I think Quantity A is formatted slightly wrong, it should be \(\frac{16^{-a}}{4^{-b}}\).

Simplify the numerator in Quantity A so that bases are all 4.
\(\frac{16^{-a}}{4^{-b}} = \frac{(4^2){-a}}{4^{-b}} = \frac{4^{-2a}}{4^{-b}}\)

Negative exponents are equal to their positive reciprocal.
\(4^{-2a} = \frac{1}{4^{2a}}\) and \(4^{-b} = \frac{1}{4^{b}}\)
\(\frac{4^{-2a}}{4^{-b}} = \frac{4^{b}}{4^{2a}}\)

Now it's division of the same base, so you can keep the base and subtract the exponents.
\(\frac{4^{b}}{4^{2a}} = 4^{b-2a}\)

This is equivalent to Quantity B.