1. The problem asks if the statement $-3^{-4} = \frac{1}{81}$ is true or false. 2. Recall the order of operations: exponents are evaluated before negation. 3. So, $-3^{-4}$ means the negative of $3^{-4}$, not $(-3)^{-4}$. 4. Calculate $3^{-4}$ using the rule $a^{-n} = \frac{1}{a^n}$: $$3^{-4} = \frac{1}{3^4} = \frac{1}{81}$$ 5. Now apply the negative sign: $$-3^{-4} = -\frac{1}{81}$$ 6. The original statement says $-3^{-4} = \frac{1}{81}$, but we found it equals $-\frac{1}{81}$. 7. Therefore, the statement is false.