Ответы к странице 223

894. Найдите значение выражения:
1) $\sqrt{(17,1)^2}$;
2) $\sqrt{(-1,17)^2}$;
3) $\frac{1}{2}\sqrt{(62)^2}$;
4) $-2,4\sqrt{(-4)^2}$;
5) $\sqrt{11^4}$;
6) $\sqrt{(-23)^4}$;
7) $\sqrt{2^6 * 7^4}$;
8) $\sqrt{(-3)^4 * 2^6 * (-0,1)^2}$.

Решение:

1) $\sqrt{(17,1)^2} = |17,1| = 17,1$

2) $\sqrt{(-1,17)^2} = |-1,17| = 1,17$

3) $\frac{1}{2}\sqrt{(62)^2} = \frac{1}{2} * |62| = \frac{1}{2} * 62 = 31$

4) $-2,4\sqrt{(-4)^2} = -2,4 * |-4| = -2,4 * 4 = -9,6$

5) $\sqrt{11^4} = \sqrt{(11^2)^2} = |11^2| = 11^2 = 121$

6) $\sqrt{(-23)^4} = \sqrt{((-23)^2)^2} = |(-23)^2| = 23^2 = 529$

7) $\sqrt{2^6 * 7^4} = \sqrt{2^6} * \sqrt{7^4} = \sqrt{(2^3)^2} * \sqrt{(7^2)^2} = |2^3| * |7^2| = 2^3 * 7^2 = 8 * 49 = 392$

8) $\sqrt{(-3)^4 * 2^6 * (-0,1)^2} = \sqrt{(-3)^4} * \sqrt{2^6} * \sqrt{(-0,1)^2} = \sqrt{((-3)^2)^2} * \sqrt{(2^3)^2} * \sqrt{(-0,1)^2} = |(-3)^2| * |2^3| * |-0,1| = 3^2 * 2^3 * 0,1 = 9 * 8 * 0,1 = 72 * 0,1 = 7,2$

895. Упростите выражение:
1) $\sqrt{q^2}$, если q > 0;
2) $\sqrt{t^2}$, если t ≤ 0;
3) $\sqrt{49m^2n^8}$, если m ≥ 0;
4) $\sqrt{0,81a^6b^{10}}$, если a ≥ 0, b ≤ 0;
5) $\frac{1}{5}x\sqrt{100x^{26}}$, если x ≤ 0;
6) $\frac{\sqrt{a^6b^{20}c^{34}}}{ab^8c^{12}}$, если a > 0, c < 0;
7) $\frac{1,2x^3}{y^5}\sqrt{\frac{y^{14}}{x^{10}}}$, если y > 0, x < 0;
8) $-0,1x^2\sqrt{1,96x^{18}y^{16}}$, если x ≤ 0.

Решение:

1) $\sqrt{q^2} = |q| = q$, если q > 0

2) $\sqrt{t^2} = |t| = -t$, если t ≤ 0

3) $\sqrt{49m^2n^8} = \sqrt{7^2m^2(n^4)^2} = |7mn^4| = 7mn^4$, если m ≥ 0

4) $\sqrt{0,81a^6b^{10}} = \sqrt{0,9^2(a^3)^2(b^{5})^2} = |0,9a^3b^5| = -0,9a^3b^5$, если a ≥ 0, b ≤ 0

5) $\frac{1}{5}x\sqrt{100x^{26}} = \frac{1}{5}x\sqrt{10^2(x^{13})^2} = \frac{1}{5}x * |10x^{13}| = \frac{1}{5}x * (-10x^{13}) = -2x^{14}$, если x ≤ 0

6) $\frac{\sqrt{a^6b^{20}c^{34}}}{ab^8c^{12}} = \frac{\sqrt{(a^3)^2(b^{10})^2(c^{17})^2}}{ab^8c^{12}} = \frac{|a^3b^{10}c^{17}|}{ab^8c^{12}} = \frac{-a^3b^{10}c^{17}}{ab^8c^{12}} = -a^2b^2c^5$, если a > 0, c < 0

