Solving for an exponent of a complex number?

[TheNascar92]

I have this problem:

x^4=4-4*(square root 3i)

I rewrote in trig form:

8cos300+isin300 then i solved by raising the 8^4 and multiplying the angle by 4...

The answer i got was 1499i. Could anyone verify my work for me? Is it correct to leave the imaginary "i" in the answer?

 

[tkhunny]

Have you considered checking the modulus?

\(\displaystyle \sqrt{4^{2} + (4\sqrt{3})^{2}} = 8\)

\(\displaystyle 8^{4} = 4096\)

How does this fit in with your 1499?

 

[DrMike]

I have this problem:

x^4=4-4*(square root 3i)

I rewrote in trig form:

8cos300+isin300

This would be correct, if you wrote

8(cos300[sup:7htxg5fi]o[/sup:7htxg5fi]+isin300[sup:7htxg5fi]o[/sup:7htxg5fi])

then i solved by raising the 8^4 and multiplying the angle by 4...

This would be correct if you were asked :

x = 4-4isqrt(3). Find x[sup:7htxg5fi]4[/sup:7htxg5fi].

But you were not. You were asked :

x[sup:7htxg5fi]4[/sup:7htxg5fi] = 4-4isqrt(3). Find x.

In other words,

Find (4-4isqrt(3))[sup:7htxg5fi]1/4[/sup:7htxg5fi].

Therefore, you need to divide the argument by 4, and take the fourth root of the modulus. Oh, and while you're doing that, remember that 300[sup:7htxg5fi]o[/sup:7htxg5fi] is not the only possible argument - you also need to consider 660[sup:7htxg5fi]o[/sup:7htxg5fi], 1020[sup:7htxg5fi]o[/sup:7htxg5fi], 1380[sup:7htxg5fi]o[/sup:7htxg5fi], .... and -60[sup:7htxg5fi]o[/sup:7htxg5fi], -420[sup:7htxg5fi]o[/sup:7htxg5fi], .... In fact, (300+360k)[sup:7htxg5fi]o[/sup:7htxg5fi] for all integers k.

Is it correct to leave the imaginary "i" in the answer?

Yes, and in general, necessary.