0.1 Radical concepts -- properties of radicals

What is $\sqrt{x^2+9}$ ? Many students will answer quickly that the answer is $(x+3)$ and have a very difficult time believing this answer is wrong. But it is wrong.

$\sqrt{x^2}$ is $x^{*}$

Why not? Remember that $\sqrt{x^2+9}$ is asking a question: “what squared gives the answer $x^2+9$ ?” So $(x+3)$ is not an answer, because ${(x+3)}^2=x^2+\text{6x+9}$ , not $x^2+9$ .

As an example, suppose $x=4$ . So $\sqrt{x^2+9}=\sqrt{4^2+9}=\sqrt{25}=5$ . But $(x+3)=7$ .

$\sqrt{x^2+9}$ cannot, in fact, be simplified at all. It is a perfectly valid function, but cannot be rewritten in a simpler form.

How about $\sqrt{9x^2}$ ? By analogy to the previous discussion, you might expect that this cannot be simplified either. But in fact, it can be simplified:

$\sqrt{9x^2}=3x$

Why? Again, $\sqrt{9x^2}$ is asking “what squared gives the answer $9x^2$ ?” The answer is $3x$ because ${\left(3x\right)}^2=9x^2$ .

Similarly, $\sqrt{\frac{9}{x^2}}=\frac{3}{x}$ , because ${\left(\frac{3}{x}\right)}^2=\frac{9}{x^2}$ .