Expand $\frac{1}{1.2}$ by multiplying both numerator and the denominator by $10$.
$$\frac{10}{12}\times \frac{3}{\sqrt{8}+1}$$
Reduce the fraction $\frac{10}{12}$ to lowest terms by extracting and canceling out $2$.
$$\frac{5}{6}\times \frac{3}{\sqrt{8}+1}$$
Factor $8=2^{2}\times 2$. Rewrite the square root of the product $\sqrt{2^{2}\times 2}$ as the product of square roots $\sqrt{2^{2}}\sqrt{2}$. Take the square root of $2^{2}$.
$$\frac{5}{6}\times \frac{3}{2\sqrt{2}+1}$$
Rationalize the denominator of $\frac{3}{2\sqrt{2}+1}$ by multiplying numerator and denominator by $2\sqrt{2}-1$.
Consider $\left(2\sqrt{2}+1\right)\left(2\sqrt{2}-1\right)$. Multiplication can be transformed into difference of squares using the rule: $\left(a-b\right)\left(a+b\right)=a^{2}-b^{2}$.
