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I am currently self-studying Intermediate Algebra by Miller, O'Neil, and Hyde. The textbook covers negative radicals ($-\sqrt{x}$), but it is not clear to me whether the negative radical notation denotes the negative root or the radical multiplied by a factor of $-1$. Here's an example from the textbook: $$\begin{aligned} \text{simplify the square root}-\sqrt144\\&=-1\cdot\sqrt144\\&=-1\cdot12\\&=-12 \end{aligned}$$ Given that $\sqrt{144}$ is equal to $12$ and $-12$, is $-1\cdot-12=12$ also a valid simplification of $-\sqrt{144}$? Or does $-\sqrt{144}$ refer exclusively to the negative square root of $144$?

williams-ji
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    No, $\sqrt{144}$ is not equal to $12$ and to $-12$. It is only equal to $12$. – Another User Nov 07 '24 at 18:09
  • The symbol $\sqrt{x}$ is intended to mean the principal square root, meaning only the plus. If, however, you introduce a square root into a problem, that's where you consider plus/minus. – Sean Roberson Nov 07 '24 at 18:11
  • The square root symbol always refers to non-negative values because we are answering the question: what side length is required to construct a square of the given area? We allow $0$ but that is degenerate. There is no such thing as a square with zero area. You are confusing $\sqrt{144}$ with the solutions to the equation $x^2=144$, which are $x=\pm\sqrt{144}=\pm 12$. – John Douma Nov 07 '24 at 19:19
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    Also, the negative square root of a number and negative one times the square root of a number are the same thing. – John Douma Nov 07 '24 at 19:20

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