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Use De Moivre' s theorem toexpress (2 + 2i)5 in polar form. (ii) find all cube roots of (V3 + i) in polar form_Pllease write neatly...

Question

Use De Moivre' s theorem toexpress (2 + 2i)5 in polar form. (ii) find all cube roots of (V3 + i) in polar form_Pllease write neatly

Use De Moivre' s theorem to express (2 + 2i)5 in polar form. (ii) find all cube roots of (V3 + i) in polar form_ Pllease write neatly



Answers

Use De Moivre s theorem to evaluate each. Leave answers in polar form. $$(1+i \sqrt{3})^{3}$$

In this problem were given under three. Plus Ayotte are to the whole power eight. Now we are asked to use the mobilized Khurram, evaluate this complex expression. Now we know that if that is equals, toe art e to the power Takita and it's a natural number. Then Zet toe the power and is equals tow R to the power and e to the power iota and Rita Now here we are given under three plus iota. Now we get our is equals toe under rule one whole square plus under three full square which gives us our is equals toe to now for theater is equals to engine in verse one over under three we get theater is equals toe by over six. Therefore, under three plus iota is equals to to eat to the power 5/6 iota Therefore, under three plus iota, the whole power eight is equals to to the power eight e to the power 8 5/6 iota is equals toe 2 56 Ito the power for over 35 iota So that's a solution in the polar form

In this problem were given under rude to e to the power 10 degrees iota. So the whole power six. Now we asked to use the more waste forum to evaluate this complex expression. Now we know that if Zen is equals toe are into the power iota Tita and any is the natural number. Then we can write this as their to the power end equals two r to the power n E to the power iota and theater. Now, in this case, under two e to the power 10 degrees iota, the whole power six is equals to under two to the power of six. He to the power 6 10 degrees iota. We get eight e to the power 60 degrees iota, but that's a solution in the polar form.

So first question you want to use the mushroom toe, figure out the answer. Andi, In order to do that, you need toe work. Out. What? Cars or Ennis on what features. So first question you and I used to Mars Freedom toe work out. Um, root free over two plus half. I water power six So and you can work out from the power straight away, which is six. So any course six are? You can walk out from the square, root off a squared plus B squared. So in this case is square root off. Excuse me. Um, fruit free, maybe two squared. Plus uh huh. Squared mhm on this equals to one on her feet are you must notice that the complex equation off the question is in the second quarter. So it is 180 minus invest time off half or if I route free of it too, Miss equals toe 150 degrees. So you're taking all this? If you plug this into equation, what you get is, uh, once in the past six. Want to buy buy because six times 150 plus I sine six times 150. Andi, If you happen into a calculator, what you get is negative one. That's the final answer.

So for this question, you want to use the master room, which is given in the top left here. Um, So what this question requires you to do is tow work, help r n and also feet up for this equation here, A negative route to over two plus 2/2. I always the power five. Yeah. So to work up, and it's quite easy. It's just what the complex equation is. Raise the power to so and equals five work. Our remember is squaring off a squared plus b squared. So square it right in this case s squaring off. Negative, too. Two squared. Plus, yeah, two by two squared on the Sequels to one on the feta. You must notice that the complex equation is in the second quarter, so ISS 180 minus in Reston Off your way. So in this case, it is the screw up to two. I had my screwed up to have to on this because 255 degrees. If you plug that into the equation off the pastoral where you get is one to the power of five. What about that? Because five times on 35 plus I sine five times 135 if you carry into the calculator. Um, what you get is route to every two minus root, too. To I just find one, sir.


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