Question

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

Explanation:

Step1: Recall the cube - root definition

The expression $(-1)^{\frac{1}{3}}$ is the cube - root of $-1$. We need to find a number $x$ such that $x^3=-1$.
Let $x = a+bi$ (in the complex - number form). Then $(a + bi)^3=-1$. Expanding $(a + bi)^3$ using the formula $(a + bi)^3=a^3 + 3a^2(bi)+3a(bi)^2+(bi)^3=a^3 + 3a^2bi-3ab^2 - b^3i=(a^3 - 3ab^2)+(3a^2b - b^3)i$.
If we consider real - number solutions (since the most common case when dealing with simple roots), we know that $(-1)\times(-1)\times(-1)=-1$.

Step2: Determine the value

The cube - root of $-1$ is $-1$ because $(-1)^3=(-1)\times(-1)\times(-1)=-1$.

Answer:

$-1$