Ответы к странице 138
536. Упростите выражение:
1) $sqrt{2}(sqrt{50} + sqrt{8})$;
2) $(sqrt{3} — sqrt{12}) * sqrt{3}$;
3) $(3sqrt{5} — 4sqrt{3}) * sqrt{5}$;
4) $2sqrt{2}(3sqrt{18} — frac{1}{4}sqrt{2} + sqrt{32})$.
Решение:
1) $sqrt{2}(sqrt{50} + sqrt{8}) = sqrt{2}(sqrt{25 * 2} + sqrt{4 * 2}) = sqrt{2}(5sqrt{2} + 2sqrt{2}) = sqrt{2} * 7sqrt{2} = 7 * 2 = 14$
2) $(sqrt{3} — sqrt{12}) * sqrt{3} = (sqrt{3} — sqrt{4 * 3}) * sqrt{3} = (sqrt{3} — 2sqrt{3}) * sqrt{3} = -sqrt{3} * sqrt{3} = -3$
3) $(3sqrt{5} — 4sqrt{3}) * sqrt{5} = 3sqrt{5} * sqrt{5} — 4sqrt{3} * sqrt{5} = 3 * 5 — 4sqrt{3 * 5} = 15 — 4sqrt{15}$
4) $2sqrt{2}(3sqrt{18} — frac{1}{4}sqrt{2} + sqrt{32}) = 2sqrt{2}(3sqrt{9 * 2} — frac{1}{4}sqrt{2} + sqrt{16 * 2}) = 2sqrt{2}(3 * 3sqrt{2} — frac{1}{4}sqrt{2} + 4sqrt{2}) = 2sqrt{2}(9sqrt{2} — frac{1}{4}sqrt{2} + 4sqrt{2}) = 2sqrt{2} * 12frac{3}{4}sqrt{2} = 2 * 2 * frac{51}{4} = 4 * frac{51}{4} = 51$
537. Упростите выражение:
1) $sqrt{7}(sqrt{7} — sqrt{28})$;
2) $(sqrt{18} + sqrt{72}) * sqrt{2}$;
3) $(4sqrt{3} — sqrt{75} + 4) * 3sqrt{3}$;
4) $(sqrt{600} + sqrt{6} — sqrt{24}) * sqrt{6}$.
Решение:
1) $sqrt{7}(sqrt{7} — sqrt{28}) = sqrt{7}(sqrt{7} — sqrt{4 * 7}) = sqrt{7}(sqrt{7} — 2sqrt{7}) = sqrt{7} * (-sqrt{7}) = -7$
2) $(sqrt{18} + sqrt{72}) * sqrt{2} = (sqrt{9 * 2} + sqrt{36 * 2}) * sqrt{2} = (3sqrt{2} + 6sqrt{2}) * sqrt{2} = 9sqrt{2} * sqrt{2} = 9 * 2 = 18$
3) $(4sqrt{3} — sqrt{75} + 4) * 3sqrt{3} = (4sqrt{3} — sqrt{25 * 3} + 4) * 3sqrt{3} = (4sqrt{3} — 5sqrt{3} + 4) * 3sqrt{3} = (4 — sqrt{3}) * 3sqrt{3} = 4 * 3sqrt{3} — sqrt{3} * 3sqrt{3} = 12sqrt{3} — 3 * 3 = 12sqrt{3} — 9$
4) $(sqrt{600} + sqrt{6} — sqrt{24}) * sqrt{6} = (sqrt{100 * 6} + sqrt{6} — sqrt{4 * 6}) * sqrt{6} = (10sqrt{6} + sqrt{6} — 2sqrt{6}) * sqrt{6} = 9sqrt{6} * sqrt{6} = 9 * 6 = 54$
538. Выполните умножение:
1) $(2 — sqrt{3})(sqrt{3} + 1)$;
2) $(sqrt{2} + sqrt{5})(2sqrt{2} — sqrt{5})$;
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) $(4sqrt{2} — 2sqrt{3})(2sqrt{3} + 4sqrt{2})$;
8) $(m + sqrt{n})^2$;
9) $(sqrt{a} — sqrt{b})^2$;
10) $(2 — 3sqrt{3})^2$.
