Question

Write the expression in a + bi form

Answer 1

(√2(cos3π/4 + isin3π/4))^4

Answer 2

Apply DeMoivre's Theorem.

Answer 3

so first im guessing we do this:
(^4√2(cos3π/4 + isin3π/4)) ?

Answer 4

\[\Large [r(\cos(\theta) + isin(\theta)]^n = r^n * [\cos(n*\theta) + isin(n*\theta)]\]

Answer 5

.. so would it be (√2)^4 this instead?

Answer 6

correct.

Answer 7

cos(theta) and sin(theta) becomes cos(n*theta) and sin(n*theta)

Answer 8

= (4(cos4 * 3π/4 * isin4 * 3π/4))

Answer 9

Yes.

Answer 10

Okay so now what do i do?

Answer 11

Simplify.
= 4{ cos(3pi) + isin(3pi) }

Answer 12

From the angle 3pi, take away as many 2pi's as you can because 2pi radians is one full circle and you end up in the same place.

Answer 13

sooo
= 4{ cos(pi) + isin(pi) }

Answer 14

correct. Substitute the values for cos(pi) and sin(pi).

Answer 15

= 4{ -1 + 0 }

Answer 16

yes. simplify.

Answer 17

-4?

Answer 18

yes!!!

Answer 19

thats not an answer tho.. :/
these are
4i
?4i
4
?4

Answer 20

idk what ?4 means??

Answer 21

Looks like the negative sign is showing up as a "?" in the answer choices.

Answer 22

I think the answer choices are probably: 4i, -4i, 4 and -4.

Answer 23

Must be yeah ill have to talk to my teacher but you really are a big help! :) i think i understand it now

Answer 24

You are welcome. Just try refreshing the browser page and see if the "?" goes away.

Answer 25

OKay i will :) just one quick question if you dont mind in this question
Which of the following is a complex Nth root of...
would the answer look like this
^n√(cos + isin)

Answer 26

with theta of course but yeah..

Answer 27

Taking the nth root is same as having the exponent (^1/n)

Answer 28

Im confused

Answer 29

It may be clearer if you type the whole question with the choices.

Answer 30

Which of the following is a complex fifth root of -3 -3√3i?

Answer 31

6(cos24 + isin24)
6(cos-24 + isin-24)
5^√6(cos24 + isin24)
5^√6(cos-24 + isin-24)

Answer 32

First put -3 -3√3i in the form r(cos(theta) + isin(theta))
Find the magnitude of -3 -3√3i and factor it out.
Magnitude of a + ib = sqrt(a^2 + b^2)

Answer 33

idk how to square this -3√3?

Answer 34

(pq)^2 = p^2 * q^2

Answer 35

still not sure what you mean..? so (-3)^2 * (3)^2?

Answer 36

(-3√3)^2 = (-3)^2 * (√3)^2 = ?

Answer 37

Oh so 9 * 3 = 27

Answer 38

yes.

Answer 39

so r = 27

Answer 40

Magnitude of a + ib = sqrt(a^2 + b^2)
don't forget "a". Don't forget to take square root.

Answer 41

r = 6 hahah forgot to do the rest :P

Answer 42

Now what?

Answer 43

Yes. factor 6 out of -3 -3√3i

Answer 44

what do you mean?

Answer 45

We want to put -3 -3√3i in the form r(cos(theta) + isin(theta))
We found r = 6. So factor 6 out of -3 -3√3i

Answer 46

you mean use actan b/a + pi?

Answer 47

arctan*

Answer 48

-3 -3√3i = 6( -1/2 - i√3/2 )

Answer 49

6( -1/2 - i√3/2 ) = r{ cos(theta) + isin(theta) }
r = 6; cos(theta) = -1/2 and sin(theta) = -√3/2. What is theta?
Both cosine and sine are negative and so theta must be in the third quadrant.

Answer 50

240!

Answer 51

Correct. +240 degrees. We will soon be applying DeMoivre's Theorem when we have to take the 5th root which is same as raising the complex number to the exponent (1/5). So each angle will get multiplied by 1/5.
1/5 * 240 = 48 degrees and none of our answer choices has 48 degrees.

So instead of +240 degrees we should choose ...?

Answer 52

-60?

Answer 53

Turning 240 degrees anti-clockwise is same as turning ? clockwise.

Answer 54

Why do we multiply by 1/5 wouldnt it be ^5?

Answer 55

-120 actually

Answer 56

square root is same as raising to the power 1/2
cube root is same as raising to the power 1/3
Fifth root is same as raising to the power 1/5

Answer 57

Yes, theta = -120 degrees.

Answer 58

Okay so -120 * 1/5 is -24!

Answer 59

yes.
Fifth root of -3 -3√3i = ( -3 -3√3i )^1/5 =
{ 6( cos(-120) + isin(-120) ) } ^1/5 = 6^1/5 * { cos(-24) + isin(-24) }

Answer 60

so how do you do the 6^1/5

Answer 61

You can't simplify it any further and you leave it as 5th root of 6:
\[\Large \sqrt[5]{6}[\cos(-24) + isin(-24)]\]

Answer 62

Awesome thank you so much! :DDDDD

Answer 63

You are welcome.