What is the value of t ?

I'm happy to help , but I disagree with the OA given.

Statement #1: The average (arithmetic mean) of t^2 and 8t is –8.
[t^2 + 8t]/2 = –8
t^2 + 8t = –16
t^2 + 8t + 16 = 0
(t + 4)^2 = 0
t = - 4
See:
https://magoosh.com/gmat/2013/three-alge ... -the-gmat/
This statement produces a unique value of t, and thus is sufficient.

Statement #2: t^1/4 = 16
I will assume what this means is t^(1/4) = 16, only because I can't think of another sensible interpretation of the ambiguity. Incidentally, on the issue of this ambiguity, and the underlying mathematical principles, read:
https://magoosh.com/gmat/2013/gmat-quant ... g-symbols/
Raising a number to the power of (1/4) is equivalent to taking the fourth root of a number. We can't take an even root of negative number, so if the fourth root of t equals anything sensible, then t absolutely must be positive. We would raise both sides to the fourth power to get a unique value of t. See:
https://magoosh.com/gmat/2012/exponent-p ... -the-gmat/
(BTW, that value would be t = 2^16 = 65,536, but absolutely no one expects you to get that without a calculator!!)
This statement also produces a unique value of t, and thus is also sufficient.

This would produce an OA of (D). I don't know whether you copied something incorrectly about statement two --- perhaps another phrasing would make that statement insufficient. Also, as this question currently stands, the two statements produce different values for t. This falls short of the standard that the GMAT keeps on DS --- statements in both statements are mathematically consistent.

Let me know if statement #2 says something else.

Mike