Proving (-1 + i)7 = -8(1 + i) Using Polar Form: Complex Number Question

[Reshma]

This is a simple problem. Show that:
(-1 + i)⁷ = -8(1 + i)
where i = sqrt(-1)

I'm able to prove this result by expanding the bracket:
[(-1 + i)³]²(-1 + i)

But please help me prove this using the polar form.

 

[HallsofIvy]

This is a simple problem. Show that:
(-1 + i)⁷ = -8(1 + i)
where i = sqrt(-1)
I'm able to prove this result by expanding the bracket:
[(-1 + i)³]²(-1 + i)
But please help me prove this using the polar form.

Okay, PUT it in polar form! Polar form is $r (cos(\theta)+ isin(\theta))$ where r is the "magnitude" of the complex number (distance from 0) which is $\sqrt{(-1)^2+ 1^2}= \sqrt{2}$ for -1+ i and $8\sqrt{2}$ for -8(i+1). You can get $\theta$ by using $\theta= arctan(\frac{Im}{Re})$ but you should be able to see simply by plotting the points. -1+ i corresponds to (-1,1) in the plane so the angle is $\frac{\3pi}{4}$. -(1+i)= -1-i corresponds to (-1, -1) so the angle is $\frac{5\pi}{4}.$.
The seventh power of a complex number corresponds to taking the seventh power of r and multiplying $\theta$ by 7.

 

[Physics Monkey]

Can you write z = -1 + i in polar form? What is the magnitude?

I see Ivy has this handled.

 

[Reshma]

Thanks, HallsofIvy! That was easy!