why is the value of $(-1)^\frac{2}{3}$ not 1?

Question

$(-1)^2$ equals 1

also $(-1)^\frac{1}{3}$ equals -1

any way you arrange it, it should be one.

how ever, typing this into Wolfram or google,gives me an irrational complex number.

I would love some explaining on this.

thank you.

Answer 1

Wolfram Alpha interprets $x^{1/3}$ as the principal complex cubic root of $x$, so $x^{2/3}$ is also a complex number. Despite of this, WA computes also the three complex roots (some rows below).

If we restrict ourselves to real numbers, an expression with a rational exponent as $a^{n/m}$ can be defined as $\left(\sqrt[m]{a}\right)^n$, where $\sqrt[m]{a}$ means the principal $m$-th root of $a$, then we have \begin{align} (-1)^{2/3}&=\left[(-1)^{1/3}\right]^2\\ &=(\sqrt[3]{-1})^2\\ &=(-1)^2\\ &=1 \end{align}

When we work with complex numbers $z^{1/n}$ is multivalued, which means that there are many complex numbers $w_k$ such that $w_k^n=z$.

Answer 2

$(-1)^{\frac{1}{3}}\neq 1$. $(-1)^{\frac{1}{3}}$ are the cube roots of unity . Now I guess you can see why there is such an answer that WA gives

Answer 3

In general, when raising numbers to non-integer powers, there is not a unique answer in the complex numbers. This is a generalization of the fact that numbers have two different square roots. It turns out that when you raise a number $z$ to a rational power $m/n$, there are $n$ different complex numbers that can be considered "values" of $z^{m/n}$. In this case, $n=3$, so there are actually $3$ different values of $(-1)^{2/3}$. One of these values is indeed $1$, but the other two are complicated-looking complex numbers. In general, there is no way to say which value is the "right" or "best" one, so Wolfram Alpha uses a somewhat arbitrary rule to pick one (called the "principal value"), and it turns out that the value of $(-1)^{2/3}$ that it picks is one of the complex values, not $1$. However, if you scroll down on https://www.wolframalpha.com/input/?i=%28-1%29%5E%282%2F3%29, you will find that it does eventually list all three possible values (under the heading "All 3rd roots of $(-1)^2$). At the top of that page, you also can click "Use the real-valued root instead" to change the rule that Wolfram Alpha uses to choose the value, so that it will choose a real value (if one exists). Clicking on it shows that with this rule, Wolfram Alpha does say that $(-1)^{2/3}$ is $1$: https://www.wolframalpha.com/input/?i=%28-1%29%5E%282%2F3%29&a=%5E_Real.

As for why there are these multiple values, that is a longer story, but I am sure there are many good explanations available online (perhaps on this very site). To find such explanations, try searching for terms like "exponentiation of complex numbers".