What expression raised to the fourth power is 81 x^12 y^8 z^16 ?

think it is (3x^3 y^2 z^4)^4

Yes, you are correct.

To find the expression raised to the fourth power, we need to take the fourth root of 81, x^12, y^8, and z^16.

First, let's find the fourth root of 81. The fourth root of a number can be found by raising the number to the power of 1/4. Therefore, the fourth root of 81 is:

∛∛(81) = ∛(3 x 3 x 3 x 3) = 3

Next, let's find the fourth root of x^12. Since the exponent is divisible by 4 (12 ÷ 4 = 3), we can simplify it as follows:

∛∛(x^12) = ∛(x^(4 x 3)) = ∛(x^4)^3 = (x^4)

Similarly, let's find the fourth root of y^8:

∛∛(y^8) = ∛(y^(4 x 2)) = ∛(y^4)^2 = (y^4)

Finally, let's find the fourth root of z^16:

∛∛(z^16) = ∛(z^(4 x 4)) = ∛(z^4)^4 = (z^4)

Combining all the fourth roots, we have:

(3 x^4 y^4 z^4)