Exact Multiplications and Divisions

When you multiply a floating point number by a power of $\beta$, the result is exact provided there is no over or underflow.

Another example, if you have two floating point numbers $x,y$ such that the leading $k_{x}$ digits of the significand of $x$ are nonzero, and likewise for $y$, and $k_{x}+k_{y}<p$, then the product of the two numbers will fit in the $p$ characters of $M$. Again, there is an assumption of no over or under flow. Note that I made no requirement on the exponents being the same.