Ответы к странице 140
549. Вынесите множитель из−под знака корня:
1) $sqrt{18x^{12}}$;
2) $sqrt{y^9}$.
Решение:
1) $sqrt{18x^{12}} = sqrt{9 * 2 * (x^6)^2} = sqrt{9} * sqrt{2} * sqrt{(x^6)^2} = 3sqrt{2}x^6$
2) $sqrt{y^9} = sqrt{y^8 * y} = sqrt{(y^4)^2 * y} = sqrt{(y^4)^2} * sqrt{y} = y^4sqrt{y}$
550. Упростите выражение:
1) $sqrt{98} — sqrt{50} + sqrt{32}$;
2) $3sqrt{8} + sqrt{128} — frac{1}{3}sqrt{162}$;
3) $0,7sqrt{300} — 7sqrt{frac{3}{49}} + frac{2}{3}sqrt{108}$;
4) $sqrt{5a} — 2sqrt{20a} + 3sqrt{80a}$;
5) $sqrt{a^3b} — frac{2}{a}sqrt{a^5b}$, если a > 0;
6) $sqrt{c^5} + 4csqrt{c^3} — 5c^2sqrt{c}$.
Решение:
1) $sqrt{98} — sqrt{50} + sqrt{32} = sqrt{49 * 2} — sqrt{25 * 2} + sqrt{16 * 2} = 7sqrt{2} — 5sqrt{2} + 4sqrt{2} = 6sqrt{2}$
2) $3sqrt{8} + sqrt{128} — frac{1}{3}sqrt{162} = 3sqrt{4 * 2} + sqrt{64 * 2} — frac{1}{3}sqrt{81 * 2} = 3 * 2sqrt{2} + 8sqrt{2} — frac{1}{3} * 9sqrt{2} = 6sqrt{2} + 8sqrt{2} — 3sqrt{2} = 11sqrt{2}$
3) $0,7sqrt{300} — 7sqrt{frac{3}{49}} + frac{2}{3}sqrt{108} = 0,7sqrt{100 * 3} — 7sqrt{frac{1}{49} * 3} + frac{2}{3}sqrt{36 * 3} = 0,7 * 10sqrt{3} — 7 * frac{1}{7}sqrt{3} + frac{2}{3} * 6sqrt{3} = 7sqrt{3} — sqrt{3} + 2 * 2sqrt{3} = 6sqrt{3} + 4sqrt{3} = 10sqrt{3}$
4) $sqrt{5a} — 2sqrt{20a} + 3sqrt{80a} = sqrt{5a} — 2sqrt{4 * 5a} + 3sqrt{16 * 5a} = sqrt{5a} — 2 * 2sqrt{5a} + 3 * 4sqrt{5a} = sqrt{5a} — 4sqrt{5a} + 12sqrt{5a} = 9sqrt{5a}$
5) $sqrt{a^3b} — frac{2}{a}sqrt{a^5b} = sqrt{a^2 * ab} — frac{2}{a}sqrt{a^4 * ab} = asqrt{ab} — frac{2}{a} * a^2sqrt{ab} = asqrt{ab} — 2asqrt{ab} = -asqrt{ab}$, если a > 0
6) $sqrt{c^5} + 4csqrt{c^3} — 5c^2sqrt{c} = sqrt{c^4 * c} + 4csqrt{c^2 * c} — 5c^2sqrt{c} = c^2sqrt{c} + 4c * csqrt{c} — 5c^2sqrt{c} = c^2sqrt{c} + 4c^2sqrt{c} — 5c^2sqrt{c} = 0$
551. Упростите выражение:
1) $0,5sqrt{12} — 3sqrt{27} + 0,4sqrt{75}$;
2) $2,5sqrt{28b} + frac{2}{3}sqrt{63b} — 10sqrt{0,07b}$;
3) $sqrt{81a^7} — 5a^3sqrt{a} + frac{6}{a}sqrt{a^9}$.
Решение:
1) $0,5sqrt{12} — 3sqrt{27} + 0,4sqrt{75} = 0,5sqrt{4 * 3} — 3sqrt{9 * 3} + 0,4sqrt{25 * 3} = 0,5 * 2sqrt{3} — 3 * 3sqrt{3} + 0,4 * 5sqrt{3} = sqrt{3} — 9sqrt{3} + 2sqrt{3} = -6sqrt{3}$
2) $2,5sqrt{28b} + frac{2}{3}sqrt{63b} — 10sqrt{0,07b} = 2,5sqrt{4 * 7b} + frac{2}{3}sqrt{9 * 7b} — 10sqrt{0,01 * 7b} = 2,5 * 2sqrt{7b} + frac{2}{3} * 3sqrt{7b} — 10 * 0,1sqrt{7b} = 5sqrt{7b} + 2sqrt{7b} — sqrt{7b} = 6sqrt{7b}$
3) $sqrt{81a^7} — 5a^3sqrt{a} + frac{6}{a}sqrt{a^9} = sqrt{81a^6 * a} — 5a^3sqrt{a} + frac{6}{a}sqrt{a^8 * a} = sqrt{81 * (a^3)^2 * a} — 5a^3sqrt{a} + frac{6}{a}sqrt{(a^4)^2 * a} = 9a^3sqrt{a} — 5a^3sqrt{a} + frac{6}{a} * a^4sqrt{a} = 4a^3sqrt{a} + 6a^3sqrt{a} = 10a^3sqrt{a}$
552. Докажите, что:
1) $sqrt{11 + 4sqrt{7}} = sqrt{7} + 2$;
2) $sqrt{14 + 8sqrt{3}} = sqrt{8} + sqrt{6}$.
Решение:
1) $sqrt{11 + 4sqrt{7}} = sqrt{7 + 4 + 4sqrt{7}} = sqrt{(sqrt{7})^2 + 2 * 2sqrt{7} + 2^2} = sqrt{(sqrt{7} + 2)^2} = |sqrt{7} + 2| = sqrt{7} + 2$
2) $sqrt{14 + 8sqrt{3}} = sqrt{8 + 6 + 2 * 4sqrt{3}} = sqrt{(sqrt{8})^2 + 2sqrt{16 * 3} + (sqrt{6})^2} = sqrt{(sqrt{8})^2 + 2sqrt{48} + (sqrt{6})^2} = sqrt{(sqrt{8})^2 + 2sqrt{8 * 6} + (sqrt{6})^2} = sqrt{(sqrt{8} + sqrt{6})^2} = |sqrt{8} + sqrt{6}| = sqrt{8} + sqrt{6}$
553. Упростите выражение:
1) $(2sqrt{3} — 1)(sqrt{27} + 2)$;
2) $(sqrt{5} — 2)^2 — (3 + sqrt{5})^2$;
3) $sqrt{sqrt{17} — 4} * sqrt{sqrt{17} + 4}$;
4) $(7 + 4sqrt{3})(2 — sqrt{3})^2$;
5) $(sqrt{6 + 2sqrt{5}} — sqrt{6 — 2sqrt{5}})^2$.
Решение:
1) $(2sqrt{3} — 1)(sqrt{27} + 2) = (2sqrt{3} — 1)(sqrt{9 * 3} + 2) = (2sqrt{3} — 1)(3sqrt{3} + 2) = 2sqrt{3} * 3sqrt{3} + 2sqrt{3} * 2 — 1 * 3sqrt{3} + (-1) * 2 = 6 * 3 + 4sqrt{3} — 3sqrt{3} — 2 = 18 + sqrt{3} — 2 = 16 + sqrt{3}$
2) $(sqrt{5} — 2)^2 — (3 + sqrt{5})^2 = (sqrt{5})^2 — 2 * 2sqrt{5} + 2^2) — (3^2 + 3 * 2sqrt{5} + (sqrt{5})^2) = 5 — 4sqrt{5} + 4 — (9 + 6sqrt{5} + 5) = 9 — 4sqrt{5} — (14 + 6sqrt{5}) = 9 — 4sqrt{5} — 14 — 6sqrt{5} = -5 — 10sqrt{5}$
3) $sqrt{sqrt{17} — 4} * sqrt{sqrt{17} + 4} = sqrt{(sqrt{17} — 4)((sqrt{17} + 4))} = sqrt{(sqrt{17})^2 — 4^2} = sqrt{17 — 16} = sqrt{1} = 1$
4) $(7 + 4sqrt{3})(2 — sqrt{3})^2 = (7 + 4sqrt{3})(2^2 — 2 * 2sqrt{3} + (sqrt{3})^2) = (7 + 4sqrt{3})(4 — 4sqrt{3} + 3) = (7 + 4sqrt{3})(7 — 4sqrt{3}) = 7^2 — (4sqrt{3})^2 = 49 — 16 * 3 = 49 — 48 = 1$
5) $(sqrt{6 + 2sqrt{5}} — sqrt{6 — 2sqrt{5}})^2 = (sqrt{6 + 2sqrt{5}})^2 — 2sqrt{6 + 2sqrt{5}}sqrt{6 — 2sqrt{5}} + (sqrt{6 — 2sqrt{5}})^2 = 6 + 2sqrt{5} — 2sqrt{(6 — 2sqrt{5})(6 + 2sqrt{5})} + 6 — 2sqrt{5} = 12 — 2sqrt{6^2 — (2sqrt{5})^2} = 12 — 2sqrt{36 — 4 * 5} = 12 — 2sqrt{36 — 20} = 12 — 2sqrt{16} = 12 — 2 * 4 = 12 — 8 = 4$
554. Найдите значение выражения:
1) $(3sqrt{2} + 1)(sqrt{8} — 2)$;
2) $(3 — 2sqrt{7})^2 + (3 + 2sqrt{7})^2$;
3) $(10 — 4sqrt{6})(2 + sqrt{6})^2$;
4) $(sqrt{9 — 4sqrt{2}} + sqrt{9 + 4sqrt{2}})^2$.
