# Understanding Square Root rules to understand an equation [closed]

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...

## closed as off-topic by TheSimpliFire, John Omielan, Lord Shark the Unknown, Chris Godsil, SongMar 2 at 14:28

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• The equation is not correct. The $\sqrt{2}$ on the right should be a $2$. – John Wayland Bales Mar 1 at 20:01

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 $$\sqrt2$$

• Thank you, this drove me crazy! – Lucky Mar 1 at 20:18

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

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 $$2$$.

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

• I think you meant $36 \mathbf - a^2$ – J. W. Tanner Mar 1 at 20:13
• Yes I meant this, thank you for your hint! – Dr. Sonnhard Graubner Mar 1 at 20:15