a) \( (-4)^{3}-(-5)^{3}-2^{2} \cdot\left[-2^{4}-(-6)\right]^{3}: \sqrt[3]{-512}= \)

We start by simplifying each component of the expression \[ (-4)^3 - (-5)^3 - 2^2\cdot\left[-2^4 - (-6)\right]^3 : \sqrt[3]{-512}. \] 1. Compute the cubes: - \( (-4)^3 = -64 \) - \( (-5)^3 = -125 \) Thus, \[ (-4)^3 - (-5)^3 = -64 - (-125) = -64 + 125 = 61. \] 2. Compute the power of 2: \[ 2^2 = 4. \] 3. Simplify the expression inside the brackets: - First, compute \(2^4 = 16\). - Then, interpret \(-2^4\) as \(-(2^4) = -16\). - Now, we have \[ -2^4 - (-6) = -16 + 6 = -10. \] 4. Raise the result to the third power: \[ \left[-2^4 - (-6)\right]^3 = (-10)^3 = -1000. \] 5. Multiply by \(2^2\): \[ 4 \cdot (-1000) = -4000. \] 6. Compute the cube root: \[ \sqrt[3]{-512} = -8 \quad \text{(since } (-8)^3 = -512\text{)}. \] 7. Perform the division: \[ \frac{-4000}{-8} = 500. \] 8. Combine the results: \[ 61 - 500 = -439. \] Thus, the final result is \[ -439. \]