Ответы к странице 138
536. Упростите выражение:
1) $\sqrt{2}(\sqrt{50} + \sqrt{8})$;
2) $(\sqrt{3} - \sqrt{12}) * \sqrt{3}$;
3) $(3\sqrt{5} - 4\sqrt{3}) * \sqrt{5}$;
4) $2\sqrt{2}(3\sqrt{18} - \frac{1}{4}\sqrt{2} + \sqrt{32})$.
Решение:
1) $\sqrt{2}(\sqrt{50} + \sqrt{8}) = \sqrt{2}(\sqrt{25 * 2} + \sqrt{4 * 2}) = \sqrt{2}(5\sqrt{2} + 2\sqrt{2}) = \sqrt{2} * 7\sqrt{2} = 7 * 2 = 14$
2) $(\sqrt{3} - \sqrt{12}) * \sqrt{3} = (\sqrt{3} - \sqrt{4 * 3}) * \sqrt{3} = (\sqrt{3} - 2\sqrt{3}) * \sqrt{3} = -\sqrt{3} * \sqrt{3} = -3$
3) $(3\sqrt{5} - 4\sqrt{3}) * \sqrt{5} = 3\sqrt{5} * \sqrt{5} - 4\sqrt{3} * \sqrt{5} = 3 * 5 - 4\sqrt{3 * 5} = 15 - 4\sqrt{15}$
4) $2\sqrt{2}(3\sqrt{18} - \frac{1}{4}\sqrt{2} + \sqrt{32}) = 2\sqrt{2}(3\sqrt{9 * 2} - \frac{1}{4}\sqrt{2} + \sqrt{16 * 2}) = 2\sqrt{2}(3 * 3\sqrt{2} - \frac{1}{4}\sqrt{2} + 4\sqrt{2}) = 2\sqrt{2}(9\sqrt{2} - \frac{1}{4}\sqrt{2} + 4\sqrt{2}) = 2\sqrt{2} * 12\frac{3}{4}\sqrt{2} = 2 * 2 * \frac{51}{4} = 4 * \frac{51}{4} = 51$
537. Упростите выражение:
1) $\sqrt{7}(\sqrt{7} - \sqrt{28})$;
2) $(\sqrt{18} + \sqrt{72}) * \sqrt{2}$;
3) $(4\sqrt{3} - \sqrt{75} + 4) * 3\sqrt{3}$;
4) $(\sqrt{600} + \sqrt{6} - \sqrt{24}) * \sqrt{6}$.
Решение:
1) $\sqrt{7}(\sqrt{7} - \sqrt{28}) = \sqrt{7}(\sqrt{7} - \sqrt{4 * 7}) = \sqrt{7}(\sqrt{7} - 2\sqrt{7}) = \sqrt{7} * (-\sqrt{7}) = -7$
2) $(\sqrt{18} + \sqrt{72}) * \sqrt{2} = (\sqrt{9 * 2} + \sqrt{36 * 2}) * \sqrt{2} = (3\sqrt{2} + 6\sqrt{2}) * \sqrt{2} = 9\sqrt{2} * \sqrt{2} = 9 * 2 = 18$
3) $(4\sqrt{3} - \sqrt{75} + 4) * 3\sqrt{3} = (4\sqrt{3} - \sqrt{25 * 3} + 4) * 3\sqrt{3} = (4\sqrt{3} - 5\sqrt{3} + 4) * 3\sqrt{3} = (4 - \sqrt{3}) * 3\sqrt{3} = 4 * 3\sqrt{3} - \sqrt{3} * 3\sqrt{3} = 12\sqrt{3} - 3 * 3 = 12\sqrt{3} - 9$
4) $(\sqrt{600} + \sqrt{6} - \sqrt{24}) * \sqrt{6} = (\sqrt{100 * 6} + \sqrt{6} - \sqrt{4 * 6}) * \sqrt{6} = (10\sqrt{6} + \sqrt{6} - 2\sqrt{6}) * \sqrt{6} = 9\sqrt{6} * \sqrt{6} = 9 * 6 = 54$
538. Выполните умножение:
1) $(2 - \sqrt{3})(\sqrt{3} + 1)$;
2) $(\sqrt{2} + \sqrt{5})(2\sqrt{2} - \sqrt{5})$;
3) $(a + \sqrt{b})(a - \sqrt{b})$;
4) $(\sqrt{b} - \sqrt{c})(\sqrt{b} + \sqrt{c})$;
5) $(4 + \sqrt{3})(4 - \sqrt{3})$;
6) $(y - \sqrt{7})(y + \sqrt{7})$;
7) $(4\sqrt{2} - 2\sqrt{3})(2\sqrt{3} + 4\sqrt{2})$;
8) $(m + \sqrt{n})^2$;
9) $(\sqrt{a} - \sqrt{b})^2$;
10) $(2 - 3\sqrt{3})^2$.
