How Do You Calculate the Fourth Root of i Cubed?

[Theofilius]

[SOLVED] calculating complex number

Homework Statement

Calculate $${i}^{\frac{3}{4}}$$

Homework Equations

The Attempt at a Solution

I tried with

$$i=cos\frac{pi}{2}+isin\frac{\pi}{2}$$

$$i^3=cos\frac{3pi}{2}+isin\frac{3\pi}{2}$$

$$i^3=-i$$

$$\sqrt[4]{i^3}=\sqrt[4]{-i}$$

I don't know where I should go out of here. Please help!

 

[Dick]

Why don't you just go all the way at once? i=e^(i*pi/2). So i^(-3/4)=(e^(i*pi/2))^(-3/4). Now use laws of exponents. Notice this answer isn't unique. i is also equal to e^(i*(pi/2+2pi)=e^(i*5pi/2). This is the same sort of nonuniqueness you get with square roots.

 

[Theofilius]

Thanks. I solve it.