing it, it's easy to find the point at which x and why intersect along the line where x is nine.
Using the equation, just replace x with nine and solve.
Change y = − 3x² + 36x - 73 to standard form, y = a (x - h)² + k.
The standard form of the given equation is: y = -3(x-6)²+35.
In mathematics, the most typical way to represent a specific element is called standard form.
A standard form is a way to express a particular mathematical idea, such as an equation, number, or expression, in prose that adheres to a set of criteria.
Given equation is: y = − 3x² + 36x - 73
In order to convert to the standard form,
y+73 = −3x² +36x
Subtract 108 from both the sides,
y+73-108 = −3x² +36x-108
y-35 = -3(x² -12x+36)
y-35 = -3(x-6)²
y = -3(x-6)²+35
This is the required standard form, corresponding to the given equation.
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the lengths of the sides of a triangle are 3, 3, 3√2. can the triangle be a right triangle? yes no
Answer:
no hdjsjfjabjsjfbsbajdisjsb
The triangle can be a right triangle
How to determine the triangle type?The side lengths are given as:
3, 3 and 3√2
If the triangle is a right triangle, then the following must be true
a^2 = b^2 + c^2
Where a is the longest side.
So, we have:
[tex](3\sqrt 2)^2 = 3^2 + 3^2[/tex]
Evaluate the expressions
18 = 18
Both sides of the equation are equal
Hence, the triangle can be a right triangle
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Which of the following describes the graph of {x | 1 < x < 2}? open circles on 1 and 2, arrows pointing out no solution open circles on 1 and 2 open circles on 1 and 2, line segment betweenWhich of the following describes the graph of {x | 1 < x < 2}? open circles on 1 and 2, arrows pointing out no solution open circles on 1 and 2 open circles on 1 and 2, line segment between
The open circles on 1 and 2 describe the graph of {x | 1 < x < 2}.
According to the statement
we have to describe the graph of {x | 1 < x < 2}.
In this type of graph there will be open circle because there is no equal sign in the statement.
there will be a line segment in between...because basically, u would put an open circle on 1, and open circle on 2 and shading in between.
so, in this statement we have given a some statements which are represent the given graph but only one. and from all of these
The condition having a open circles 1 and 2 represent the graphical representation than others because except this condition all are incapable to represent the graph.
So, The open circles on 1 and 2 describe the graph of {x | 1 < x < 2}.
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What is the length of AC?
Answer:
Step-by-step explanation:
you are right AC is 16 because A-C are the same length
Total area=
Help me please thanks so much
Answer:
1954
Step-by-step explanation:
11*7*50
Which ordered pair is generated from the equation shown below y = 3x - 1 OA) (6, 18) O B) (6, 17) OC) (4, 12) OD) (9,8)
Answer:
B) (6, 17)
Step-by-step explanation:
y = 3x - 1
To find a solution to the equation
Let x = 6
y = 3*6 -1
y = 18-1
y = 17
One possible solution is ( 6,17)
This is choice B
make x the subject. help please
Answer:
[tex]\sf x = \dfrac{n - 2mp}{2m}[/tex]
Step-by-step explanation:
[tex]\sf 3m = m + \dfrac{n}{p + x}\\[/tex]
[tex]\sf 3m - m = \dfrac{n}{p +x}[/tex]
[tex]\sf 2m = \dfrac{n}{p +x}[/tex]
Now cross multiply,
2m*(p +x) = n
To open the parenthesis, multiply 2m by p and x.
2mp + 2mx = n
Isolate the term with variable 'x'
2mx = n - 2mp
Now, isolate 'x', by dividing both sides by 2m
[tex]\sf x = \dfrac{n-2mp}{2m}[/tex]
Answer:
x = [tex]\frac{N-2MP}{2M}[/tex]
Step-by-step explanation:
3M = M + [tex]\frac{N}{P+x}[/tex] ( subtract M from both sides )
2M = [tex]\frac{N}{P+x}[/tex] ( multiply both sides by P + x to clear the fraction )
2M(P + x) = N ← distribute parenthesis on left side
2MP + 2Mx = N ( subtract 2MP from both sides )
2Mx = N - 2MP ( divide both sides by 2M )
x = [tex]\frac{N-2MP}{2M}[/tex]
Let z = 3(cos(15°) + i sin(15°)) and w = 5(cos(90°) + i sin(90°)). What best describes the geometric construction of the quotient z/w on the complex plane?
z is scaled by a factor of One-fifth and rotated 90 degrees clockwise.
z is scaled by a factor of 1 and rotated 90 degrees clockwise.
z is scaled by a factor of 1 and rotated 90 degrees counterclockwise.
z is scaled by a factor of One-fifth and rotated 90 degrees counterclockwise.
The best description of the geometric construction of the quotient z/w on the complex plane is; Option A; z is scaled by a factor of One-fifth and rotated 90 degrees clockwise
How to find Complex Trigonometric numbers?
We are given;
z = 3(cos(15°) + i sin(15°))
w = 5(cos(90°) + i sin(90°))
Now, if we want to find the quotient z/w, it is clear that in geometric construction, the procedure will be to scale z by a factor of 1/5 and thereafter we will rotate by 90° clockwise.
Thus, option A is the correct answer.
