Simple Algebra Problem

Hello, I am very weak in Mathematics, so I would appreciate any help on this equation I've been thinking about lately:
Equation: 1/4 - 3/16 = x.
(1/4)^2 - 3/16 = x.
1/16 - 3/16 = x.
*x = -2/16*?
OR,
1/4 - 3/16 = x.
4/16 - 3/16 = x.
*x = 1/16*?
What I did was try to establish a common denominator (turning 4 into 16), so in the first "method", I square rooted the whole thing to get a denominator of 16, because 4^2 = 16. However, 1 has a square root of 1, so theoretically, is it just wrong to square root any fraction with 1 as a numerator? Because then, you'd have fractions like 1/2 = 1/4 = 1/8 = 1/16, which doesn't sound very mathematically correct.
I understand the second part where x = 1/16 is likely the correct answer, but I can't figure out where the first part went wrong. If this sort of question has been answered already, I would appreciate a link to the post as this forum is rather big.
Thanks!

As you say, the first method is wrong and the second one is correct. The explanation is very simple. You can't square only one term. If you square something, that has to be both members of the equation, that is to say, (1/4 - 3/16)^2 = x^2. That will be correct. But when you are working with fractions, is as you were working like decimal numbers. In the first case, you have 1/4 - 3/16 = x --> 0.25 - 0.1875 = x. If you square the (1/4), then you'll have 0.0625 - 0.1875 = x, which is not actually the same expression as the last one. However, in a fraction, you can multiply numerator and denominator because the fraction result will not change. Then, 1/4 = 4/16 = 0.25. 1/4 is not equal to (1/4)^2= 1/16.

I hope I've been helpful.

Regards!!!

$\dfrac 1 4 - \dfrac{3}{16} = x$
$\dfrac{4}{16}-\dfrac{3}{16} = x$
$\dfrac{1}{16}=x$