Решение:
1) $(2 - \sqrt{3})(\sqrt{3} + 1) = 2 * \sqrt{3} + 2 * 1 - \sqrt{3} * \sqrt{3} - \sqrt{3} * 1 = 2\sqrt{3} + 2 - 3 - \sqrt{3} = \sqrt{3} - 1$
2) $(\sqrt{2} + \sqrt{5})(2\sqrt{2} - \sqrt{5}) = \sqrt{2} * 2\sqrt{2} + \sqrt{2} * (-\sqrt{5}) + \sqrt{5} * 2\sqrt{2} + \sqrt{5} * (-\sqrt{5}) = 2 * 2 - \sqrt{2 * 5} + 2\sqrt{2 * 5} - 5 = 4 - \sqrt{10} + 2\sqrt{10} - 5 = \sqrt{10} - 1$
3) $(a + \sqrt{b})(a - \sqrt{b}) = (a - \sqrt{b})(a + \sqrt{b}) = a^2 - (\sqrt{b})^2 = a^2 - b$
4) $(\sqrt{b} - \sqrt{c})(\sqrt{b} + \sqrt{c}) = (\sqrt{b})^2 - (\sqrt{c})^2 = b - c$
5) $(4 + \sqrt{3})(4 - \sqrt{3}) = (4 - \sqrt{3})(4 + \sqrt{3}) = 4^2 - (\sqrt{3})^2 = 16 - 3 = 13$
6) $(y - \sqrt{7})(y + \sqrt{7}) = y^2 - (\sqrt{7})^2 = y^2 - 7$
7) $(4\sqrt{2} - 2\sqrt{3})(2\sqrt{3} + 4\sqrt{2}) = (4\sqrt{2} - 2\sqrt{3})(4\sqrt{2} + 2\sqrt{3}) = (4\sqrt{2})^2 - (2\sqrt{3})^2 = 16 * 2 - 4 * 3 = 32 - 12 = 20$
8) $(m + \sqrt{n})^2 = m^2 + 2m\sqrt{n} + (\sqrt{n})^2 = m^2 + 2m\sqrt{n} + n$
9) $(\sqrt{a} - \sqrt{b})^2 = (\sqrt{a})^2 - 2\sqrt{a}\sqrt{b} + (\sqrt{b})^2 = a - 2\sqrt{ab} + b$
10) $(2 - 3\sqrt{3})^2 = 2^2 - 2 * 2 * 3\sqrt{3} + (3\sqrt{3})^2 = 4 - 12\sqrt{3} + 9 * 3 = 4 - 12\sqrt{3} + 27 = 31 - 12\sqrt{3}$
539. Выполните умножение:
1) $(\sqrt{7} + 3)(3\sqrt{7} - 1)$;
2) $(4\sqrt{2} - \sqrt{3})(2\sqrt{2} + 5\sqrt{3})$;
3) $(\sqrt{p} - q)(\sqrt{p} + q)$;
4) $(6 - \sqrt{13})(6 + \sqrt{13})$;
5) $(\sqrt{5} - x)(\sqrt{5} + x)$;
6) $(\sqrt{19} + \sqrt{17})(\sqrt{19} - \sqrt{17})$;
7) $(\sqrt{6} + \sqrt{2})^2$;
8) $(3 - 2\sqrt{15})^2$.
