Looking for irrational Numbers Proof

Question

$a,b,c,~$and $d$ are rational numbers. $b>0$ and $d>0$ the $\sqrt{b}$ and the $\sqrt{d}$ are both irrational.

if $a+\sqrt{b}=c+\sqrt{d}$

show that $a = c$ and $b = d$.

I know that a=c and b=d intuitively, but I'm not sure how to prove it.

Answer 1

$a-c=\sqrt(d)-\sqrt(b)$ If this is true, then that means $\sqrt(d)-\sqrt(b)$ is rational because rational numbers are closed under subtraction. The only way for this to be true is that $\sqrt(d)-\sqrt(b)=0$.