Сокращение рациональных дробей. Практика

<div class="test"><pre><textarea>{"questions":[{"content":"Сократите дробь:$$\\frac{36xy}{2x}$$[[fill_choice_big-1]]","widgets":{"fill_choice_big-1":{"type":"fill_choice_big","options":["$18y$","$72xy$","$\\frac{18y}{x}$"],"placeholder":0,"answer":0}},"step":1,"hints":["Дробь можно сократить на одинаковые множители $\\textcolor{blue}{x}$ и $\\textcolor{coral}{2}.$","$$\\frac{{}^{18}\\cancel{\\textcolor{coral}{36}}\\cancel{\\textcolor{blue}{x}}y}{{}^{1}\\cancel{\\textcolor{coral}{2}}\\cancel{\\textcolor{blue}{x}}}=18y$$"],"id":"0"},{"content":"Сократите дробь:$$\\frac{40b}{80ab}$$[[fill_choice_big-14]]","widgets":{"fill_choice_big-14":{"type":"fill_choice_big","options":["$\\frac{1}{2a}$","$\\frac{1a}{40b}$","$a$"],"placeholder":0,"answer":0}},"step":1,"hints":["Дробь можно сократить на одинаковые множители $\\textcolor{blue}{b}$ и $\\textcolor{coral}{40}.$","$$\\frac{{}^{1}\\cancel{\\textcolor{coral}{40}\\textcolor{blue}{b}}}{{}^{2}\\cancel{\\textcolor{coral}{80}}a\\cancel{\\textcolor{blue}{b}}}=\\frac{1}{2a}$$"],"id":"0"},{"content":"Сократите дробь:$$\\frac{3n}{9n^2}$$[[fill_choice_big-25]]","widgets":{"fill_choice_big-25":{"type":"fill_choice_big","options":["$\\frac{1}{3n}$","$\\frac{3}{n}$","$n^2$"],"placeholder":0,"answer":0}},"step":1,"hints":["Дробь можно сократить на одинаковые множители $\\textcolor{blue}{n}$ и $\\textcolor{coral}{3}.$","$$\\frac{{}^{1}\\cancel{\\textcolor{coral}{3}\\textcolor{blue}{n}}}{{}^{3}\\cancel{\\textcolor{coral}{9}}\\textcolor{blue}{n}^{{}^{1}\\cancel{2}}}=\\frac{1}{3n}$$"],"id":"0"},{"content":"Представьте частное в виде дроби и сократите ее:$$4x^3 :3 x^2y^3$$[[fill_choice_big-332]]","widgets":{"fill_choice_big-332":{"type":"fill_choice_big","options":["$\\frac{4x}{3y^3}$","$\\frac{4}{3}$","$\\frac{4}{3y^2}$"],"placeholder":0,"answer":0}},"step":1,"hints":["Представим частное в виде дроби:$$4x^3 :3 x^2y^3=\\frac{4x^3 }{3 x^2y^3}$$","Сократим дробь на одинаковые множители $\\textcolor{blue}{x}$:$$\\frac{4\\textcolor{blue}{x}^{\\cancel{3}} }{3 \\cancel{\\textcolor{blue}{x}^2}y^3}=\\frac{4x}{3y^3}$$"],"id":"1"},{"content":"Представьте частное в виде дроби и сократите ее:$$7 nm^8: 11m^5$$[[fill_choice_big-339]]","widgets":{"fill_choice_big-339":{"type":"fill_choice_big","options":["$\\frac{7nm^3}{11}$","$\\frac{7}{11}$","$\\frac{7m^2}{11}$"],"placeholder":0,"answer":0}},"step":1,"hints":["Представим частное в виде дроби:$$7 nm^8: 11m^5=\\frac{7 nm^8}{11m^5}$$","Сократим дробь на одинаковые множители $\\textcolor{blue}{m}$:$$\\frac{7 n\\textcolor{blue}{m}^{{}^{3}\\cancel{8}}}{11\\cancel{\\textcolor{blue}{m}^5}}=\\frac{7nm^3}{11}$$"],"id":"1"},{"content":"Представьте частное в виде дроби и сократите ее:$$m^3(1+n) : 5m^4$$[[fill_choice_big-346]]","widgets":{"fill_choice_big-346":{"type":"fill_choice_big","options":["$\\frac{1+n}{5m}$","$\\frac{m(1+n)}{5m^2}$","$\\frac{1+n}{5m^3}$"],"placeholder":0,"answer":0}},"step":1,"hints":["Представим