Divide $1$ by $\frac{3}{\sqrt{8}+1}$ by multiplying $1$ by the reciprocal of $\frac{3}{\sqrt{8}+1}$.
$$\frac{\sqrt{8}+1}{3}$$
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 $2^{2}$.
