Let α by a cycle of length 3 and β a cycle of length 9. What is the order of αβ. Is the permutation αβ
even or odd?
b) Show that for every positive integer n we have
(αβ)^n = α^n*β^n

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,

Proof:

Expanding from left, we have

Since  α and β commutes, we rearrange  to get:

We simplify on the right to get:

hope this !

step-by-step explanation:

d = distance between moon and sun, which in this problem is the hypotenuse of a right triangle.  

the distance y is the side opposite to angle x in the right triangle.   use the definition of the sine function from basic trigonometry.

sin(x) = y/d ⇒

d = y/sin(x)

answer: b.48 for

step-by-step explanation: