Express $156$ as a product of primes.

Prove that: $\frac{2+\sqrt{3}}{\sqrt{2}+\sqrt{2-\sqrt{3}}}+\frac{2-\sqrt{3}}{\sqrt{2}-\sqrt{2+\sqrt{3}}}=\sqrt{2}$

State whether the following rational number has a terminating decimal expansion or not. If it has a terminating decimal expansion,find it: $\frac{64}{455}$

Show that the square of any positive integer cannot be of the form $6m+2$ or $6m+5$ for any integer $m$.

The length,breadth,and height of a prayer room are $15\,m$,$12\,m$,and $21\,m$ respectively. Determine the longest rod which can measure the three dimensions of the room exactly. (in $,m$)

The following real number is expressed in decimal form. Determine whether it is rational or not. If it is rational,express it in the form of $\frac{p}{q}$. $3.127$