1. **State the problem:** Simplify the expression $\left(\sqrt{64} - \sqrt{2} \cdot 5\right)^2$. 2. **Recall the formula:** The square of a difference is given by $$(a - b)^2 = a^2 - 2ab + b^2$$ where $a = \sqrt{64}$ and $b = \sqrt{2} \cdot 5$. 3. **Calculate each term:** - $a = \sqrt{64} = 8$ - $b = \sqrt{2} \cdot 5 = 5\sqrt{2}$ 4. **Apply the formula:** $$\left(8 - 5\sqrt{2}\right)^2 = 8^2 - 2 \cdot 8 \cdot 5\sqrt{2} + \left(5\sqrt{2}\right)^2$$ 5. **Simplify each term:** - $8^2 = 64$ - $-2 \cdot 8 \cdot 5\sqrt{2} = -80\sqrt{2}$ - $\left(5\sqrt{2}\right)^2 = 5^2 \cdot (\sqrt{2})^2 = 25 \cdot 2 = 50$ 6. **Combine all terms:** $$64 - 80\sqrt{2} + 50 = 114 - 80\sqrt{2}$$ **Final answer:** $$\boxed{114 - 80\sqrt{2}}$$