Why is a 4th root always positive? [duplicate]

$\begingroup$

I am now teaching mathematics and in the math book for the students, it says that a 4th root is always positive. I have never thought about this rule before but actually, say for example:

$\sqrt[4]{16} = 2 $ because $ 2^2=4$.

But also

$(-2)^4 = 16$

But the rule for even number roots are clear. It says that even number roots are always positive, without giving any explanation on why. Could someone here answer the reason for this rule?

$\endgroup$ 6

2 Answers

$\begingroup$

As others said in the comments, it is from the definition that $\sqrt{{}\cdot{}}$ must assign one value of $\sqrt{x}$ for each $x$. Think of the function $x = y^2$, if you were to graph it would look like a sideways parabola, however, in practice this function actually takes the value of $y = \sqrt{x}$ where $y \geq 0$.

$\endgroup$ $\begingroup$

$$x^4=16\implies 16e^{2ik\pi} \implies x_k=2 e^{(2ik\pi)/4}, k=0,1,2,3$$So all the roots are $2, 2i, -2, -2i$, only the first one is real and positive.

$\endgroup$