частное в виде дроби:$$m^3(1+n) : 5m^4=\\frac{m^3(1+n)}{5m^4}$$","Сократим дробь на одинаковые множители $\\textcolor{blue}{m}$:$$\\frac{\\cancel{\\textcolor{blue}{m}^3}(1+n)}{5\\textcolor{blue}{m}^{{}^{1}\\cancel{4}}}=\\frac{1+n}{5m}$$"],"id":"1"},{"content":"Разложите на множители числитель и знаменатель дроби и сократите ее:$$\\frac{6x^2}{2x-3xy}$$[[fill_choice_big-1825]]","widgets":{"fill_choice_big-1825":{"type":"fill_choice_big","options":["$\\frac{6x}{2-3y}$","$\\frac{6}{2-3y}$","$\\frac{6}{2-3xy}$"],"placeholder":0,"answer":0}},"step":1,"hints":["В знаменателе вынесем общий множитель $\\textcolor{blue}{x}.$","$$\\frac{6x^2}{2x-3xy}=\\frac{6x^2}{\\textcolor{blue}{x}(2-3y)}$$ $$\\frac{6\\textcolor{coral}{x}^{\\cancel{2}}}{\\cancel{\\textcolor{coral}{x}}(2-3y)}=\\frac{6x}{2-3y}$$"],"id":"2"},{"content":"Разложите на множители числитель и знаменатель дроби и сократите ее:$$\\frac{5y+xy}{y^2-3y}$$[[fill_choice_big-2002]]","widgets":{"fill_choice_big-2002":{"type":"fill_choice_big","options":["$\\frac{5+x}{y-3}$","$\\frac{5+y}{x-3}$","$\\frac{5}{3}$"],"placeholder":0,"answer":0}},"step":1,"hints":["В числителе и знаменателе вынесем общий множитель $\\textcolor{blue}{y}.$","$$\\frac{5y+xy}{y^2-3y}=\\frac{\\textcolor{blue}{y}(5+x)}{\\textcolor{blue}{y}(y-3)}$$ $$\\frac{\\cancel{\\textcolor{coral}{y}}(5+x)}{\\cancel{\\textcolor{coral}{y}}(y-3)}=\\frac{5+x}{y-3}$$"],"id":"2"},{"content":"Разложите на множители числитель и знаменатель дроби и сократите ее:$$\\frac{2x+4}{18-2y}$$[[fill_choice_big-2193]]","widgets":{"fill_choice_big-2193":{"type":"fill_choice_big","options":["$\\frac{x+2}{9-y}$","$\\frac{x+1}{9}$","$\\frac{2}{9x-y}$"],"placeholder":0,"answer":0}},"step":1,"hints":["В числителе и знаменателе вынесем общий множитель $\\textcolor{blue}{2}.$","$$\\frac{2x+4}{18-2y}=\\frac{\\textcolor{blue}{2}(x+2)}{\\textcolor{blue}{2}(9-y)}$$ $$\\frac{\\cancel{\\textcolor{coral}{2}}(x+2)}{\\cancel{\\textcolor{coral}{2}}(9-y)}=\\frac{x+2}{9-y}$$"],"id":"2"},{"content":"Сократите дробь, применив формулы сокращенного умножения:$$\\frac{x^2+6x+9}{(x+3)^5}$$[[fill_choice_big-1259]]","widgets":{"fill_choice_big-1259":{"type":"fill_choice_big","options":["$\\frac{1}{(x+3)^3}$","$x+3$","$1$"],"placeholder":0,"answer":0}},"step":1,"hints":["Свернем числитель по формуле сокращенного умножения «квадрат суммы»: $$(\\textcolor{coral}{x}+\\textcolor{darkgreen}{y})^2=\\textcolor{coral}{x}^2+2\\textcolor{coral}{x}\\textcolor{darkgreen}{y}+\\textcolor{darkgreen}{y}^2$$","$$\\frac{x^2+6x+9}{(x+3)^5}=\\frac{(x+3)^2}{(x+3)^5}$$ $$\\frac{\\cancel{(\\textcolor{blue}{x+3})^2}}{(\\textcolor{blue}{x+3})^{{}^{3}\\cancel{5}}}=\\frac{1}{(x+3)^3}$$"],"id":"3"},{"content":"Сократите дробь, применив формулы сокращенного умножения:$$\\frac{7-y}{49-14y+y^2}$$[[fill_choice_big-1418]]","widgets":{"fill_choice_big-1418":{"type":"fill_choice_big","options":["$\\frac{1}{7-y}$","$7-y$","$1$"],"placeholder":0,"answer":0}},"step":1,"hints":["Свернем