1. **State the problem:** Simplify the expression $$0.52 - 117 + 4 \sqrt{13} \sqrt{224} - \sqrt{176} - \sqrt{56}$$. 2. **Rewrite the expression:** $$0.52 - 117 + 4 \sqrt{13} \sqrt{224} - \sqrt{176} - \sqrt{56}$$ 3. **Simplify the square roots where possible:** - Factor inside the roots: $$\sqrt{224} = \sqrt{16 \times 14} = 4 \sqrt{14}$$ $$\sqrt{176} = \sqrt{16 \times 11} = 4 \sqrt{11}$$ $$\sqrt{56} = \sqrt{4 \times 14} = 2 \sqrt{14}$$ 4. **Substitute back:** $$0.52 - 117 + 4 \sqrt{13} \times 4 \sqrt{14} - 4 \sqrt{11} - 2 \sqrt{14}$$ 5. **Multiply the terms with square roots:** $$4 \sqrt{13} \times 4 \sqrt{14} = 16 \sqrt{13 \times 14} = 16 \sqrt{182}$$ 6. **Rewrite the expression:** $$0.52 - 117 + 16 \sqrt{182} - 4 \sqrt{11} - 2 \sqrt{14}$$ 7. **Combine the constants:** $$0.52 - 117 = -116.48$$ 8. **Final simplified expression:** $$-116.48 + 16 \sqrt{182} - 4 \sqrt{11} - 2 \sqrt{14}$$ This is the simplified form since the radicals are not like terms and cannot be combined further.