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In Exercises 17 through 23, find the number of elements in theindicated cyclic group.22. The cyclic subgroup <p^10> of D24...

Question

In Exercises 17 through 23, find the number of elements in theindicated cyclic group.22. The cyclic subgroup <p^10> of D24

In Exercises 17 through 23, find the number of elements in the indicated cyclic group. 22. The cyclic subgroup <p^10> of D24



Answers

In Exercises $13-16,$ use the translation.
$$
(\mathrm{x}, \mathrm{y}) \rightarrow(\mathrm{x} \mp 8, \mathrm{y}+4)
$$
What is the preimage of $\mathrm{C}^{\prime}(-3,-10) ?$

Sorry about the video being sideways but hopefully you can see it if you can't let me know. So first we have to find the like terms so there's P. And P. Those are both multiplication. And then we have our regular integers. So we're going to want to get everything um on one side and everything on the other. So we're gonna want all of our variables on one side and all are integers on the other. So the first way we're going to start doing that is we're going to start moving things over. So we're gonna do the inverse operation here which is addition. So we're gonna add 23 when we do that minus 20 negative 23 plus 23 is equal to zero and then negative seven plus 23 is going to be and a subtraction problem. And it's going to equal out to a positive 16. So we're gonna put that 16 down here and then we're going to bring our six P. Down. And because this is positive the operation in between is going to be plus. So let's bring our six P. Down Equals right here. And then on the other side that's cancelled out. So we're going to bring down our 14 pete. So now we have to work on getting all of our variables to one side. So we're gonna take this six P. And move it on over. So six P. Or minus six P. That's gonna cancel each other out and everything we do one side we have to do to the other. So those cancel we're going to bring our 16 down to the bottom and there's nothing else over here so we can just write are equal sign and then 14 p minus six P. Is going to be an 18. So let's go ahead and divide out here. So we're going to the opposite of multiplication, which is division, divide that by eight, eight x 8, cancelling each other out. So we're left with P equals 16 divided by eight, which is to thank you so much.

Sorry about the video being sideways but hopefully you can see it if you can't let me know. So first we have to find the like terms so there's P. And P. Those are both multiplication. And then we have our regular integers. So we're going to want to get everything um on one side and everything on the other. So we're gonna want all of our variables on one side and all are integers on the other. So the first way we're going to start doing that is we're going to start moving things over. So we're gonna do the inverse operation here which is addition. So we're gonna add 23 when we do that minus 20 negative 23 plus 23 is equal to zero and then negative seven plus 23 is going to be and a subtraction problem. And it's going to equal out to a positive 16. So we're gonna put that 16 down here and then we're going to bring our six P. Down. And because this is positive the operation in between is going to be plus. So let's bring our six P. Down Equals right here. And then on the other side that's cancelled out. So we're going to bring down our 14 pete. So now we have to work on getting all of our variables to one side. So we're gonna take this six P. And move it on over. So six P. Or minus six P. That's gonna cancel each other out and everything we do one side we have to do to the other. So those cancel we're going to bring our 16 down to the bottom and there's nothing else over here so we can just write are equal sign and then 14 p minus six P. Is going to be an 18. So let's go ahead and divide out here. So we're going to the opposite of multiplication, which is division, divide that by eight, eight x 8, cancelling each other out. So we're left with P equals 16 divided by eight, which is to thank you so much.

In discussion. Were able to form a three digit number and all the digits must be order. So first of all we know that the all digits that are available are 135799 Okay. Now we have to form a three digit number or a third agent in teacher from this digits here. So it can be done by making arrangement of three digits. Okay, So we have to find their number of our enemies. Oh radiates out of this available five minutes. Okay. Out of this available five minutes. So to do this, we know the formula is The formula says that if you have to arrange are objects out of an object. So the formula is for that arrangements and B. And it's well, you can be given as N. Factorial divided well and minus our factory. Okay. So I have like this form layers. So we get here five P. Three means five factorial divided by five minus three factorial. Ok, now I further solve it. So I get here. Five factory divided by two. Factory. And it can be written as five and 24 in 23 That is It was too 60. Okay, so the answer is or the total numbers that can be formed are 60. Now, I'm going to explain you it by another method also. So I'm using method number two Matter number two says that I am constructing your three boxes. Okay, these are my three boxes. And if I am able to find the number of ways of selling all of this three, then I can get the required three digit interior. So this is my first books And this is my second books. And this is my third box. Okay, So first of all, we have to find for the first books and second box And third box the number of ways of killing Okay. Or options available. Also, I can say that. So you can see that if we are feeling the first works, then you have five different options available. The options are 135799 Okay. After filling this first box, you move towards your second box. Then you're first in one or number is used here. So for second books, the remaining options available are four and for the third box after killing. Second books also, you have only three options because you are not able to repeat the digits here. Okay. So finally now to construct the three digit number, I can say that I have to fill all these boxes here. And finally I got the three digit number that should be equals two. Given by multiplication of the available options. That is 54 and three. And I get here 60 so 60 is my final answer. Okay, thank you.

For this problem. We're looking at the pre image of C Prime, which is located the cord. It's negative three and negative 10. Now, in order to find that we're going to use the translation rule that we've been given now the translation rule, as I have in a coordinate C and I'm going to go ahead and take that coordinate. And it is moving eight units to the left and then going to go to the white coordinate and move it. Four units up. Well, we're working backwards. So what we Because we're looking for the pre image. So right here, That guy, the pre image. So what did it look like before I transformed him? Well, it's actually pretty easy to do. What you're gonna do is go ahead and take your rule and said it equal to the X component of C prime. So X minus eight is equal to negative. Three. Go ahead and saw for X, and you get X is equal to five and saw for why Brenda's same steps. So why plus four is equal to negative. 10 at four to both sides. Sorry. Subtract four from both sides and you get lies equals negative. 14. So your see, the original coordinates C was located at five. Negative 14. That is the pre image of C Prime.


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