Деление рациональных дробей. Практика

<div class="test"><pre><textarea>{"questions":[{"content":"Выполните действие:$$\\frac{2}{3a}:\\frac{3}{a}=$$[[fill_choice_big-1]]","widgets":{"fill_choice_big-1":{"type":"fill_choice_big","options":["$\\frac{2}{9}$","$\\frac{6}{3a^2}$","$\\frac{2}{9a^2}$"],"placeholder":0,"answer":0}},"step":1,"hints":["Заменим деление умножением, а вторую дробь перевернем:$$\\frac{2}{3a}:\\frac{\\textcolor{blue}{3}}{\\textcolor{darkgreen}{a}}=\\frac{2}{3a}\\cdot \\frac{\\textcolor{darkgreen}{a}}{\\textcolor{blue}{3}}$$","Произведем вычисления:$$\\frac{2}{3\\cancel{\\textcolor{orange}{a}}}\\cdot \\frac{\\cancel{\\textcolor{orange}{a}}}{3}=\\frac{2 \\cdot 1}{3 \\cdot 3}=\\frac{2}{9}$$"],"id":"0"},{"content":"Выполните действие:$$\\frac{8}{y}:\\frac{x^2}{y}=$$[[fill_choice_big-31]]","widgets":{"fill_choice_big-31":{"type":"fill_choice_big","options":["$\\frac{8}{x^2}$","$\\frac{8x^2}{y^2}$","$\\frac{8y}{y^2}$"],"placeholder":0,"answer":0}},"step":1,"hints":["Заменим деление умножением, а вторую дробь перевернем:$$\\frac{8}{y}:\\frac{\\textcolor{blue}{x^2}}{\\textcolor{darkgreen}{y}}=\\frac{8}{y} \\cdot \\frac{\\textcolor{darkgreen}{y}}{\\textcolor{blue}{x^2}}$$","Произведем вычисления:$$\\frac{8}{\\cancel{\\textcolor{orange}{y}}} \\cdot \\frac{\\cancel{\\textcolor{orange}{y}}}{x^2}=\\frac{8}{x^2}$$"],"id":"0"},{"content":"Выполните действие:$$\\frac{10b}{3}:\\frac{5}{7a}=$$[[fill_choice_big-87]]","widgets":{"fill_choice_big-87":{"type":"fill_choice_big","options":["$\\frac{14ab}{3}$","$\\frac{50b}{21a}$","$\\frac{3}{14ab}$"],"placeholder":0,"answer":0}},"step":1,"hints":["Заменим деление умножением, а вторую дробь перевернем:$$\\frac{10b}{3}:\\frac{\\textcolor{blue}{5}}{\\textcolor{darkgreen}{7a}}=\\frac{10b}{3} \\cdot \\frac{\\textcolor{darkgreen}{7a}}{\\textcolor{blue}{5}}$$","Произведем вычисления:$$\\frac{{}^{2}\\cancel{\\textcolor{orange}{10}}b}{3} \\cdot \\frac{7a}{\\cancel{\\textcolor{orange}{5}}}=\\frac{14ab}{3}$$"],"id":"0"},{"content":"Разделите $\\frac{2x}{3}$ на $\\frac{y-4}{9}$.[[fill_choice_big-183]]","widgets":{"fill_choice_big-183":{"type":"fill_choice_big","options":["$\\frac{6x}{y-4}$","$\\frac{2xy-8}{27}$","$\\frac{y-4}{3x}$"],"placeholder":0,"answer":0}},"step":1,"hints":["Запишем пример:$$\\frac{2x}{3}:\\frac{y-4}{9}$$","Заменим деление умножением, а вторую дробь перевернем:$$\\frac{2x}{3}:\\frac{\\textcolor{blue}{y-4}}{\\textcolor{darkgreen}{9}}=\\frac{2x}{3} \\cdot \\frac{\\textcolor{darkgreen}{9}}{\\textcolor{blue}{y-4}}$$","Сократим и произведем вычисления:$$\\frac{2x}{\\cancel{\\textcolor{orange}{3}}} \\cdot \\frac{{}^{3}\\cancel{\\textcolor{orange}{9}}}{y-4}=\\frac{6x}{y-4}$$"],"id":"1"},{"content":"Разделите $\\frac{2+x}{8y}$ на $\\frac{3}{4y}$.[[fill_choice_big-240]]","widgets":{"fill_choice_big-240":{"type":"fill_choice_big","options":["$\\frac{2+x}{6}$","$\\frac{6+3x}{32y^2}$","$\\frac{8y}{2+x}$"],"placeholder":0,"answer":0}},"step":1,"hints":["Запишем пример:$$\\frac{2+x}{8y}:\\frac{3}{4y}$$","Заменим деление умножением, а вторую дробь перевернем:$$\\frac{2+x}{8y}:\\frac{\\textcolor{blue}{3}}{\\textcolor{darkgreen}{4y}}=\\frac{2+x}{8y} \\cdot \\frac{\\textcolor{darkgreen}{4y}}{\\textcolor{blue}{3}}$$","Сократим и произведем вычисления:$$\\frac{2+x}{{}^{2}\\cancel{\\textcolor{orange}{8y}}} \\cdot \\frac{\\cancel{\\textcolor{orange}{4y}}}{3}=\\frac{2+x}{6}$$"],"id":"1"},{"content":"Разделите $\\frac{18}{5m+n}$ на $\\frac{3}{5m+n}$.