How to take square root of $x^2 + y^2 = 144$?

Question

I've to take square root of the equation. Which of the following two is correct way and why?

Taking square root of $x^2+y^2=144$

1. $x + y = 12$

   OR

2. $\sqrt{x^2+y^2} = 12$

   OR

3. $\sqrt{x^2} + \sqrt{y^2} = 12$

Answer 1

Because $$(x+y)^2=x^2+2xy+y^2$$ $$x^2+2xy+y^2\neq x^2+y^2$$ $$\sqrt{x^2+2xy+y^2}\neq \sqrt{x^2+y^2}$$ $$x+y\neq \sqrt{x^2+y^2}$$

Answer 2

In general, the square root of a sum is not the sum of the square roots, so

$$\sqrt{x^2 + y^2} \neq x+y.$$

Answer 3

Just documenting here based on all the answers and comments.

It's option 3 in the question.

i.e $\sqrt{x^2} + \sqrt{y2} = 12$

Here's why :

Let's put two sample numbers to test above form, it should hold the equation true. Sample numbers : 10 and 2. Notice sum of these two numbers is 12.

Putting values in our equation:

$\sqrt{10^2} + \sqrt{2^2} = \sqrt{144}$

$\sqrt{100} + \sqrt{4}$ = 12

10 + 2 = 12

12 = 12

Other two forms mentioned in the question won't hold this test, hence are invalid.