Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Evaluate $(-1)^{\frac{1}{3}}$.

I've tried to answer it by letting it be $x$ so that $x^3+1=0$. But by this way, I'll get $3$ roots, how do I get the actual answer of $(-1)^{\frac{1}{3}}$??

share|improve this question
18  
Is there a reason your accept rate is 0%? –  dtldarek Mar 17 '12 at 11:30
2  
What is your definition of $(\cdot)^{\frac{1}{3}}$? –  dtldarek Mar 17 '12 at 11:32
    
This is a related question: math.stackexchange.com/questions/109484/… –  xD13G0x Mar 17 '12 at 15:16

4 Answers 4

up vote 4 down vote accepted

It depends whether you are trying to solve it as an equation over the reals ($\mathbb{R}$) or over the complex plane ($\mathbb{C}$).
The polynomial $x^3+1$ factors as $x^3+1=(x+1)(x^2-x+1)$. So in the first case, you have exactly one solution: $x=-1$, since the second polynomial has no real roots. If you're looking for roots over $\mathbb{C}$, then you'll have three roots, since the $\sqrt[3]t$ is not a function over $\mathbb{C}$, hence all the 3 roots have equal rights to be called "the root".

share|improve this answer
    
It is also ture for 1^(1/3), I think. –  eccstartup Mar 17 '12 at 13:28
    
@eccstartup: Sure, it's true for any $z\in\mathbb{C}$ –  Dennis Gulko Mar 17 '12 at 13:44

Have you tried looking for factors of $(x+y)^3$ ??

$(x+y)^3 = (x+y)(x^2-xy+y^2)$ therefore $(x+1)^3 = (x+1)(x^2-x+1) \Rightarrow x=-1$ is one of the root, and apply methods you learnt to find roots of quadratic equation to find other roots.

share|improve this answer

They are all equally considered to be values of $\sqrt[3]{-1}$. There is no unique cube root, just as there is no unique square root.

share|improve this answer

Just put it like complex numbers:

We know that $z=\sqrt[k]{m_\theta}$, so

$z=\sqrt[3]{-1}$

$-1=1_{\pi}$

$\alpha_n=\dfrac{\theta+k\pi}{n}$

$\alpha_0=60$

$\alpha_1=180$

$\alpha_2=300$

So the answers are:

$z_1=1_{\pi/3}$

$z_2=1_{\pi}$

$z_3=1_{5\pi/3}$

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.