7) $\frac{1,2x^3}{y^5}\sqrt{\frac{y^{14}}{x^{10}}} = \frac{1,2x^3}{y^5} * \frac{\sqrt{y^{14}}}{\sqrt{x^{10}}} = \frac{1,2x^3}{y^5} * \frac{\sqrt{(y^{7})^2}}{\sqrt{(x^{5})^2}} = \frac{1,2x^3}{y^5} * \frac{|y^7|}{|x^5|} = \frac{1,2x^3}{y^5} * (-\frac{y^7}{x^5}) = \frac{1,2}{1} * (-\frac{y^2}{x^2}) = -\frac{1,2y^2}{x^2}$, если y > 0, x < 0

8) $-0,1x^2\sqrt{1,96x^{18}y^{16}} = -0,1x^2\sqrt{1,4^2(x^{9})^2(y^{8})^2} = -0,1x^2 * |1,4x^9y^8| = -0,1x^2 * (-1,4x^9y^8) = 0,14x^{11}y^8$, если x ≤ 0

896. Упростите выражение:
1) $\sqrt{(10 - \sqrt{11})^2}$;
2) $\sqrt{(\sqrt{10} - 11)^2}$;
3) $\sqrt{(\sqrt{10} - \sqrt{11})^2}$;
4) $\sqrt{(3 - \sqrt{6})^2} + \sqrt{(2 - \sqrt{6})^2}$;
5) $\sqrt{(\sqrt{24} - 5)^2} - \sqrt{(\sqrt{24} - 4)^2}$.

Решение:

1) $\sqrt{(10 - \sqrt{11})^2} = |10 - \sqrt{11}| = 10 - \sqrt{11}$

2) $\sqrt{(\sqrt{10} - 11)^2} = |\sqrt{10} - 11| = -\sqrt{10} + 11 = 11 - \sqrt{10}$

3) $\sqrt{(\sqrt{10} - \sqrt{11})^2} = |\sqrt{10} - 11| = -\sqrt{10} + \sqrt{11} = \sqrt{11} - \sqrt{10}$

4) $\sqrt{(3 - \sqrt{6})^2} + \sqrt{(2 - \sqrt{6})^2} = |3 - \sqrt{6}| + |2 - \sqrt{6}| = 3 - \sqrt{6} - (2 - \sqrt{6}) = 3 - \sqrt{6} - 2 + \sqrt{6} = 1$

5) $\sqrt{(\sqrt{24} - 5)^2} - \sqrt{(\sqrt{24} - 4)^2} = |\sqrt{24} - 5| - |\sqrt{24} - 4| = -\sqrt{24} + 5 - \sqrt{24} + 4 = 9 - \sqrt{24}$

897. Упростите выражение:
1) $\sqrt{18 + 8\sqrt{2}}$;
2) $\sqrt{38 - 12\sqrt{2}}$;
3) $\sqrt{16 + 6\sqrt{7}} + \sqrt{23 - 8\sqrt{7}}$;
4) $\sqrt{26 - 6\sqrt{17}} - \sqrt{66 - 14\sqrt{17}}$;
5) $\sqrt{46 + 10\sqrt{21}} + \sqrt{46 - 10\sqrt{21}}$.

Решение:

1) $\sqrt{18 + 8\sqrt{2}} = \sqrt{16 + 2 + 2 * 4 * \sqrt{2}} = \sqrt{4^2 + 2 * 4 * \sqrt{2} + (\sqrt{2})^2} = \sqrt{(4 + \sqrt{2})^2} = |4 + \sqrt{2}| = 4 + \sqrt{2}$

2) $\sqrt{38 - 12\sqrt{2}} = \sqrt{36 + 2 - 2 * 6 * \sqrt{2}} = \sqrt{6^2 - 2 * 6 * \sqrt{2} + (\sqrt{2})^2} = \sqrt{(6 - \sqrt{2})^2} = |6 - \sqrt{2}| = 6 - \sqrt{2}$

