Series expansion for small ratio of variables

Question

I have a messy expression of variables that I would like to simplify under the assumption that certain ratios of the variables are small. For example, consider $\sqrt{x+y}\;$ expanded for small $\frac{y}{x}$. That is, $x >> y$. This is simple enough to do by hand (pull out a $\sqrt{x}$ and then define a new variable to be $\frac{y}{x}$, then expand this variable about $0$). But, I cannot figure out how to do it in Mathematica.

EDIT 1

I've since realized that what I actually want to do is more complicated than the example above. Consider the following expression $\sqrt{a+d} + \sqrt{b+d}\;$ expanded for small $\frac{d}{a}$ and small $\frac{d}{b}$ (that is $a>>d$ and $b>>d$). Following Artes' answer below, the solution is

Series[ (a + d)^(1/2) + (b + d)^(1/2) /. {d -> z1 a, d -> z2}, {z1, 0, 2},{z2, 0, 2}] /. 
{z1 -> d/a, z2 -> d/b}

But this solution doesn't work as intended due to the order operations Mathematica performs upon this command. What needs to be done is to replace d with z1 a, then expand z1 about 0, replace z1 with d/a, then replace d with z2 b, expand z2 about 0, and finally replace z2 with d/b. How can one do this in Mathematica?
(Obviously the example I gave can be separated into two expressions when can be expanded separately, but this cannot be done for the more complicated expression I am actually trying to expand).

Answer 1

There are options in Series (Analytic, Assumptions) which you might exploit. Nonetheless let's use a bit more straightforward approach, simply change the variable y (y -> z x) and then back z -> y/x:

Series[(x + y)^(1/2) /. y -> z x, {z, 0, 5}] /. z -> y/x // Quiet // TraditionalForm

Edit

Since the question has been edited we should add a few remarks.
In order to transform appropriately the Series results (its head is SeriesData) we should get rid of it with Normal (then we loose terms with powers (y/x)), e.g.

Normal @ Series[(x + y)^(1/2) /. y -> z x, {z, 0, 5}] /. z -> y/x // TraditionalForm

Nonetheless we can transform appropriately expressions of the form (a + d)^(1/2) + (b + d)^(1/2) working with SeriesData if we define a simple function, e.g.

f[x_, y_, k_] := Series[(x + y)^k /. y -> z x, {z, 0, 5}] /. z -> y/x

(a + d)^(1/2) + (b + d)^(1/2) /. (x_ + y_)^k_ -> f[x, y, k] // Quiet // TraditionalForm

Alternatively we can expand a given expression around infinity, e.g.

Series[(a + d)^(1/2) + (b + d)^(1/2) /. {a -> v d, b -> w d},
       {v, Infinity, 5}, {w, Infinity, 5}] /.
{v -> a/ d, w -> b/d} // Quiet // TraditionalForm