1. **State the problem:** Simplify the expression $$2^{4}\sqrt[4]{405} - 3^{4}\sqrt[4]{80} - 3\sqrt{192}$$. 2. **Recall the rules:** - The notation $a^{n}\sqrt[m]{b}$ means $a^n \times \sqrt[m]{b}$. - Simplify radicals by factoring out perfect powers. - Combine like terms if possible. 3. **Simplify each term:** - First term: $$2^{4}\sqrt[4]{405} = 16 \times \sqrt[4]{405}$$ since $2^4=16$. - Factor 405: $$405 = 3^4 \times 5$$ so $$\sqrt[4]{405} = \sqrt[4]{3^4 \times 5} = 3 \sqrt[4]{5}$$. - So first term becomes $$16 \times 3 \sqrt[4]{5} = 48 \sqrt[4]{5}$$. - Second term: $$3^{4}\sqrt[4]{80} = 81 \times \sqrt[4]{80}$$ since $3^4=81$. - Factor 80: $$80 = 16 \times 5 = 2^4 \times 5$$ so $$\sqrt[4]{80} = \sqrt[4]{2^4 \times 5} = 2 \sqrt[4]{5}$$. - So second term becomes $$81 \times 2 \sqrt[4]{5} = 162 \sqrt[4]{5}$$. - Third term: $$3\sqrt{192} = 3 \times \sqrt{192}$$. - Factor 192: $$192 = 64 \times 3 = 8^2 \times 3$$ so $$\sqrt{192} = \sqrt{8^2 \times 3} = 8 \sqrt{3}$$. - So third term becomes $$3 \times 8 \sqrt{3} = 24 \sqrt{3}$$. 4. **Rewrite the expression:** $$48 \sqrt[4]{5} - 162 \sqrt[4]{5} - 24 \sqrt{3}$$ 5. **Combine like terms:** - Combine $$48 \sqrt[4]{5} - 162 \sqrt[4]{5} = (48 - 162) \sqrt[4]{5} = -114 \sqrt[4]{5}$$. 6. **Final simplified expression:** $$-114 \sqrt[4]{5} - 24 \sqrt{3}$$. **Answer:** $$\boxed{-114 \sqrt[4]{5} - 24 \sqrt{3}}$$