3) $\sqrt{16 + 6\sqrt{7}} + \sqrt{23 - 8\sqrt{7}} = \sqrt{9 + 7 + 2 * 3 * \sqrt{7}} + \sqrt{16 + 7 - 2 * 4 * \sqrt{7}} = \sqrt{3^2 + 2 * 3 * \sqrt{2} + (\sqrt{7})^2} + \sqrt{4^2 - 2 * 4 * \sqrt{7} + (\sqrt{7})^2} = \sqrt{(3 + \sqrt{7})^2} + \sqrt{(4 - \sqrt{7})^2} = |3 + \sqrt{7}| + |4 - \sqrt{7}| = 3 + \sqrt{7} + 4 - \sqrt{7} = 7$

4) $\sqrt{26 - 6\sqrt{17}} - \sqrt{66 - 14\sqrt{17}} = \sqrt{9 + 17 - 2 * 3 * \sqrt{17}} - \sqrt{49 + 17 - 2 * 7 * \sqrt{17}} = \sqrt{3^2 - 2 * 3 * \sqrt{17} + (\sqrt{17})^2} - \sqrt{7^2 - 2 * 7 * \sqrt{17} + (\sqrt{17})^2} = \sqrt{(3 - \sqrt{17})^2} - \sqrt{(7 - \sqrt{17})^2} = |3 - \sqrt{17}| - |7 - \sqrt{17}| = -3 + \sqrt{17} - 7 + \sqrt{17} = 2\sqrt{17} - 10$

5) $\sqrt{46 + 10\sqrt{21}} + \sqrt{46 - 10\sqrt{21}} = \sqrt{25 + 21 + 2 * 5 * \sqrt{21}} + \sqrt{25 + 21 - 2 * 5\sqrt{21}} = \sqrt{5^2 + 2 * 5 * \sqrt{21} + (\sqrt{21})^2} + \sqrt{5^2 - 2 * 5\sqrt{21} + (\sqrt{21})^2} = \sqrt{(5 + \sqrt{21})^2} + \sqrt{(5 - \sqrt{21})^2} = |5 + \sqrt{21}| + |5 - \sqrt{21}| = 5 + \sqrt{21} + 5 - \sqrt{21} = 10$

898. Вынесите множитель из−под знака корня:
1) $\sqrt{24}$;
2) $\sqrt{63}$;
3) $\sqrt{700}$;
4) $\sqrt{0,32}$;
5) $\frac{1}{7}\sqrt{196}$;
6) $-2,4\sqrt{600}$;
7) $-1,6\sqrt{50}$;
8) $\frac{5}{8}\sqrt{3\frac{21}{25}}$.

Решение:

1) $\sqrt{24} = \sqrt{4 * 6} = \sqrt{4} * \sqrt{6} = 2\sqrt{6}$

2) $\sqrt{63} = \sqrt{9 * 7} = \sqrt{9} * \sqrt{7} = 3\sqrt{7}$

3) $\sqrt{700} = \sqrt{100 * 7} = \sqrt{100} * \sqrt{7} = 10\sqrt{7}$

4) $\sqrt{0,32} = \sqrt{0,16 * 2} = \sqrt{0,16} * \sqrt{2} = 0,4\sqrt{2}$

5) $\frac{1}{7}\sqrt{196} = \frac{1}{7} * 14 = 2$

6) $-2,4\sqrt{600} = -2,4 * \sqrt{100 * 6} = -2,4 * \sqrt{100} * \sqrt{6} = -2,4 * 10 * \sqrt{6} = -24\sqrt{6}$

7) $-1,6\sqrt{50} = -1,6 * \sqrt{25 * 2} = -1,6 * \sqrt{25} * \sqrt{2} = -1,6 * 5\sqrt{2} = -8\sqrt{2}$

8) $\frac{5}{8}\sqrt{3\frac{21}{25}} = \frac{5}{8}\sqrt{\frac{96}{25}} = \frac{5}{8} * \frac{\sqrt{96}}{\sqrt{25}} = \frac{5}{8} * \frac{\sqrt{16 * 6}}{5} = \frac{\sqrt{16} * \sqrt{6}}{8} = \frac{4\sqrt{6}}{8} = \frac{\sqrt{6}}{2}$