Question

Find the fifth roots of 243(cos 300° + i sin 300°).

Answer 1

Please help me!

Answer 2

@SithsAndGiggles Can you help me?

Answer 3

Do you mean
\[\sqrt[5]{243\left(\cos300^\circ+i\sin300^\circ\right)}~~?\]

Answer 4

Yes

Answer 5

Well, \(243=3^5\), and using Euler's formula, you know that \(\cos300^\circ+i\sin300^\circ=e^{300^\circ i}\).

So, the above number is equivalently
\[3\sqrt[5]{e^{300 i}}=3\left(e^{300 i}\right)^{1/5}=3e^{60 i}\]
Applying Euler's formula again, you have
\[3\left(\cos60^\circ+i\sin60^\circ\right)\]
Simplify that and you get your answer.

Answer 6

I don't think this is the approach that I'm supposed to be taking :/ ... For example, for this example problem that I found it says:
Find the fifth roots of 32(cos 280° + i sin 280°).

and the answers are:

2*[ cos(56) + isin(56) ]
2*[ cos(128) + isin(128) ]
2*[ cos(200) + isin(200) ]
2*[ cos(272) + isin(272) ]
2*[ cos(344) + isin(344) ]

Answer 7

Using the same method as for the last one:
\[\sqrt[5]{32\left(\cos280^\circ+i\sin280^\circ\right)}=2\left(e^{280i}\right)^{1/5}=2e^{56i}=2\left(\cos56^\circ+i\sin56^\circ\right)\]

Answer 8

And that list implies a multiple choice question, since none of the other options are equal to each other.

Answer 9

But it's not a multiple choice problem :/

Answer 10

@SithsAndGiggles

Answer 11

@Preetha

Answer 12

In that case, I'm clearly missing some information that only you have access to. I don't know how you may have been taught this material; I only did what I thought was right for this problem. Sorry I couldn't help.

Answer 13

It could also be the case that you didn't simplify your answer.

When you have \(3\left(\cos60^\circ+i\sin60^\circ\right)\), you can simplify the sine and cosine:
\[\cos60^\circ=\frac{1}{2}\\
\sin60^\circ=\frac{\sqrt3}{2}\]