Решение:
1) $(3sqrt{2} + 1)(sqrt{8} — 2) = (3sqrt{2} + 1)(sqrt{4 * 2} — 2) = (3sqrt{2} + 1)(2sqrt{2} — 2) = 3sqrt{2} * 2sqrt{2} — 3sqrt{2} * 2 + 1 * 2sqrt{2} — 1 * 2 = 6 * 2 — 6sqrt{2} + 2sqrt{2} — 2 = 12 — 4sqrt{2} — 2 = 10 — 4sqrt{2}$
2) $(3 — 2sqrt{7})^2 + (3 + 2sqrt{7})^2 = 3^2 — 2 * 3 * 2sqrt{7} + (2sqrt{7})^2 + 3^2 + 2 * 3 * 2sqrt{7} + (2sqrt{7})^2 = 9 — 12sqrt{7} + 4 * 7 + 9 + 12sqrt{7} + 4 * 7 = 18 + 28 + 28 = 74$
3) $(10 — 4sqrt{6})(2 + sqrt{6})^2 = (10 — 4sqrt{6})(2^2 + 2 * 2sqrt{6} + (sqrt{6})^2) = (10 — 4sqrt{6})(4 + 4sqrt{6} + 6) = (10 — 4sqrt{6})(10 + 4sqrt{6}) = 10^2 — (4sqrt{6})^2 = 100 — 16 * 6 = 100 — 96 = 4$
4) $(sqrt{9 — 4sqrt{2}} + sqrt{9 + 4sqrt{2}})^2 = (sqrt{9 — 4sqrt{2}})^2 + 2sqrt{9 — 4sqrt{2}}sqrt{9 + 4sqrt{2}} + (sqrt{9 + 4sqrt{2}})^2 = 9 — 4sqrt{2} + 2sqrt{(9 — 4sqrt{2})(9 + 4sqrt{2})} + 9 + 4sqrt{2} = 18 + 2sqrt{9^2 — (4sqrt{2})^2} = 18 + 2sqrt{81 — 16 * 2} = 18 + 2sqrt{81 — 32} = 18 + 2sqrt{49} = 18 + 2 * 7 = 18 + 14 = 32$
555. Сократите дробь:
1) $frac{4a + 4sqrt{5}}{a^2 — 5}$;
2) $frac{sqrt{28} — 2sqrt{2a}}{6a — 21}$;
3) $frac{a + 4sqrt{ab} + 4b}{a — 4b}$, если a > 0, b > 0;
4) $frac{x^2 — 6y}{x^2 + 6y — xsqrt{24y}}$;
5) $frac{sqrt{a} + sqrt{b}}{sqrt{a^3} + sqrt{b^3}}$;
6) $frac{msqrt{m} — 27}{sqrt{m} — 3}$.