[[fill_choice_big-337]]","widgets":{"fill_choice_big-337":{"type":"fill_choice_big","options":["$6$","$\\frac{25m+n^2}{54}$","$\\frac{54}{(5m+1)^2}$"],"placeholder":0,"answer":0}},"step":1,"hints":["Запишем пример:$$\\frac{18}{5m+n}:\\frac{3}{5m+n}$$","Заменим деление умножением, а вторую дробь перевернем:$$\\frac{18}{5m+n}:\\frac{\\textcolor{blue}{3}}{\\textcolor{darkgreen}{5m+n}}=\\frac{18}{5m+n} \\cdot \\frac{\\textcolor{darkgreen}{5m+n}}{\\textcolor{blue}{3}}$$","Сократим и произведем вычисления:$$\\frac{18}{\\cancel{\\textcolor{orange}{5m+n}}} \\cdot \\frac{\\cancel{\\textcolor{orange}{5m+n}}}{3}=\\frac{18}{3}=6$$"],"id":"1"},{"content":"Вычислите:$$\\frac{x}{2}:(m-15)=$$[[fill_choice_big-1199]]","widgets":{"fill_choice_big-1199":{"type":"fill_choice_big","options":["$\\frac{x}{2m-30}$","$\\frac{mx-15x}{2}$","$mx+15$"],"placeholder":0,"answer":0}},"step":1,"hints":["Представим делитель в виде дроби:$$\\frac{x}{2}:(m-15)=\\frac{x}{2}:\\frac{m-15}{1}$$","Заменим деление умножением, а вторую дробь перевернем:$$\\frac{x}{2}:\\frac{\\textcolor{blue}{m-15}}{\\textcolor{darkgreen}{1}}=\\frac{x}{2}\\cdot \\frac{\\textcolor{darkgreen}{1}}{\\textcolor{blue}{m-15}}$$","Произведем вычисления:$$\\frac{x \\cdot 1}{2 \\cdot (m-15)}=\\frac{x}{2m-30}$$"],"id":"2"},{"content":"Вычислите:$$\\frac{22}{3y}:2y=$$[[fill_choice_big-1335]]","widgets":{"fill_choice_big-1335":{"type":"fill_choice_big","options":["$\\frac{11}{3y^2}$","$\\frac{44}{3}$","$\\frac{22}{3y}$"],"placeholder":0,"answer":0}},"step":1,"hints":["Представим делитель в виде дроби:$$\\frac{22}{3y}:2y=\\frac{22}{3y}:\\frac{2y}{1}$$","Заменим деление умножением, а вторую дробь перевернем:$$\\frac{22}{3y}:\\frac{\\textcolor{blue}{2y}}{\\textcolor{darkgreen}{1}}=\\frac{22}{3y}\\cdot \\frac{\\textcolor{darkgreen}{1}}{\\textcolor{blue}{2y}}$$","Произведем вычисления:$$\\frac{{}^{11}\\cancel{\\textcolor{orange}{22}} \\cdot 1}{3y \\cdot \\cancel{\\textcolor{orange}{2}}y}=\\frac{11}{3y^2}$$"],"id":"2"},{"content":"Вычислите:$$(3z+2):\\frac{15t^2}{2}=$$[[fill_choice_big-1541]]","widgets":{"fill_choice_big-1541":{"type":"fill_choice_big","options":["$\\frac{6z+4}{15t^2}$","$\\frac{15t^2}{6z+4}$","$\\frac{45t^2z+30t^2}{2}$"],"placeholder":0,"answer":0}},"step":1,"hints":["Представим делимое в виде дроби:$$(3z+2):\\frac{15t^2}{2}=\\frac{3z+2}{1}:\\frac{15t^2}{2}$$","Заменим деление умножением, а вторую дробь перевернем:$$\\frac{3z+2}{1}:\\frac{\\textcolor{blue}{15t^2}}{\\textcolor{darkgreen}{2}}=\\frac{3z+2}{1} \\cdot \\frac{\\textcolor{darkgreen}{2}}{\\textcolor{blue}{15t^2}}$$","Произведем вычисления:$$\\frac{(3z+2)\\cdot 2}{1 \\cdot 15t^2}=\\frac{6z+4}{15t^2}$$"],"id":"2"},{"content":"Запишите деление в виде дроби и