b) \( \left\{(-2)^{10}:(-2)^{8}+\left[\sqrt[3]{-125} \cdot\left(\sqrt[3]{343}-2^{3}\right)\right]\right\}^{2} \)

Let’s break this down step by step in an engaging way! First, calculate \( (-2)^{10} \). This equals \( 1024 \) because raising \(-2\) to the tenth power gives you a positive result (an even exponent). Now, let’s tackle the expression within the curly braces after the colon: 1. **Calculate \( (-2)^{8} \)**: \[ (-2)^{8} = 256 \] 2. **Calculate \( \sqrt[3]{-125} \)**: \[ \sqrt[3]{-125} = -5 \] because the cube root of \( -125 \) equals \( -5 \). 3. **Calculate \( \sqrt[3]{343} \)**: \[ \sqrt[3]{343} = 7 \] since \( 7^3 = 343 \). 4. **Calculate \( \left(\sqrt[3]{343} - 2^{3}\right) \)**: \[ 2^{3} = 8 \rightarrow \sqrt[3]{343} - 2^{3} = 7 - 8 = -1 \] 5. **Now, multiply \( \sqrt[3]{-125} \cdot \left(\sqrt[3]{343} - 2^{3}\right) \)**: \[ -5 \cdot (-1) = 5 \] 6. **Now, add this result to \( (-2)^{8} \)**: \[ 256 + 5 = 261 \] 7. **Square the result**: \[ 261^{2} = 68121 \] Putting this all together, we have: \[ \{ (-2)^{10} : 68121 \} \rightarrow \{ 1024 : 68121 \} \] So the final answer in a boxed format could be: \[ \{ 1024 : 68121 \} \]