Long Roots

The 1st equation can be written as -

$\sqrt{x+y} = y$

And, the Second Equation as -

$\sqrt{y+5} = 5$

$\textbf{squaring both the sides, we get}$

$y+5 = 25$ So, $y=20$

$\textbf{Which means,}$

$\sqrt{x+y} = \boxed{20}$

Let us first find the value of $y$ . We can do that by squaring the second equation, getting:

$\sqrt{y+\sqrt{y+\sqrt{y+\sqrt{\dots}}}}=5 (^{2})$

$y+\sqrt{y+\sqrt{y+\sqrt{y+\sqrt{\dots}}}}=25 \Rightarrow \sqrt{y+\sqrt{y+\sqrt{y+\sqrt{\dots}}}}=25-y \Rightarrow$

$\Rightarrow 5=25-y \Rightarrow \boxed{y=20}$ .

Similarly, we can now acquire the value of $x$ as well:

$\sqrt{x+\sqrt{x+\sqrt{x+\sqrt{\dots}}}}=20 (^{2})$

$x+\sqrt{x+\sqrt{x+\sqrt{x+\sqrt{\dots}}}}=400 \Rightarrow \sqrt{x+\sqrt{x+\sqrt{x+\sqrt{\dots}}}}=400-x \Rightarrow$

$\Rightarrow 20=400-x \Rightarrow \boxed{x=380}$ .

From here, we can clearly see that $\sqrt{x+y}=\sqrt{400}=\boxed{20}$ .

its a infinite series question even if u remove one x and one y from beginning the expressions remain same,,,,,,, do urself after this hint

square the second equation yields y + 5 = 25, therefore y = 20.

square the first equation yields x + y = y^2 or x + 20 = 400, hence x = 380.

finally sqrt(x+y) = sqrt(380+20) = sqrt(400) = 20,

This is an approach from middle school. more sophisticated solution can be found from Karthik Sharma :)

Square both sides of the second equation.

y + sqrt( y + sqrt( y + sqrt( ....))) = 25

=> y + 5 = 25

=> y = 20

Substitute y = 20 in first equation. Square both sides.

x + sqrt( x + sqrt( x + sqrt( ....))) = 400

=> x + y = 400

=> sqrt(x + y) = 20