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If she was vaping every 15 minutes, assuming she was awake from 7 am- 11 pm (16 hours) how many times in a day would she vape?
Answer:
64 times
Step-by-step explanation:
16 hours to minutes: 16*60 = 960 minutes
vaping every 15 minutes: 960/15 = 64
Please help me im stuckkm
Answer:
Below in bold.
Step-by-step explanation:
(a) Using the Pythagoras theorem:
30^2 = 21^2 + p^2
p^2 = 30^2 - 21^2 = 459
p = √459
= 21.4 cm to nearest tenth.
(b) cos P = 21/30 = 0.70
m < P = 45.57 degrees to nearest hundredth.
Answer:
a) p = 21.4 cm (nearest tenth)
B) P = 45.6° (nearest tenth)
Step-by-step explanation:
As triangle PQR is a right triangle, we can use Pythagoras Theorem to find the length of p.
Pythagoras Theorem
[tex]a^2+b^2=c^2[/tex]
where:
a and b are the legs of the right trianglec is the hypotenuse (longest side) of the right triangleFrom inspection of the triangle:
a = PQ = 21b = QR = pc = PR = 30Substitute the given values into the formula and solve for p:
[tex]\implies 21^2+p^2=30^2[/tex]
[tex]\implies 441+p^2=900[/tex]
[tex]\implies 441+p^2-441=900-441[/tex]
[tex]\implies p^2=459[/tex]
[tex]\implies \sqrt{p^2}=\sqrt{459}[/tex]
[tex]\implies p=\pm 3\sqrt{51}[/tex]
As p is the length, p can be positive only.
Therefore, p = 21.4 cm (nearest tenth)
To find the angle P, use the cosine trigonometric ratio.
Cosine trigonometric ratio
[tex]\sf \cos(\theta)=\dfrac{A}{H}[/tex]
where:
[tex]\theta[/tex] is the angleA is the side adjacent the angleH is the hypotenuse (the side opposite the right angle)From inspection of the triangle:
[tex]\theta[/tex] = PA = PQ = 21H = PR = 30Substitute the given values into the formula and solve for P:
[tex]\implies \sf \cos P=\dfrac{21}{30}[/tex]
[tex]\implies \sf P=\cos^{-1}\left(\dfrac{21}{30}\right)[/tex]
[tex]\implies \sf P=45.572996^{\circ}[/tex]
Therefore, the measure of angle P is 45.6° (nearest tenth).
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The function a(b) relates the area of a trapezoid with a given height of 12 and
one base length of 9 with the length of its other base.
It takes as input the other base value, and returns as output the area of the
trapezoid.
2.b19
a(b)-12..
Which equation below represents the inverse function b(a), which takes the
trapezoid's area as input and returns as output the length of the other base?
A. b(a)-2-9
B. b(a)-+9
OC. b(a)- +6
OD. b(a)-9-6
The inverse function of [tex]A(b) = 12 \cdot \frac{b + 9}2[/tex] is b(a) = a/6 - 9
How to determine the inverse function?The function is given as:
[tex]A(b) = 12 \cdot \frac{b + 9}2[/tex]
Divide 12 by 2
[tex]A(b) = 6 \cdot (b + 9)[/tex]
Rewrite the function as
[tex]a = 6 \cdot (b + 9)[/tex]
Divide both sides by 6
a/6 = b + 9
Subtract 9 from both sides
b = a/6 - 9
Express as a function
b(a) = a/6 - 9
Hence, the inverse function of [tex]A(b) = 12 \cdot \frac{b + 9}2[/tex] is b(a) = a/6 - 9
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ANSWER BOTH QUESTIONS PLEASE REALLY IMPORTANT AND URGENT! WILL GIVE BRAINLIEST! THANK YOU!!!
Answer:
10. Area: ((6+4) * 8)/2 = 40 ft^2
Perimeter: 6 + 4 + 8 + 12.8 = 30.8 ft
8. 22 + 22 + 22 + 38 + 27.2 = 131.2 ft
(-1,1) (0,-1) (5,-11) (10,-21) what is the domain of the set of ordered pairs above?
Answer:
domain =(-1,0,5,10)to find domain just take the first values of each ordered pairs.
range=(1,-1,-11,-21)to find range take the second values of the ordered pairs.
Which equation is the inverse of 5y+4= (x+3)²+?
0 y = x ² +
O
5
Oy=3±15x+
7
O y=-3±5x
11
10
O-5y-4--(x+3)²-1
72
4
Answer:eq
Step-by-step explanation:
Which equation describes how the parent function, y = x cubed, is vertically stretched by a factor of 4?
The equation that describes how the parent function, y = x³, is vertically stretched by a factor of 4 is y = 4x³.
We can find how the equation is vertically stretched below:When an equation is vertically stretched, it means that the parent equation has been multiplied by a number that is more than one.
This means that the equation is multiplied by some factor.
It is given that we should describe how the parent function is vertically stretched by a factor of 4.
The parent function is given as y = x³.
Since the equation is said to be vertically stretched by a factor of 4, we must multiply the parent equation by 4 to stretch it.
Therefore, we have found the equation that describes how the parent function, y = x³, is vertically stretched by a factor of 4 to be y = 4x³.
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Answer: C
Step-by-step explanation:
Just C