To simplify the expression 81⁴.(z²-y²)-z²+y², we can start by expanding the power of 81⁴:

81⁴ = (3²)⁴ = 3⁸

Now, let's substitute this value back into the expression:

3⁸.(z²-y²)-z²+y²

Next, we can simplify the expression within the parentheses:

3⁸.(z²-y²) = (3⁴)².(z²-y²) = 81².(z²-y²)

Now, let's substitute this value back into the expression:

81².(z²-y²)-z²+y²

Next, we can simplify the expression within the parentheses:

81².(z²-y²) = (9²)².(z²-y²) = 9⁴.(z²-y²)

Now, let's substitute this value back into the expression:

9⁴.(z²-y²)-z²+y²

Next, we can simplify the expression within the parentheses:

9⁴.(z²-y²) = (3²)⁴.(z²-y²) = 3⁸.(z²-y²)

Now, let's substitute this value back into the expression:

3⁸.(z²-y²)-z²+y²

Finally, we can simplify the expression:

3⁸.(z²-y²)-z²+y² = 81⁴.(z²-y²)-z²+y²

Therefore, the simplified expression is 81⁴.(z²-y²)-z²+y².