Решение:
1) $(\sqrt{7} + 3)(3\sqrt{7} - 1) = \sqrt{7} * 3\sqrt{7} + \sqrt{7} * (-1) + 3 * 3\sqrt{7} + 3 * (-1) = 3 * 7 - \sqrt{7} + 9\sqrt{7} - 3 = 21 + 8\sqrt{7} - 3 = 18 + 8\sqrt{7}$
2) $(4\sqrt{2} - \sqrt{3})(2\sqrt{2} + 5\sqrt{3}) = 4\sqrt{2} * 2\sqrt{2} + 4\sqrt{2} * 5\sqrt{3} - \sqrt{3} * 2\sqrt{2} - \sqrt{3} * 5\sqrt{3} = 8 * 2 + 20\sqrt{2 * 3} - 2\sqrt{2 * 3} - 5 * 3 = 16 + 20\sqrt{6} - 2\sqrt{6} - 15 = 1 + 18\sqrt{6}$
3) $(\sqrt{p} - q)(\sqrt{p} + q) = (\sqrt{p})^2 - q^2 = p - q^2$
4) $(6 - \sqrt{13})(6 + \sqrt{13}) = 6^2 - (\sqrt{13})^2 = 36 - 13 = 23$
5) $(\sqrt{5} - x)(\sqrt{5} + x) = (\sqrt{5})^2 - x^2 = 5 - x^2$
6) $(\sqrt{19} + \sqrt{17})(\sqrt{19} - \sqrt{17}) = (\sqrt{19} - \sqrt{17})(\sqrt{19} + \sqrt{17} = (\sqrt{19})^2 - (\sqrt{17})^2 = 19 - 17 = 2$
7) $(\sqrt{6} + \sqrt{2})^2 = (\sqrt{6})^2 + 2\sqrt{6} * \sqrt{2} + (\sqrt{2})^2 = 6 + 2\sqrt{6 * 2} + 2 = 8 + 2\sqrt{12} = 8 + 2\sqrt{4 * 3} = 8 + 2 * 2\sqrt{3} = 8 + 4\sqrt{3}$
8) $(3 - 2\sqrt{15})^2 = 3^2 - 2 * 3 * 2\sqrt{15} + (2\sqrt{15})^2 = 9 - 12\sqrt{15} + 4 * 15 = 9 - 12\sqrt{15} + 60 = 69 - 12\sqrt{15}$
540. Чему равно значение выражения:
1) $(2 + \sqrt{7})^2 - 4\sqrt{7}$;
2) $(\sqrt{6} - \sqrt{3})^2 + 6\sqrt{2}$?
Решение:
1) $(2 + \sqrt{7})^2 - 4\sqrt{7} = 2^2 + 2 * 2\sqrt{7} + (\sqrt{7})^2 - 4\sqrt{7} = 4 + 4\sqrt{7} + 7 - 4\sqrt{7} = 11$
2) $(\sqrt{6} - \sqrt{3})^2 + 6\sqrt{2} = (\sqrt{6})^2 - 2 * \sqrt{6} * \sqrt{3} + \sqrt{3})^2 + 6\sqrt{2} = 6 - 2\sqrt{6 * 3} + 3 + 6\sqrt{2} = 9 - 2\sqrt{18} + 6\sqrt{2} = 9 - 2\sqrt{9 * 2} + 6\sqrt{2} = 9 - 2 * 3\sqrt{2} + 6\sqrt{2} = 9 - 6\sqrt{2} + 6\sqrt{2} = 9$
541. Найдите значение выражения:
1) $(3 + \sqrt{5})^2 - 6\sqrt{5}$;
2) $(\sqrt{12} - 2\sqrt{2})^2 + 8\sqrt{6}$.
Решение:
1) $(3 + \sqrt{5})^2 - 6\sqrt{5} = 3^2 + 2 * 3\sqrt{5} + (\sqrt{5})^2 - 6\sqrt{5} = 9 + 6\sqrt{5} + 5 - 6\sqrt{5} = 14$
2) $(\sqrt{12} - 2\sqrt{2})^2 + 8\sqrt{6} = (\sqrt{12})^2 - 2 * 2\sqrt{12} * \sqrt{2} + (2\sqrt{2})^2 + 8\sqrt{6} = 12 - 4\sqrt{12 * 2} + 4 * 2 + 8\sqrt{6} = 12 - 4\sqrt{24} + 8 + 8\sqrt{6} = 20 - 4\sqrt{4 * 6} + 8\sqrt{6} = 20 - 4 * 2\sqrt{6} + 8\sqrt{6} = 20 - 8\sqrt{6} + 8\sqrt{6} = 20$
542. Освободитесь от иррациональности в знаменателе дроби:
1) $\frac{4}{\sqrt{2}}$;
2) $\frac{12}{\sqrt{6}}$;
3) $\frac{18}{\sqrt{5}}$;
4) $\frac{m}{\sqrt{n}}$;
5) $\frac{a}{b\sqrt{b}}$;
6) $\frac{5}{\sqrt{15}}$;
7) $\frac{7}{\sqrt{7}}$;
8) $\frac{24}{5\sqrt{3}}$.
