415. Упростите выражение: a) √8p-√2p + √18p; б) √160c + 2√40c-3√90c; в) 5√27m-4√48m-2√12m; г) √54-√24+√150; д) 3√2 + √32-√200; e) 2√72-√50-2√8.

Решение:

1. а) \( \sqrt{8p} - \sqrt{2p} + \sqrt{18p} = \sqrt{4 \cdot 2p} - \sqrt{2p} + \sqrt{9 \cdot 2p} = 2\sqrt{2p} - \sqrt{2p} + 3\sqrt{2p} = (2-1+3)\sqrt{2p} = 4\sqrt{2p} \)
2. б) \( \sqrt{160c} + 2\sqrt{40c} - 3\sqrt{90c} = \sqrt{16 \cdot 10c} + 2\sqrt{4 \cdot 10c} - 3\sqrt{9 \cdot 10c} = 4\sqrt{10c} + 2 \cdot 2\sqrt{10c} - 3 \cdot 3\sqrt{10c} = 4\sqrt{10c} + 4\sqrt{10c} - 9\sqrt{10c} = (4+4-9)\sqrt{10c} = -\sqrt{10c} \)
3. в) \( 5\sqrt{27m} - 4\sqrt{48m} - 2\sqrt{12m} = 5\sqrt{9 \cdot 3m} - 4\sqrt{16 \cdot 3m} - 2\sqrt{4 \cdot 3m} = 5 \cdot 3\sqrt{3m} - 4 \cdot 4\sqrt{3m} - 2 \cdot 2\sqrt{3m} = 15\sqrt{3m} - 16\sqrt{3m} - 4\sqrt{3m} = (15-16-4)\sqrt{3m} = -5\sqrt{3m} \)
4. г) \( \sqrt{54} - \sqrt{24} + \sqrt{150} = \sqrt{9 \cdot 6} - \sqrt{4 \cdot 6} + \sqrt{25 \cdot 6} = 3\sqrt{6} - 2\sqrt{6} + 5\sqrt{6} = (3-2+5)\sqrt{6} = 6\sqrt{6} \)
5. д) \( 3\sqrt{2} + \sqrt{32} - \sqrt{200} = 3\sqrt{2} + \sqrt{16 \cdot 2} - \sqrt{100 \cdot 2} = 3\sqrt{2} + 4\sqrt{2} - 10\sqrt{2} = (3+4-10)\sqrt{2} = -3\sqrt{2} \)
6. е) \( 2\sqrt{72} - \sqrt{50} - 2\sqrt{8} = 2\sqrt{36 \cdot 2} - \sqrt{25 \cdot 2} - 2\sqrt{4 \cdot 2} = 2 \cdot 6\sqrt{2} - 5\sqrt{2} - 2 \cdot 2\sqrt{2} = 12\sqrt{2} - 5\sqrt{2} - 4\sqrt{2} = (12-5-4)\sqrt{2} = 3\sqrt{2} \)

Ответ: а) 4√2p; б) -√10c; в) -5√3m; г) 6√6; д) -3√2; е) 3√2.