What are some of the practical applications of functions that extract the exponent and mantissa of a floating point number?
They are among other things very useful for calculating logarithms, or square roots and cubic roots. 

For example: log x = log ($2^e * m$) = e * log 2 + log m. With 0.5 ≤ m ≤ 1, you can approximate log m using a polynomial, which wouldn't work for x which can be in the complete range of floating point numbers. 

$x^{1/3}$ = $(2^e * m)^{1/3}$. Let e = 3k, 3k+1, or 3k+2, then this equals $2^k * m^{1/3}$ for 0.5 ≤ m ≤ 4. You can either use approximation by a polynomial for m, or you use Newton iteration which will be much faster for m in the small interval.