Решение:
1) $frac{4a + 4sqrt{5}}{a^2 — 5} = frac{4(a + sqrt{5})}{(a — sqrt{5})(a + sqrt{5})} = frac{4}{a -sqrt{5}}$
2) $frac{sqrt{28} — 2sqrt{2a}}{6a — 21} = frac{sqrt{4 * 7} — 2sqrt{2a}}{3(2a — 7)} = frac{2sqrt{7} — 2sqrt{2a}}{3((sqrt{2a})^2 — (sqrt{7})^2)} = frac{2(sqrt{7} — sqrt{2a})}{3(sqrt{2a} — sqrt{7})(sqrt{2a} + sqrt{7})} = -frac{2(sqrt{2a} — sqrt{7})}{3(sqrt{2a} — sqrt{7})(sqrt{2a} + sqrt{7})} = -frac{2}{3(sqrt{2a} + sqrt{7})}$
3) $frac{a + 4sqrt{ab} + 4b}{a — 4b} = frac{(sqrt{a})^2 + 2 * sqrt{a} * 2sqrt{b} + (2sqrt{b})^2}{(sqrt{a})^2 — (2sqrt{b})^2} = frac{(sqrt{a} + 2sqrt{b})^2}{(sqrt{a} — 2sqrt{b})(sqrt{a} + 2sqrt{b})} = frac{sqrt{a} + 2sqrt{b}}{sqrt{a} — 2sqrt{b}}$, если a > 0, b > 0
4) $frac{x^2 — 6y}{x^2 + 6y — xsqrt{24y}} = frac{x^2 — (sqrt{6y})^2}{x^2 — xsqrt{4 * 6y} + (sqrt{6y})^2} = frac{(x — sqrt{6y})(x + sqrt{6y})}{x^2 — 2xsqrt{6y} + (sqrt{6y})^2} = frac{(x — sqrt{6y})(x + sqrt{6y})}{(x — sqrt{6y})^2} = frac{x + sqrt{6y}}{x — sqrt{6y}}$
5) $frac{sqrt{a} + sqrt{b}}{sqrt{a^3} + sqrt{b^3}} = frac{sqrt{a} + sqrt{b}}{(sqrt{a})^3 + (sqrt{b})^3} = frac{sqrt{a} + sqrt{b}}{(sqrt{a} + sqrt{b})((sqrt{a})^2 — sqrt{a}sqrt{b} + (sqrt{b})^2)} = frac{sqrt{a} + sqrt{b}}{(sqrt{a} + sqrt{b})(a — sqrt{a}sqrt{b} + b)} = frac{1}{a — sqrt{a}sqrt{b} + b}$
6) $frac{msqrt{m} — 27}{sqrt{m} — 3} = frac{sqrt{m^2 * m} — 27}{sqrt{m} — 3} = frac{sqrt{m^3} — 27}{sqrt{m} — 3} = frac{(sqrt{m})^3 — 3^3}{sqrt{m} — 3} = frac{(sqrt{m} — 3)((sqrt{m})^2) + 3sqrt{m} + 3^2)}{sqrt{m} — 3} = m + 3sqrt{m} + 9$
556. Сократите дробь:
1) $frac{a — b}{sqrt{11b} — sqrt{11a}}$;
2) $frac{2a + 10sqrt{2ab} + 25b}{6a — 75b}$, если a > 0, b > 0;
3) $frac{a — 2sqrt{a} + 4}{asqrt{a} + 8}$.
Решение:
1) $frac{a — b}{sqrt{11b} — sqrt{11a}} = frac{(sqrt{a})^2 — (sqrt{b})^2}{sqrt{11} * sqrt{b} — sqrt{11} * sqrt{a}} = frac{(sqrt{a} — sqrt{b})(sqrt{a} + sqrt{b})}{sqrt{11}(sqrt{b} — sqrt{a})} = -frac{(sqrt{a} — sqrt{b})(sqrt{a} + sqrt{b})}{sqrt{11}(sqrt{a} — sqrt{b})} = -frac{sqrt{a} + sqrt{b}}{sqrt{11}}$
2) $frac{2a + 10sqrt{2ab} + 25b}{6a — 75b} = frac{(sqrt{2a})^2 + 10sqrt{2ab} + (sqrt{25b})^2}{3(2a — 25b)} = frac{(sqrt{2a})^2 + 2 * sqrt{2a} * 5sqrt{b} + (5sqrt{b})^2}{3((sqrt{2a})^2 — (sqrt{25b})^2)} = frac{(sqrt{2a} + 5sqrt{b})^2}{3((sqrt{2a})^2 — (5sqrt{b})^2)} = frac{(sqrt{2a} + 5sqrt{b})^2}{3(sqrt{2a} — 5sqrt{b})(sqrt{2a} + 5sqrt{b})} = frac{sqrt{2a} + 5sqrt{b}}{3(sqrt{2a} — 5sqrt{b})}$, если a > 0, b > 0
3) $frac{a — 2sqrt{a} + 4}{asqrt{a} + 8} = frac{a — 2sqrt{a} + 4}{sqrt{a^2 * a} + 8} = frac{a — 2sqrt{a} + 4}{sqrt{a^3} + 8} = frac{a — 2sqrt{a} + 4}{(sqrt{a})^3 + 2^3} = frac{a — 2sqrt{a} + 4}{(sqrt{a} + 2)((sqrt{a})^2 — 2sqrt{a} + 2^2)} = frac{a — 2sqrt{a} + 4}{(sqrt{a} + 2)(a — 2sqrt{a} + 4)} = frac{1}{sqrt{a} + 2}$