Решение:
1) $(2 — sqrt{3})(sqrt{3} + 1) = 2 * sqrt{3} + 2 * 1 — sqrt{3} * sqrt{3} — sqrt{3} * 1 = 2sqrt{3} + 2 — 3 — sqrt{3} = sqrt{3} — 1$
2) $(sqrt{2} + sqrt{5})(2sqrt{2} — sqrt{5}) = sqrt{2} * 2sqrt{2} + sqrt{2} * (-sqrt{5}) + sqrt{5} * 2sqrt{2} + sqrt{5} * (-sqrt{5}) = 2 * 2 — sqrt{2 * 5} + 2sqrt{2 * 5} — 5 = 4 — sqrt{10} + 2sqrt{10} — 5 = sqrt{10} — 1$
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) $(4sqrt{2} — 2sqrt{3})(2sqrt{3} + 4sqrt{2}) = (4sqrt{2} — 2sqrt{3})(4sqrt{2} + 2sqrt{3}) = (4sqrt{2})^2 — (2sqrt{3})^2 = 16 * 2 — 4 * 3 = 32 — 12 = 20$
8) $(m + sqrt{n})^2 = m^2 + 2msqrt{n} + (sqrt{n})^2 = m^2 + 2msqrt{n} + n$
9) $(sqrt{a} — sqrt{b})^2 = (sqrt{a})^2 — 2sqrt{a}sqrt{b} + (sqrt{b})^2 = a — 2sqrt{ab} + b$
10) $(2 — 3sqrt{3})^2 = 2^2 — 2 * 2 * 3sqrt{3} + (3sqrt{3})^2 = 4 — 12sqrt{3} + 9 * 3 = 4 — 12sqrt{3} + 27 = 31 — 12sqrt{3}$
539. Выполните умножение:
1) $(sqrt{7} + 3)(3sqrt{7} — 1)$;
2) $(4sqrt{2} — sqrt{3})(2sqrt{2} + 5sqrt{3})$;
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) $(3 — 2sqrt{15})^2$.
Решение:
1) $(sqrt{7} + 3)(3sqrt{7} — 1) = sqrt{7} * 3sqrt{7} + sqrt{7} * (-1) + 3 * 3sqrt{7} + 3 * (-1) = 3 * 7 — sqrt{7} + 9sqrt{7} — 3 = 21 + 8sqrt{7} — 3 = 18 + 8sqrt{7}$
2) $(4sqrt{2} — sqrt{3})(2sqrt{2} + 5sqrt{3}) = 4sqrt{2} * 2sqrt{2} + 4sqrt{2} * 5sqrt{3} — sqrt{3} * 2sqrt{2} — sqrt{3} * 5sqrt{3} = 8 * 2 + 20sqrt{2 * 3} — 2sqrt{2 * 3} — 5 * 3 = 16 + 20sqrt{6} — 2sqrt{6} — 15 = 1 + 18sqrt{6}$
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) $(sqrt{6} + sqrt{2})^2 = (sqrt{6})^2 + 2sqrt{6} * sqrt{2} + (sqrt{2})^2 = 6 + 2sqrt{6 * 2} + 2 = 8 + 2sqrt{12} = 8 + 2sqrt{4 * 3} = 8 + 2 * 2sqrt{3} = 8 + 4sqrt{3}$
8) $(3 — 2sqrt{15})^2 = 3^2 — 2 * 3 * 2sqrt{15} + (2sqrt{15})^2 = 9 — 12sqrt{15} + 4 * 15 = 9 — 12sqrt{15} + 60 = 69 — 12sqrt{15}$
540. Чему равно значение выражения:
1) $(2 + sqrt{7})^2 — 4sqrt{7}$;
2) $(sqrt{6} — sqrt{3})^2 + 6sqrt{2}$?
