Understanding Square Root rules to understand an equation

Question

So this is an equation from one of the solutions in my textbook that I am trying to understand as part of solving a cholesky-factorization problem:

$$\sqrt{18-(\frac{a}{\sqrt2})^2} = \sqrt{\frac{36-a^2}{\sqrt2}} $$

Which square root rule applies here? Feels like I am missing some basics...

Answer 1

That's because it isn't true. $$\sqrt{18-\left(\frac a{\sqrt2}\right)^2} = \sqrt{\frac{36-a^2}2}$$

Notice the $2$ in the right hand side as opposed to $\sqrt2$

Answer 2

$$\sqrt{18-\left(\frac{a}{\sqrt2}\right)^2} = \sqrt{18-\frac{a^2}{2}} = \sqrt{\frac{36-a^2}{2}};$$

as pointed out by @John Wayland Bales in a comment to the question, $\sqrt2$ on the right side of the equation in the question should be $2$.

Answer 3

It is $$18-\left(\frac{a}{\sqrt{2}}\right)^2=18-\frac{a^2}{2}=\frac{36-a^2}{2}$$