(4X^5Y/16XY^4)^2

To simplify the expression (4X^5Y/16XY^4)^2, we can start by simplifying the numerator and denominator separately before squaring the entire fraction.

Numerator: 4X^5Y
We only have X^5 in the numerator, so we can write it as X * X^4:
4X^5Y = (4XY) * X^4

Denominator: 16XY^4
We only have Y^4 in the denominator, so we can write it as Y * Y^3:
16XY^4 = 16XY * Y^3

Now, let's rewrite the entire fraction:
(4X^5Y/16XY^4)^2 = [(4XY) * X^4] / [(16XY) * Y^3]

Now, let's square the fraction:
[(4XY) * X^4 / (16XY) * Y^3]^2 = [(4XY * X^4)^2 / (16XY * Y^3)^2]

To simplify this completely, we can cancel out the common factors in the numerator and denominator:
[(4XY * X^4)^2 / (16XY * Y^3)^2] = [(4^2 * X^2 * Y^2 * X^4) / (4^2 * X^2 * Y^2 * Y^6)]

Next, we simplify the remaining expression:
[(4^2 * X^2 * Y^2 * X^4) / (4^2 * X^2 * Y^2 * Y^6)] = (16X^6) / (16Y^8)

Finally, we see that the simplified expression is 16X^6 / 16Y^8, but since the numerator and denominator have a common factor of 16, we can cancel it out and simplify further:
16X^6 / 16Y^8 = X^6 / Y^8

Therefore, the simplified form of (4X^5Y/16XY^4)^2 is X^6 / Y^8.