Calculating Powers of Complex Numbers: Is My Answer Correct?

[Trail_Builder]

not sure if this is in the right section as i havn't got to calculus yet lol. anyways, was wondering whether you could check i have calculated this right as I'm learning complex numbers at the moment and have no way of checking my working. thnx

Homework Statement

Calculate $$(\surd5 - 2i)^{11}$$

Homework Equations

The Attempt at a Solution

Find polar form of $$(\surd5 - 2i)$$

$$r = complex modulus$$

$$r = \sqrt{5 + 4}$$
$$r = 3$$

$$Arg(z) = sin^{-1}(\frac{2}{3})$$

sub all that into $$r(cosArg(z) + i*sinArg(z))$$

but before actual calculate take the power of 11 into consideration so...

$$(\surd5 - 2i)^{11} = r^{11}(11*cosArg(z) + 11i*sinArg(z))$$

I then calculate all that as

$$177147(8.19..+7.33..i)$$

$$1452413.3..+1299078i$$



hope that's right :D

thnx

 

[neutrino]

The argument of a complex number (x+iy) = ?

 

[Trail_Builder]

ermmm, did i just work out the angle of the triangle instead of working out the angle counterclockwise from the x-axis? (on the argand diagram i think its called/complex plane)

think i might of

would the argument be instead $$360 - sin^{-1}(\frac{2}{3})$$?

 

[HallsofIvy]

Why not just sin^-1(-2/3)? (Admittedly, that is the same as 360- sin^-1(2/3).)