Решение:
1) $\frac{4}{\sqrt{2}} = \frac{4 * \sqrt{2}}{\sqrt{2} * \sqrt{2}} = \frac{4\sqrt{2}}{(\sqrt{2})^2} = \frac{4\sqrt{2}}{2} = 2\sqrt{2}$
2) $\frac{12}{\sqrt{6}} = \frac{12 * \sqrt{6}}{\sqrt{6} * \sqrt{6}} = \frac{12\sqrt{6}}{(\sqrt{6})^2} = \frac{12\sqrt{6}}{6} = 2\sqrt{6}$
3) $\frac{18}{\sqrt{5}} = \frac{18 * \sqrt{5}}{\sqrt{5} * \sqrt{5}} = \frac{18\sqrt{5}}{(\sqrt{5})^2} = \frac{18\sqrt{5}}{5}$
4) $\frac{m}{\sqrt{n}} = \frac{m * \sqrt{n}}{\sqrt{n} * \sqrt{n}} = \frac{m\sqrt{n}}{(\sqrt{n})^2} = \frac{m\sqrt{n}}{n}$
5) $\frac{a}{b\sqrt{b}} = \frac{a * \sqrt{b}}{b\sqrt{b} * \sqrt{b}} = \frac{a * \sqrt{b}}{b(\sqrt{b})^2} = \frac{a * \sqrt{b}}{b * b} = \frac{a\sqrt{b}}{b^2}$
6) $\frac{5}{\sqrt{15}} = \frac{5 * \sqrt{15}}{\sqrt{15} * \sqrt{15}} = \frac{5\sqrt{15}}{(\sqrt{15})^2} = \frac{5\sqrt{15}}{15} = \frac{\sqrt{15}}{3}$
7) $\frac{7}{\sqrt{7}} = \frac{7 * \sqrt{7}}{\sqrt{7} * \sqrt{7}} = \frac{7\sqrt{7}}{(\sqrt{7})^2} = \frac{7\sqrt{7}}{7} = \sqrt{7}$
8) $\frac{24}{5\sqrt{3}} = \frac{24 * \sqrt{3}}{5\sqrt{3} * \sqrt{3}} = \frac{24\sqrt{3}}{5(\sqrt{3})^2} = \frac{24\sqrt{3}}{5 * 3} = \frac{8\sqrt{3}}{5}$
543. Освободитесь от иррациональности в знаменателе дроби:
1) $\frac{a}{\sqrt{11}}$;
2) $\frac{18}{\sqrt{6}}$;
3) $\frac{5}{\sqrt{10}}$;
4) $\frac{13}{\sqrt{26}}$;
5) $\frac{30}{\sqrt{15}}$;
6) $\frac{2}{3\sqrt{x}}$.
