Even-even-odd rule with multiple terms in the radicand

Question

I found one definition of the even-even-odd rule by the peeps at planetmath.

They say that if

• a real variable to an even exponent is under a radical
• and the radical has an even index
• and, when the radical is eliminated, the resulting exponent on the variable is odd

then absolute value signs must be placed around the variable.

My question is how to apply the rule when the radicand and result after eliminating the radical have multiple terms.

Do you wrap the whole radicand with absolute value signs or how does that work?

Here's the problem that provoked this question:

Rationalize the expression:

$\sqrt{x^2 + x} - \sqrt{x^2 - x}$

Answer 1

you can multiply $$\sqrt{x^2+x}-\sqrt{x^2-x}$$ by $$\frac{\sqrt{x^2+x}+\sqrt{x^2+x}}{\sqrt{x^2+x}+\sqrt{x^2+x}}$$

Answer 2

The exponent in question is the exponent on the whole expression under the radical. The point is that in $\sqrt{a^2}$ we are guaranteed that $a^2 \ge 0$ so taking the square root makes sense. As $\sqrt {a^2}$ is defined to be the positive square root, we need an absolute value sign on the $a$. $\sqrt{a^2}=a$ is incorrect for $a \le 0$.

In your case, if we are just given $$\sqrt{(x^2+x)^2}-\sqrt{(x^2-x)^2}$$ we need to put absolute value signs around the expressions because $x^2+x, x^2-x$ might be less than zero, so we get $$|x^2+x|-|x^2-x|$$

However if we multiply $$(\sqrt{x^2+x}-\sqrt{x^2-x})(\sqrt{x^2+x}+\sqrt{x^2-x})=\sqrt{(x^2+x)^2}-\sqrt{(x^2-x)^2}$$ the original problem tells us that $x^2+x \ge 0, x^2-x \ge 0$ or the original expression does not make sense. We can use that to avoid the absolute value signs and get $$(x^2+x)-(x^2-x)=2x$$ Note that for $x=\frac 12$, $$\sqrt{(x^2+x)^2}-\sqrt{(x^2-x)^2}=\sqrt{(\frac 34)^2}-\sqrt{(\frac {-1}4)^2}=|\frac 34|-|\frac {-1}4|=\frac 12 \neq 2x$$ but here we had nothing to tell us $x^2-x \ge 0$