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.