Complex Numbers Obtained Two Different Ways

I was recently grading my students' tests for a college algebra course I teach. Students were to solve $9x^2+1=0$. So the correct approach would be to say

$$9x^2+1=0 \implies 9x^2=-1 \implies x^2 = - \frac{1}{9}$$

$$\implies x = \pm \frac{\sqrt{-1}}{\sqrt{9}} \implies x= \pm \frac{1}{3}i$$

But my student approached it differently. They said

$$9x^2+1=0 \implies 9x^2=-1 \implies x^2 = - \frac{1}{9}$$

$$\implies x = \pm \frac{\sqrt{1}}{\sqrt{-9}} \implies x= \pm \frac{1}{3i}$$

So, what's going on here? I can't see a mistake with my student's line of reasoning, since the negative sign can be placed arbitrarily in a rational expression. But clearly in general $a+bi \neq \frac{1}{a+bi}$. Is there something I'm missing here?

Your student is not wrong.

$\displaystyle \frac 1i = \frac {1\cdot i} {i\cdot i} = \frac i{-1} = - i$.

$\displaystyle\pm \frac 1{3i} = \mp \frac 13 i$. So it is equivalent to your answer.

Now that having been said, it is proper to give every complex number in the canonical form $a+bi$, so leaving $i$ in the denominator might be worth docking a half point or so. But it's not mathematically incorrect.

Observe that $1/i=-i$. So the second expression equals $-\pm \frac1{3 i}=\mp \frac13 i$