1. **State the problem:** Evaluate the expression $$\sqrt{9^3 - 3^4} \times 2 + \frac{64}{2^3} - 5$$. 2. **Recall the order of operations:** Parentheses, exponents, multiplication and division (from left to right), addition and subtraction (from left to right). 3. **Calculate the exponents:** $$9^3 = 9 \times 9 \times 9 = 729$$ $$3^4 = 3 \times 3 \times 3 \times 3 = 81$$ $$2^3 = 2 \times 2 \times 2 = 8$$ 4. **Substitute these values back into the expression:** $$\sqrt{729 - 81} \times 2 + \frac{64}{8} - 5$$ 5. **Simplify inside the square root:** $$729 - 81 = 648$$ 6. **Calculate the square root:** $$\sqrt{648}$$ Since $$648 = 81 \times 8$$, and $$\sqrt{81} = 9$$, we have: $$\sqrt{648} = \sqrt{81 \times 8} = 9 \sqrt{8}$$ 7. **Simplify $$\sqrt{8}$$:** $$\sqrt{8} = \sqrt{4 \times 2} = 2 \sqrt{2}$$ 8. **So, $$\sqrt{648} = 9 \times 2 \sqrt{2} = 18 \sqrt{2}$$** 9. **Multiply by 2:** $$18 \sqrt{2} \times 2 = 36 \sqrt{2}$$ 10. **Calculate the division:** $$\frac{64}{8} = 8$$ 11. **Put it all together:** $$36 \sqrt{2} + 8 - 5$$ 12. **Simplify the addition and subtraction:** $$8 - 5 = 3$$ 13. **Final answer:** $$36 \sqrt{2} + 3$$