Dividing a whole number by another fraction that includes a Root?

Question

The question is simplifying $$\frac{9}{\frac{9\sqrt{97}}{97}}$$ The program has told me the answer found is $\sqrt{97}$, but I cannot figure out how this answer is found. I also do not have a calculator that shows roots in the solution, so I need to figure this out myself. I have searched all over the internet for video explanations on dividing a rational by a whole number and can't find anything. I am studying for a placement test after taking a gap year so I figure there's some cancelling I might of forgot, or a forgotten strategy here.

Answer 1

$$\frac{\not9}{\cfrac{\not9\sqrt{97}}{97}}=\frac{97}{\sqrt{97}}=\sqrt{97}.$$

Answer 2

$\color{green}{\text{This is why we multiply by the reciprocal when dividing fractions}}$ $$\frac{9}{\frac{9\sqrt{97}}{97}}=\frac{\frac{9}{1}}{\frac{9\sqrt{97}}{97}}=\frac{\frac{9}{1}}{\frac{9\sqrt{97}}{97}}\cdot\color{green}{\frac{\frac{97}{9\sqrt{97}}}{\frac{97}{9\sqrt{97}}}}=\frac{\frac{9\cdot97}{9\sqrt{97}}}{\color{green}{1}}=\frac{9\cdot97}{9\cdot\sqrt{97}}=\frac{\sqrt{97}^2}{\sqrt{97}}=\sqrt{97}$$

Answer 3

You could do it like this $$\frac 9{\left(\frac{9\sqrt{97}}{97}\right)}\cdot\frac {\sqrt {97}}{\sqrt {97}}=\frac {9\sqrt {97}}{\left(\frac {9\cdot 97}{97}\right)}=\frac {9\sqrt{97} }{9}=\sqrt {97}$$

Answer 4

Why is everyone here complicating this... Here is a straightforward answer: $$ \frac{9}{\frac{9\sqrt{97}}{97}} = 9 \div \frac{9\sqrt{97}}{97} = 9 \times \frac {97} {9\sqrt{97}} = \frac {97}{\sqrt {97}} = \frac {(\sqrt {97}) ^2 } {\sqrt {97}} = \sqrt {97} $$

No reciprocal multiplying or confusing shortcuts, straight to the point :3