Ambigous question regarding how to view surds with numbers infront

Question

Say I want to multiply 2 by 5$\sqrt3$ . Do I firstly do 2 * 5, then 2 * 3? I'm not sure about the order of operations here. Such a dumb question, I know.

Edit - can someone show me the systematic way of rationalizing this:

$\frac{\sqrt5 + \sqrt2}{4\sqrt5 - 3\sqrt3}$

When I look at this, I think of the difference of two squares at the bottom, but I also think of multiplying by the conjuguate. I'm not sure how to approach this.

Answer 1

In general, $a\sqrt b = a\cdot \sqrt b.\;$ Here, $5\sqrt 3 = 5\cdot \sqrt 3$. So multiplying by $2$ gives $$2\cdot 5\cdot \sqrt 3 = 10\cdot\sqrt 3 = 10\sqrt 3$$

Answer 2

Using $a(bc)=(ab)c,$ we have $2\cdot 5 \sqrt{3}=10 \sqrt{3}$

Answer 3

You are thinking of the distribution rule: $2(5+\sqrt{3})$ then you multiply the 2 times the 5 and the 2 times the $\sqrt{3}$. This is the not the case for $2\cdot5\sqrt{3}$.