Does a square root come out plus/minus even if there is a negative sign outside?

Question

For example: $-\sqrt{100x^{20}y^{10}}$.

Would that give $\pm10x^{10}y^5$ or just $-10x^{10}y^5$?

Answer 1

The symbol $\sqrt{x}$ refers to the principal square root of $x$, which means it only refers to the positive square root. For example, $\sqrt{9}=3$, not $\pm3$.

In your example, we have $-\sqrt{100x^{20}y^{10}}$. The principal square root of $100x^{20}y^{10}$ is $|10x^{10}y^5|$ (which is $10x^{10}y^5$ if $y$ is positive (because a positive value of $y$ would make the entire expression positive) and $-10x^{10}y^5$ if $y$ is negative). Thus, $-\sqrt{100x^{20}y^{10}}=-|10x^{10}y^5|$

Answer 2

$$-\sqrt{100x^{20} \cdot y^{10}} = -\sqrt{(10 \cdot y^5 \cdot x^{10})^2} = -\operatorname{sgn}(10 \cdot y^5 \cdot x^{10}) \cdot (10 \cdot y^5 \cdot x^{10})$$