Решение:
1) $\frac{a}{\sqrt{11}} = \frac{a * \sqrt{11}}{\sqrt{11} * \sqrt{11}} = \frac{a\sqrt{11}}{(\sqrt{11})^2} = \frac{a\sqrt{11}}{11}$
2) $\frac{18}{\sqrt{6}} = \frac{18 * \sqrt{6}}{\sqrt{6} * \sqrt{6}} = \frac{18\sqrt{6}}{(\sqrt{6})^2} = \frac{18\sqrt{6}}{6} = 3\sqrt{6}$
3) $\frac{5}{\sqrt{10}} = \frac{5 * \sqrt{10}}{\sqrt{10} * \sqrt{10}} = \frac{5\sqrt{10}}{(\sqrt{10})^2} = \frac{5\sqrt{10}}{10} = \frac{\sqrt{10}}{2}$
4) $\frac{13}{\sqrt{26}} = \frac{13 * \sqrt{26}}{\sqrt{26} * \sqrt{26}} = \frac{13\sqrt{26}}{(\sqrt{26})^2} = \frac{13\sqrt{26}}{26} = \frac{\sqrt{26}}{2}$
5) $\frac{30}{\sqrt{15}} = \frac{30 * \sqrt{15}}{\sqrt{15} * \sqrt{15}} = \frac{30\sqrt{15}}{(\sqrt{15})^2} = \frac{30\sqrt{15}}{15} = 2\sqrt{15}$
6) $\frac{2}{3\sqrt{x}} = \frac{2 * \sqrt{x}}{3\sqrt{x} * \sqrt{x}} = \frac{2\sqrt{x}}{3(\sqrt{x})^2} = \frac{2\sqrt{x}}{3x}$
544. Разложите на множители выражение:
1) $a^2 - 3$;
2) $4b^2 - 2$;
3) $5 - 6c^2$;
4) a − 9, если a ≥ 0;
5) m − n, если m ≥ 0, n ≥ 0;
6) 16x − 25y, если x ≥ 0, y ≥ 0;
7) $a - 2\sqrt{a} + 1$;
8) $4m - 28\sqrt{mn} + 49n$, если m ≥ 0, n ≥ 0;
9) $b + 6\sqrt{b} + 9$;
10) $3 + 2\sqrt{3c} + c$;
11) $2 + \sqrt{2}$;
12) $6\sqrt{7} - 7$;
13) $a - \sqrt{a}$;
14) $\sqrt{b} + \sqrt{3b}$;
15) $\sqrt{15} - \sqrt{5}$.
Решение:
1) $a^2 - 3 = a^2 - (\sqrt{3})^2 = (a - \sqrt{3})(a + \sqrt{3})$
2) $4b^2 - 2 = (2b)^2 - (\sqrt{2})^2 = (2b - \sqrt{2})(2b + \sqrt{2})$
3) $5 - 6c^2 = (\sqrt{5})^2 - (\sqrt{6}c)^2 = (\sqrt{5} - \sqrt{6}c)(\sqrt{5} + \sqrt{6}c)$
4) $a - 9 = (\sqrt{a})^2 - 3^2 = (\sqrt{a} - 3)(\sqrt{a} + 3)$
5) $m - n = (\sqrt{m})^2 - (\sqrt{n})^2 = (\sqrt{m} - \sqrt{n})(\sqrt{m} + \sqrt{n})$
6) $16x - 25y = (4\sqrt{x})^2 - (5\sqrt{y})^2 = (4\sqrt{x} - 5\sqrt{y})(4\sqrt{x} + 5\sqrt{y})$
7) $a - 2\sqrt{a} + 1 = (\sqrt{a})^2 - 2 * \sqrt{a} * 1 + 1^2 = (\sqrt{a} - 1)^2$
8) $4m - 28\sqrt{mn} + 49n = (2\sqrt{m})^2 - 2 * 2\sqrt{m} * 7\sqrt{n} + (7\sqrt{n})^2 = (2\sqrt{m} - 7\sqrt{n})^2$
9) $b + 6\sqrt{b} + 9 = (\sqrt{b})^2 + 2 * 3 * \sqrt{b} + 3^2 = (\sqrt{b} + 3)^2$
10) $3 + 2\sqrt{3c} + c = \sqrt{3} + 2 * \sqrt{3} * \sqrt{c} + \sqrt{c} = (\sqrt{3} + \sqrt{c})^2$
11) $2 + \sqrt{2} = (\sqrt{2})^2 + \sqrt{2} = \sqrt{2}(\sqrt{2} + 1)$
12) $6\sqrt{7} - 7 = 6\sqrt{7} - (\sqrt{7})^2 = \sqrt{7}(6 - \sqrt{7})$
13) $a - \sqrt{a} = (\sqrt{a})^2 - \sqrt{a} = \sqrt{a}(\sqrt{a} - 1)$
14) $\sqrt{b} + \sqrt{3b} = \sqrt{b} + \sqrt{3} * \sqrt{b} = \sqrt{b}(1 + \sqrt{3})$
15) $\sqrt{15} - \sqrt{5} = \sqrt{5 * 3} - \sqrt{5} = \sqrt{5} * \sqrt{3} - \sqrt{5} = \sqrt{5}(\sqrt{3} - 1)$