Решение:
1) $(2 + sqrt{7})^2 — 4sqrt{7} = 2^2 + 2 * 2sqrt{7} + (sqrt{7})^2 — 4sqrt{7} = 4 + 4sqrt{7} + 7 — 4sqrt{7} = 11$
2) $(sqrt{6} — sqrt{3})^2 + 6sqrt{2} = (sqrt{6})^2 — 2 * sqrt{6} * sqrt{3} + sqrt{3})^2 + 6sqrt{2} = 6 — 2sqrt{6 * 3} + 3 + 6sqrt{2} = 9 — 2sqrt{18} + 6sqrt{2} = 9 — 2sqrt{9 * 2} + 6sqrt{2} = 9 — 2 * 3sqrt{2} + 6sqrt{2} = 9 — 6sqrt{2} + 6sqrt{2} = 9$
541. Найдите значение выражения:
1) $(3 + sqrt{5})^2 — 6sqrt{5}$;
2) $(sqrt{12} — 2sqrt{2})^2 + 8sqrt{6}$.
Решение:
1) $(3 + sqrt{5})^2 — 6sqrt{5} = 3^2 + 2 * 3sqrt{5} + (sqrt{5})^2 — 6sqrt{5} = 9 + 6sqrt{5} + 5 — 6sqrt{5} = 14$
2) $(sqrt{12} — 2sqrt{2})^2 + 8sqrt{6} = (sqrt{12})^2 — 2 * 2sqrt{12} * sqrt{2} + (2sqrt{2})^2 + 8sqrt{6} = 12 — 4sqrt{12 * 2} + 4 * 2 + 8sqrt{6} = 12 — 4sqrt{24} + 8 + 8sqrt{6} = 20 — 4sqrt{4 * 6} + 8sqrt{6} = 20 — 4 * 2sqrt{6} + 8sqrt{6} = 20 — 8sqrt{6} + 8sqrt{6} = 20$
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) $frac{a}{bsqrt{b}}$;
6) $frac{5}{sqrt{15}}$;
7) $frac{7}{sqrt{7}}$;
8) $frac{24}{5sqrt{3}}$.
Решение:
1) $frac{4}{sqrt{2}} = frac{4 * sqrt{2}}{sqrt{2} * sqrt{2}} = frac{4sqrt{2}}{(sqrt{2})^2} = frac{4sqrt{2}}{2} = 2sqrt{2}$
2) $frac{12}{sqrt{6}} = frac{12 * sqrt{6}}{sqrt{6} * sqrt{6}} = frac{12sqrt{6}}{(sqrt{6})^2} = frac{12sqrt{6}}{6} = 2sqrt{6}$
3) $frac{18}{sqrt{5}} = frac{18 * sqrt{5}}{sqrt{5} * sqrt{5}} = frac{18sqrt{5}}{(sqrt{5})^2} = frac{18sqrt{5}}{5}$
4) $frac{m}{sqrt{n}} = frac{m * sqrt{n}}{sqrt{n} * sqrt{n}} = frac{msqrt{n}}{(sqrt{n})^2} = frac{msqrt{n}}{n}$
5) $frac{a}{bsqrt{b}} = frac{a * sqrt{b}}{bsqrt{b} * sqrt{b}} = frac{a * sqrt{b}}{b(sqrt{b})^2} = frac{a * sqrt{b}}{b * b} = frac{asqrt{b}}{b^2}$
6) $frac{5}{sqrt{15}} = frac{5 * sqrt{15}}{sqrt{15} * sqrt{15}} = frac{5sqrt{15}}{(sqrt{15})^2} = frac{5sqrt{15}}{15} = frac{sqrt{15}}{3}$
7) $frac{7}{sqrt{7}} = frac{7 * sqrt{7}}{sqrt{7} * sqrt{7}} = frac{7sqrt{7}}{(sqrt{7})^2} = frac{7sqrt{7}}{7} = sqrt{7}$
8) $frac{24}{5sqrt{3}} = frac{24 * sqrt{3}}{5sqrt{3} * sqrt{3}} = frac{24sqrt{3}}{5(sqrt{3})^2} = frac{24sqrt{3}}{5 * 3} = frac{8sqrt{3}}{5}$
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) $frac{2}{3sqrt{x}}$.
