How do you know that a positive algebraic radical refers to a nonnegative root?

Question

The online course I am taking says that the 4th root of an equation refers to the nonnegative root (see attached screenshot). But how can you know that it is not a negative root, I thought that that was always a possibility when there was a positive index of a radical? Is this just some aspect of mathematic notation that I am missing?

Answer 1

Recall that $\sqrt{x}$ only has positive values. You can graph this in desmos to see this. An easier way to discern why this is true is to think of the square root of -1. As in, 2 numbers that are the same that multiply into -1. There are no such numbers. $\sqrt[4]{7x+11}$ is a fourth root, but it can be written as $(\sqrt{7x + 11})^{\frac{1}{2}}$. It should now be easy to see that it is impossible to ever get this to be negative, hence it cannot equal -3.

Answer 2

Let $x$ be a positive real number. "$y$ is a fourth root of $x$" means that $y^4=x$. Each positive real number has two real fourth roots, one of which is positive, and the other is negative. The positive one is denoted by $\sqrt[4]{x}$, and the negative one is $-\sqrt[4]{x}$. So by definition, $\sqrt[4]{x}\ge0$ for all real $x$.