Quantity A
A negative exponent is equal to the reciprocal of the positive exponent.
Thus, \(2^{-2} = \dfrac{1}{2^2} = \frac{1}{4}\)

\(\sqrt{\frac{9}{4}} = \sqrt{\frac{3^2}{2^2}} = \frac{\sqrt{3^2}}{\sqrt{2^2}} = \frac{3}{2}\)

Combine to find that \(2^{-2} * \sqrt{\frac{9}{4}} = \frac{1}{4} * \frac{3}{2} = \frac{3}{8}\)

Quantity B
Again, since a negative exponent is equal to the reciprocal of the positive exponent, \(4^{-1} = \dfrac{1}{4^1}\)

Sub back in to find that \(\frac{2}{3} * 4^{-1} = \frac{2}{3}* \dfrac{1}{4^1} = \frac{2}{12}\)


At this point, I'd save time by plugging both into the calculator to find their decimal equivalents to compare the quantities.
Quantity A = 0.375
Quantity B = 0.167

Since 0.375 > 0.167, the answer is A.

If you wanted to compare by making the quantities' denominators equivalent,
Quantity A: \(\frac{3}{8} = \frac{3}{8} * \frac{3}{3} = \frac{9}{24}\)
Quantity B: \(\frac{2}{12} = \frac{2}{12} * \frac{2}{2} = \frac{4}{24}\)

Again, since 9 > 4 and their denominators are the same, Quantity A is greater.