знаменатель по формуле сокращенного умножения «квадрат разности»:$$(\\textcolor{coral}{x}-\\textcolor{darkgreen}{y})^2=\\textcolor{coral}{x}^2-2\\textcolor{coral}{x}\\textcolor{darkgreen}{y}+\\textcolor{darkgreen}{y}^2$$","$$\\frac{7-y}{49-14y+y^2}=\\frac{7-y}{(7-y)^2}$$ $$\\frac{\\cancel{\\textcolor{blue}{7-y}}}{(\\textcolor{blue}{7-y})^{{}^{1}\\cancel{2}}}=\\frac{1}{7-y}$$"],"id":"3"},{"content":"Сократите дробь, применив формулы сокращенного умножения:$$\\frac{x^2-25}{2(x+5)}$$[[fill_choice_big-1564]]","widgets":{"fill_choice_big-1564":{"type":"fill_choice_big","options":["$\\frac{x-5}{2}$","$2(x-5)$","$2$"],"placeholder":0,"answer":0}},"step":1,"hints":["Развернем числитель по формуле сокращенного умножения «разность квадратов»:$$\\textcolor{coral}{x}^2-\\textcolor{darkgreen}{y}^2=(\\textcolor{coral}{x}+\\textcolor{darkgreen}{y})(\\textcolor{coral}{x}-\\textcolor{darkgreen}{y})$$","$$\\frac{x^2-25}{2(x+5)}=\\frac{(x+5)(x-5)}{2(x+5)}$$ $$\\frac{(\\cancel{\\textcolor{blue}{x+5}})(x-5)}{2(\\cancel{\\textcolor{blue}{x+5}})}=\\frac{x-5}{2}$$"],"id":"3"},{"content":"Вынесите «минус» за скобки и сократите дробь:$$\\frac{x+3}{2(-x-3)}$$[[fill_choice_big-2385]]","widgets":{"fill_choice_big-2385":{"type":"fill_choice_big","options":["$-0.5$","$2$","$-1$"],"placeholder":0,"answer":0}},"step":1,"hints":["Вынесем «минус» за скобки, поменяв знаки у слагаемых, и сократим дробь.","$$\\frac{x+3}{2(-x-3)}=\\frac{x+3}{-2(x+3)}$$ $$\\frac{\\cancel{\\textcolor{coral}{x+3}}}{-2(\\cancel{\\textcolor{coral}{x+3}})}=\\frac{1}{-2}=-0.5$$"],"id":"4"},{"content":"Вынесите «минус» за скобки и сократите дробь:$$-\\frac{n(m-8)}{8-m}$$[[fill_choice_big-1006]]","widgets":{"fill_choice_big-1006":{"type":"fill_choice_big","options":["$n$","$-n$","$-\\frac{1}{n}$"],"placeholder":0,"answer":0}},"step":1,"hints":["Вынесем «минус» за скобки, поменяв знаки у слагаемых, и сократим дробь.","$$-\\frac{n(m-8)}{8-m}=-\\frac{n(m-8)}{-(-8+m)}$$ $$-\\frac{n\\cancel{(\\textcolor{coral}{m-8})}}{-\\cancel{(\\textcolor{coral}{m-8})}}=-\\frac{n}{-1}$$","Занесем «минус», стоящий перед дробью, в знаменатель:$$-\\frac{n}{-1}=\\frac{n}{--1}=n$$"],"id":"4"},{"content":"Вынесите «минус» за скобки и сократите дробь:$$-\\frac{11(11-5t)}{121(5t-11)}$$[[fill_choice_big-897]]","widgets":{"fill_choice_big-897":{"type":"fill_choice_big","options":["$\\frac{1}{11}$","$-\\frac{1}{121}$","$-11$"],"placeholder":0,"answer":0}},"step":1,"hints":["Вынесем «минус» за скобки, поменяв знаки у слагаемых, и сократим дробь.","$$-\\frac{11(11-5t)}{121(5t-11)}=$$ $$-\\frac{11(11-5t)}{-121(-5t+11)}$$ $$-\\frac{{}^{1}\\cancel{\\textcolor{blue}{11}}\\cancel{(\\textcolor{coral}{11-5t})}}{-{}^{11}\\cancel{\\textcolor{blue}{121}}\\cancel{(\\textcolor{coral}{11-5t})}}=-\\frac{1}{-11}$$","Занесем «минус», стоящий перед дробью, в знаменатель:$$-\\frac{1}{-11}=\\frac{1}{--11}=\\frac{1}{11}$$"],"id":"4"}]}</textarea></pre></div>