What's the advantage of rationalizing radical fractions?

Question

"What's the advantage of rationalizing radical fractions?"
It was a question in all years of my teaching experience. I make an example to show the advantage(s) of rationalizing the denominator as below. Please help me improve my reasoning for students.
Remark: I think it is a hard example to show, and I am looking for a simple example or Idea to show the advantage(s) of rationalizing the denominator. I start by writing $$x=\frac{\sqrt{15}}{\sqrt{5}+\sqrt{3}} \ \to \frac{x+5\sqrt3}{x+\sqrt{5}}=?\\ x\sim0.976025$$ then ask student to use calculator and put $$x\sim0.9 \\x\sim0.97\\x\sim0.976$$ into $\frac{x+5\sqrt3}{x+\sqrt{5}}$
then we can see $$x\sim0.9 \to \frac{x+5\sqrt3}{x+\sqrt{5}}\sim 3.048 \\x\sim0.97 \to \frac{x+5\sqrt3}{x+\sqrt{5}}\sim 3.0037 \\x\sim0.976 \to \frac{x+5\sqrt3}{x+\sqrt{5}}\sim 3.000015$$ then use rationalizing as follows $$x=\frac{\sqrt{15}}{\sqrt{5}+\sqrt{3}}\times \frac{\sqrt{5}-\sqrt{3}}{\sqrt{5}-\sqrt{3}}=\frac{5\sqrt3 -3\sqrt5}{2}$$ and put it into $\frac{x+5\sqrt3}{x+\sqrt{5}}$ after simplifying we(they)can see it becomes $\frac{x+5\sqrt3}{x+\sqrt{5}}=3$
Then I will say take a look, If we can use a calculator the answer might be wrong. It is one of the advantages of rationalizing. The second example is to find $$\frac{1}{\sqrt 3}+\frac{1}{\sqrt 6+\sqrt3}$$ in every method they like. For example some try this $$\frac{\sqrt 6 +\sqrt 3 +\sqrt 3}{(\sqrt 6 +\sqrt 3 )\sqrt 3}=\frac{2\sqrt 3 +\sqrt6 }{(\sqrt 6 +\sqrt 3 )\sqrt 3} \sim 0.83$$ by using one digit point . then try to rationalize $$\frac{1}{\sqrt 3}+\frac{1}{\sqrt 6+\sqrt3}=\\ \frac{\sqrt 3}{3}+\frac{\sqrt 6-\sqrt3}{3}=\frac {\sqrt 6}3\sim 0.8164$$
sometimes I say find the floor of $$A=\frac{10}{\sqrt{25} -\sqrt{24}}$$ they will find $$\sqrt{24}\sim 4.8 \to A=50\\ \sqrt{24}\sim 4.89 \to A=90.9090\\ \sqrt{24}\sim 4.898 \to A=98.0392\\ and \\ \text{true value of }A\sim 98.9897$$ theد do rationalizing and show them we can go wrong even by a calculator instead of rationalizing.
Thanks in advance for any idea