Решение:
1) $frac{a}{sqrt{11}} = frac{a * sqrt{11}}{sqrt{11} * sqrt{11}} = frac{asqrt{11}}{(sqrt{11})^2} = frac{asqrt{11}}{11}$
2) $frac{18}{sqrt{6}} = frac{18 * sqrt{6}}{sqrt{6} * sqrt{6}} = frac{18sqrt{6}}{(sqrt{6})^2} = frac{18sqrt{6}}{6} = 3sqrt{6}$
3) $frac{5}{sqrt{10}} = frac{5 * sqrt{10}}{sqrt{10} * sqrt{10}} = frac{5sqrt{10}}{(sqrt{10})^2} = frac{5sqrt{10}}{10} = frac{sqrt{10}}{2}$
4) $frac{13}{sqrt{26}} = frac{13 * sqrt{26}}{sqrt{26} * sqrt{26}} = frac{13sqrt{26}}{(sqrt{26})^2} = frac{13sqrt{26}}{26} = frac{sqrt{26}}{2}$
5) $frac{30}{sqrt{15}} = frac{30 * sqrt{15}}{sqrt{15} * sqrt{15}} = frac{30sqrt{15}}{(sqrt{15})^2} = frac{30sqrt{15}}{15} = 2sqrt{15}$
6) $frac{2}{3sqrt{x}} = frac{2 * sqrt{x}}{3sqrt{x} * sqrt{x}} = frac{2sqrt{x}}{3(sqrt{x})^2} = frac{2sqrt{x}}{3x}$
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) $a — 2sqrt{a} + 1$;
8) $4m — 28sqrt{mn} + 49n$, если m ≥ 0, n ≥ 0;
9) $b + 6sqrt{b} + 9$;
10) $3 + 2sqrt{3c} + c$;
11) $2 + sqrt{2}$;
12) $6sqrt{7} — 7$;
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) $16x — 25y = (4sqrt{x})^2 — (5sqrt{y})^2 = (4sqrt{x} — 5sqrt{y})(4sqrt{x} + 5sqrt{y})$
7) $a — 2sqrt{a} + 1 = (sqrt{a})^2 — 2 * sqrt{a} * 1 + 1^2 = (sqrt{a} — 1)^2$
8) $4m — 28sqrt{mn} + 49n = (2sqrt{m})^2 — 2 * 2sqrt{m} * 7sqrt{n} + (7sqrt{n})^2 = (2sqrt{m} — 7sqrt{n})^2$
9) $b + 6sqrt{b} + 9 = (sqrt{b})^2 + 2 * 3 * sqrt{b} + 3^2 = (sqrt{b} + 3)^2$
10) $3 + 2sqrt{3c} + c = sqrt{3} + 2 * sqrt{3} * sqrt{c} + sqrt{c} = (sqrt{3} + sqrt{c})^2$
11) $2 + sqrt{2} = (sqrt{2})^2 + sqrt{2} = sqrt{2}(sqrt{2} + 1)$
12) $6sqrt{7} — 7 = 6sqrt{7} — (sqrt{7})^2 = sqrt{7}(6 — sqrt{7})$
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) $sqrt{15} — sqrt{5} = sqrt{5 * 3} — sqrt{5} = sqrt{5} * sqrt{3} — sqrt{5} = sqrt{5}(sqrt{3} — 1)$