Answer:
a) The order of αβ is 9 and αβ is an even permutation
b) See proof
Step-by-step explanation:
a) Given that: α is a cycle of length 3 and β a cycle of length 9.
We want to find the order of αβ .
The order of αβ is the LCM of the length of the disjoint cycles.
Therefore the order is the LCM of 3 and 9 which is 9.
Since α is a cycle of length 3 and β a cycle of length 9, and (3-1)+(9-1)=10, the permutation αβ is even.
b) We want to show that, for every positive integer n,
![(\alpha \cdot \beta)^n=\alpha^n \cdot \beta^n](https://tex.z-dn.net/?f=%28%5Calpha%20%5Ccdot%20%5Cbeta%29%5En%3D%5Calpha%5En%20%5Ccdot%20%5Cbeta%5En)
Proof:
Expanding from left, we have
![(\alpha \cdot \beta)^n=\alpha \cdot \beta \cdot \alpha \cdot \beta \cdot \alpha \cdot \beta ....\alpha \cdot \beta \:n-times](https://tex.z-dn.net/?f=%28%5Calpha%20%5Ccdot%20%5Cbeta%29%5En%3D%5Calpha%20%5Ccdot%20%5Cbeta%20%5Ccdot%20%5Calpha%20%5Ccdot%20%5Cbeta%20%5Ccdot%20%5Calpha%20%5Ccdot%20%5Cbeta%20....%5Calpha%20%5Ccdot%20%5Cbeta%20%5C%3An-times)
Since α and β commutes, we rearrange to get:
![(\alpha\cdot \beta)^n=\alpha\cdot \alpha \cdot \alpha...n-times \cdot \beta \cdot \beta \cdot \beta-----n\:times](https://tex.z-dn.net/?f=%28%5Calpha%5Ccdot%20%5Cbeta%29%5En%3D%5Calpha%5Ccdot%20%5Calpha%20%5Ccdot%20%5Calpha...n-times%20%5Ccdot%20%5Cbeta%20%5Ccdot%20%5Cbeta%20%5Ccdot%20%5Cbeta-----n%5C%3Atimes)
We simplify on the right to get:
![(\alpha \cdot \beta)^n=\alpha^n \cdot \beta^n](https://tex.z-dn.net/?f=%28%5Calpha%20%5Ccdot%20%5Cbeta%29%5En%3D%5Calpha%5En%20%5Ccdot%20%5Cbeta%5En)
![Q\cdot E\cdot D](https://tex.z-dn.net/?f=Q%5Ccdot%20E%5Ccdot%20D)
Answer:
6 cars per hour
Step-by-step explanation:
The rate can be expressed several ways. One of them is "cars per hour". To find that rate, divide cars by hours:
(24 cars)/(4 hours) = 6 cars/hour
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If 6 cars were washed in 60 minutes, another way to express the rate is ...
(60 minutes)/(6 cars) = 10 minutes/car
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When you ask "what is the rate?", you need to be specific as to the rate you're interested in. Different expressions of the rate are useful for different purposes.
Answer:
5 feet / 6 feet = 0.8333
Step-by-step explanation:
400,000 is the nearest ten thousand
Given that the ship was traveling for 4 mph faster heading west than south then let the speed of the ship heading south be x mph, the speed of the ship heading west will be (x+4) mph
the distance is given by:
distance=speed*time
distance heading south was:
x*4=4x miles
distance heading west was:
5(x+4)=5x+20
thus the total distance will be given by:
4x+(5x+20)=191
9x+20=191
9x=191-20
9x=171
x=171/9
x=19 mph
the speed the ship used to travel west was:
x+4
=19+4
=21 mph