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) $-2,4sqrt{(-4)^2}$;
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) $-2,4sqrt{(-4)^2} = -2,4 * |-4| = -2,4 * 4 = -9,6$
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) $frac{1}{5}xsqrt{100x^{26}}$, 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) $-0,1x^2sqrt{1,96x^{18}y^{16}}$, 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) $frac{1}{5}xsqrt{100x^{26}} = frac{1}{5}xsqrt{10^2(x^{13})^2} = frac{1}{5}x * |10x^{13}| = frac{1}{5}x * (-10x^{13}) = -2x^{14}$, 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 < 07) $frac{1,2x^3}{y^5}sqrt{frac{y^{14}}{x^{10}}} = frac{1,2x^3}{y^5} * frac{sqrt{y^{14}}}{sqrt{x^{10}}} = frac{1,2x^3}{y^5} * frac{sqrt{(y^{7})^2}}{sqrt{(x^{5})^2}} = frac{1,2x^3}{y^5} * frac{|y^7|}{|x^5|} = frac{1,2x^3}{y^5} * (-frac{y^7}{x^5}) = frac{1,2}{1} * (-frac{y^2}{x^2}) = -frac{1,2y^2}{x^2}$, если y > 0, x < 08) $-0,1x^2sqrt{1,96x^{18}y^{16}} = -0,1x^2sqrt{1,4^2(x^{9})^2(y^{8})^2} = -0,1x^2 * |1,4x^9y^8| = -0,1x^2 * (-1,4x^9y^8) = 0,14x^{11}y^8$, 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) $sqrt{18 + 8sqrt{2}}$;
2) $sqrt{38 — 12sqrt{2}}$;
3) $sqrt{16 + 6sqrt{7}} + sqrt{23 — 8sqrt{7}}$;
4) $sqrt{26 — 6sqrt{17}} — sqrt{66 — 14sqrt{17}}$;
5) $sqrt{46 + 10sqrt{21}} + sqrt{46 — 10sqrt{21}}$.
Решение:
1) $sqrt{18 + 8sqrt{2}} = sqrt{16 + 2 + 2 * 4 * sqrt{2}} = sqrt{4^2 + 2 * 4 * sqrt{2} + (sqrt{2})^2} = sqrt{(4 + sqrt{2})^2} = |4 + sqrt{2}| = 4 + sqrt{2}$
2) $sqrt{38 — 12sqrt{2}} = sqrt{36 + 2 — 2 * 6 * sqrt{2}} = sqrt{6^2 — 2 * 6 * sqrt{2} + (sqrt{2})^2} = sqrt{(6 — sqrt{2})^2} = |6 — sqrt{2}| = 6 — sqrt{2}$
3) $sqrt{16 + 6sqrt{7}} + sqrt{23 — 8sqrt{7}} = sqrt{9 + 7 + 2 * 3 * sqrt{7}} + sqrt{16 + 7 — 2 * 4 * sqrt{7}} = sqrt{3^2 + 2 * 3 * sqrt{2} + (sqrt{7})^2} + sqrt{4^2 — 2 * 4 * sqrt{7} + (sqrt{7})^2} = sqrt{(3 + sqrt{7})^2} + sqrt{(4 — sqrt{7})^2} = |3 + sqrt{7}| + |4 — sqrt{7}| = 3 + sqrt{7} + 4 — sqrt{7} = 7$
4) $sqrt{26 — 6sqrt{17}} — sqrt{66 — 14sqrt{17}} = sqrt{9 + 17 — 2 * 3 * sqrt{17}} — sqrt{49 + 17 — 2 * 7 * sqrt{17}} = sqrt{3^2 — 2 * 3 * sqrt{17} + (sqrt{17})^2} — sqrt{7^2 — 2 * 7 * sqrt{17} + (sqrt{17})^2} = sqrt{(3 — sqrt{17})^2} — sqrt{(7 — sqrt{17})^2} = |3 — sqrt{17}| — |7 — sqrt{17}| = -3 + sqrt{17} — 7 + sqrt{17} = 2sqrt{17} — 10$
5) $sqrt{46 + 10sqrt{21}} + sqrt{46 — 10sqrt{21}} = sqrt{25 + 21 + 2 * 5 * sqrt{21}} + sqrt{25 + 21 — 2 * 5sqrt{21}} = sqrt{5^2 + 2 * 5 * sqrt{21} + (sqrt{21})^2} + sqrt{5^2 — 2 * 5sqrt{21} + (sqrt{21})^2} = sqrt{(5 + sqrt{21})^2} + sqrt{(5 — sqrt{21})^2} = |5 + sqrt{21}| + |5 — sqrt{21}| = 5 + sqrt{21} + 5 — sqrt{21} = 10$
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) $-2,4sqrt{600}$;
7) $-1,6sqrt{50}$;
8) $frac{5}{8}sqrt{3frac{21}{25}}$.
Решение:
1) $sqrt{24} = sqrt{4 * 6} = sqrt{4} * sqrt{6} = 2sqrt{6}$
2) $sqrt{63} = sqrt{9 * 7} = sqrt{9} * sqrt{7} = 3sqrt{7}$
3) $sqrt{700} = sqrt{100 * 7} = sqrt{100} * sqrt{7} = 10sqrt{7}$
4) $sqrt{0,32} = sqrt{0,16 * 2} = sqrt{0,16} * sqrt{2} = 0,4sqrt{2}$
5) $frac{1}{7}sqrt{196} = frac{1}{7} * 14 = 2$
6) $-2,4sqrt{600} = -2,4 * sqrt{100 * 6} = -2,4 * sqrt{100} * sqrt{6} = -2,4 * 10 * sqrt{6} = -24sqrt{6}$
7) $-1,6sqrt{50} = -1,6 * sqrt{25 * 2} = -1,6 * sqrt{25} * sqrt{2} = -1,6 * 5sqrt{2} = -8sqrt{2}$
8) $frac{5}{8}sqrt{3frac{21}{25}} = frac{5}{8}sqrt{frac{96}{25}} = frac{5}{8} * frac{sqrt{96}}{sqrt{25}} = frac{5}{8} * frac{sqrt{16 * 6}}{5} = frac{sqrt{16} * sqrt{6}}{8} = frac{4sqrt{6}}{8} = frac{sqrt{6}}{2}$