сократите:$$(x^2-12x+36):(x-6)$$[[fill_choice_big-995]][[image-1723]]","widgets":{"fill_choice_big-995":{"type":"fill_choice_big","options":["$x-6$","$1$","$\\frac{1}{x-6}$"],"placeholder":0,"answer":0},"image-1723":{"type":"image","url":"https://obrazavr.ru/wp-content/uploads/2023/04/chitaet-knigi-sidit-za-partoj-parta.svg","width":"400"}},"step":1,"hints":["Запишем деление в виде дроби:$$\\frac{x^2-12x+36}{x-6}$$","Свернем числитель по формуле сокращенного умножения «квадрат разности»:$$(\\textcolor{blue}{x}-\\textcolor{darkgreen}{y})^\\textcolor{coral}{2}=\\textcolor{blue}{x}^\\textcolor{coral}{2}-2\\textcolor{blue}{x}\\textcolor{darkgreen}{y}+\\textcolor{darkgreen}{y}^\\textcolor{coral}{2}$$ $$\\frac{\\textcolor{blue}{x}^\\textcolor{coral}{2}-2 \\cdot \\textcolor{darkgreen}{6}\\cdot \\textcolor{blue}{x}+\\textcolor{darkgreen}{6}^\\textcolor{coral}{2}}{x-6}=\\frac{(\\textcolor{blue}{x}-\\textcolor{darkgreen}{6})^\\textcolor{coral}{2}}{x-6}$$","Сократим:$$\\frac{(\\textcolor{orange}{x-6})^{\\cancel{2}}}{\\cancel{\\textcolor{orange}{x-6}}}=x-6$$"],"id":"3"},{"content":"Запишите деление в виде дроби и сократите:$$(2+y):(4+4y+y^2)$$[[fill_choice_big-799]][[image-1780]]","widgets":{"fill_choice_big-799":{"type":"fill_choice_big","options":["$\\frac{1}{2+y}$","$1$","$2+y$"],"placeholder":0,"answer":0},"image-1780":{"type":"image","url":"https://obrazavr.ru/wp-content/uploads/2022/01/uchenik-01-01-1.svg","width":"400"}},"step":1,"hints":["Запишем деление в виде дроби:$$\\frac{2+y}{4+4y+y^2}$$","Свернем нижнее уравнение по формуле сокращенного умножения «квадрат суммы»:$$(\\textcolor{blue}{x}+\\textcolor{darkgreen}{y})^\\textcolor{coral}{2}=\\textcolor{blue}{x}^\\textcolor{coral}{2}+2\\textcolor{blue}{x}\\textcolor{darkgreen}{y}+\\textcolor{darkgreen}{y}^\\textcolor{coral}{2}$$ $$\\frac{2+y}{\\textcolor{blue}{2}^\\textcolor{coral}{2}+2\\cdot \\textcolor{blue}{2}\\textcolor{darkgreen}{y}+\\textcolor{darkgreen}{y}^\\textcolor{coral}{2}}=\\frac{2+y}{(\\textcolor{blue}{2}+\\textcolor{darkgreen}{y})^\\textcolor{coral}{2}}$$","Сократим:$$\\frac{\\cancel{\\textcolor{orange}{2+y}}}{(\\textcolor{orange}{2+y})^{\\cancel{2}}}=\\frac{1}{2+y}$$"],"id":"3"},{"content":"Запишите деление в виде дроби и сократите:$$(n^2-9m^2):(n-3m)(n+3m)$$[[fill_choice_big-676]][[image-1670]]","widgets":{"fill_choice_big-676":{"type":"fill_choice_big","options":["$1$","$n-3m$","$\\frac{1}{n+3m}$"],"placeholder":0,"answer":0},"image-1670":{"type":"image","url":"https://obrazavr.ru/wp-content/uploads/2023/11/obrazavr_angl_3_klass_papa_uchitel.svg","width":"400"}},"step":1,"hints":["Запишем деление в виде дроби:$$\\frac{n^2-9m^2}{(n-3m)(n+3m)}$$","Свернем нижнее уравнение по формуле сокращенного умножения «разность квадратов»:$$(\\textcolor{blue}{x}-\\textcolor{darkgreen}{y})(\\textcolor{blue}{x}+\\textcolor{darkgreen}{y})=\\textcolor{blue}{x}^\\textcolor{coral}{2}-\\textcolor{darkgreen}{y}^\\textcolor{coral}{2}$$ $$\\frac{n^2-9m^2}{(\\textcolor{blue}{n}-\\textcolor{darkgreen}{3m})(\\textcolor{blue}{n}+\\textcolor{darkgreen}{3m})}=\\frac{n^2-9m^2}{\\textcolor{blue}{n}^\\textcolor{coral}{2}-\\textcolor{darkgreen}{9m}^\\textcolor{coral}{2}}$$","Сократим:$$\\frac{\\cancel{n^2-9m^2}}{\\cancel{n^2-9m^2}}=1$$"],"id":"3"}]}</textarea></pre></div>