Comparing the magnitudes of expressions of surds
Comparing $\sqrt{10}$ and $\sqrt2+\sqrt3$ is the same as comparing $10$ and $(\sqrt2+\sqrt3)^2=5+2\sqrt6$. That's the same as comparing $5$ and $2\sqrt6$. Which of these is bigger?
Likewise comparing $\sqrt{10}$ and $5-\sqrt3$ is the same as comparing $10$ and $(5-\sqrt3)^2=28-10\sqrt3$. That's the same as comparing $10\sqrt3$ and $18$.
Which of these is bigger?
You can use:
(1) the fact that $f(x)=x^2$ is a monotonically increasing function when $x\geq0$ and
(2) the arithmetic-geometric mean inequality $\sqrt{ab}\leq\frac{a+b}{2}$, when $a, b\geq0$. Hence, $$ (\sqrt{2}+\sqrt{3})^2=5+2\sqrt{2\cdot3}\leq5+2\frac{2+3}{2}=5+5=10=(\sqrt{10})^2 $$ Therefore, using (1), we obtain $\sqrt{2}+\sqrt{3}\leq 10$. I forgot about this: $$ 5-\sqrt{3}=3+2-\sqrt{3}=3+\frac{1}{2+\sqrt{3}}\geq3+\frac{1}{2+2}=3.25 $$ One can easily verify that $(3+1/4)^2>10.5>10$. One also finds that $10.5^2>110>101$.
Then, performing argument (1) twice, one finds that $5-\sqrt{3}>(101)^{1/4}$.