# Thread: Polar form of complex numbers

1. ## Polar form of complex numbers

hi guys , i really need help in these two questions :
Express each of the following in the form
a + ib and also in the polar form r\µ,
where the angle is the principal value.
(a)
(-sqrt(3)- i )^7

(b)(1 + i)^3(sqrt(3) + i)^3

2. Start by using this, where $z = x+yi$ then $(x+yi)^n = z^n = r^n(\cos n\theta +i\sin n\theta)$

given $r = \sqrt{x^2+y^2}$ and $\theta = \tan^{-1}\frac{y}{x}$

3. ## hi

i did start with that but im notgetting the right answer i dont know what im doing wrong

4. Originally Posted by sandy
i did start with that but im notgetting the right answer i dont know what im doing wrong
We won't know either unless you show all your working.

5. Originally Posted by mr fantastic
We won't know either unless you show all your working.
Hi ,i tried to do it is this correct?

(-sqrt(3) -i)^7=2^7(cos 7pi/6 + i sin 7pi/6)

and how do we do it with the form a+ib

6. Originally Posted by sandy

and how do we do it with the form a+ib
That's exactly what we are doing. you have $-\sqrt{3}-i$ giving $a = -\sqrt{3}$ and $b = -1$

Originally Posted by sandy
Hi ,i tried to do it is this correct?

(-sqrt(3) -i)^7=2^7(cos 7pi/6 + i sin 7pi/6)
$-\pi < \theta < \pi$

7. Hi ,i tried to do it is this correct?

(-sqrt(3) -i)^7=2^7(cos 7pi/6 + i sin 7pi/6)

and how do we do it with the form a+ib

Originally Posted by pickslides
Start by using this, where $z = x+yi$ then $(x+yi)^n = z^n = r^n(\cos n\theta +i\sin n\theta)$

given $r = \sqrt{x^2+y^2}$ and $\theta = \tan^{-1}\frac{y}{x}$

8. So do we expand a+ib to the power of 7 by to the power of 2 multiplied by 3 and then multiply that by a+ib again?

Originally Posted by pickslides
That's exactly what we are doing. you have $-\sqrt{3}-i$ giving $a = -\sqrt{3}$ and $b = -1$

$-\pi < \theta < \pi$

9. Originally Posted by sandy
So do we expand a+ib to the power of 7 by to the power of 2 multiplied by 3 and then multiply that by a+ib again?
i got it !!
z^7= 64*sqrt(3) - i64