<div class="test"><pre><textarea>{"questions":[{"content":"[[speech-10]][[fill_choice_big-297]]","widgets":{"speech-10":{"type":"speech","char":"0","text":"Найдите значение выражения: $\\sqrt{0.25 \\cdot 144}$"},"fill_choice_big-297":{"type":"fill_choice_big","options":["$6$","$12$","$1$"],"placeholder":0,"answer":0}},"step":1,"hints":["Найдем корень из каждого множителя по отдельности, а затем перемножим результаты: $$\\sqrt{\\textcolor{blue}{0.25} \\cdot \\textcolor{coral}{144}}=\\sqrt{\\textcolor{blue}{0.25}} \\cdot \\sqrt{\\textcolor{coral}{144}}$$","$$\\sqrt{0.25}=0.5$$ $$\\sqrt{144}=12$$ $$0.5 \\cdot 12 = 6$$"],"id":"0"},{"content":"[[speech-21]][[fill_choice_big-372]]","widgets":{"speech-21":{"type":"speech","char":"1","text":"Найдите значение выражения: $\\sqrt{0.01 \\cdot 196}$"},"fill_choice_big-372":{"type":"fill_choice_big","options":["$1.4$","$1.96$","$1960$"],"placeholder":0,"answer":0}},"step":1,"hints":["Найдем корень из каждого множителя по отдельности, а затем перемножим результаты: $$\\sqrt{\\textcolor{blue}{0.01} \\cdot \\textcolor{coral}{196}}=\\sqrt{\\textcolor{blue}{0.01}} \\cdot \\sqrt{\\textcolor{coral}{196}}$$","$$\\sqrt{0.01}=0.1$$ $$\\sqrt{196}=14$$ $$0.1 \\cdot 14 = 1.4$$"],"id":"0"},{"content":"[[speech-38]][[fill_choice_big-449]]","widgets":{"speech-38":{"type":"speech","char":"2","text":"Найдите значение выражения: $\\sqrt{900 \\cdot 36}$"},"fill_choice_big-449":{"type":"fill_choice_big","options":["$180$","$300$","$360$"],"placeholder":0,"answer":0}},"step":1,"hints":["Найдем корень из каждого множителя по отдельности, а затем перемножим результаты: $$\\sqrt{\\textcolor{blue}{900}\\cdot \\textcolor{coral}{36}}=\\sqrt{\\textcolor{blue}{900}} \\cdot \\sqrt{\\textcolor{coral}{36}}$$","$$\\sqrt{900}=30$$ $$\\sqrt{36}=6$$ $$30 \\cdot 6 = 180$$"],"id":"0"},{"content":"Вычислите:$$\\sqrt{2} \\cdot \\sqrt{18}$$[[fill_choice_big-617]]","widgets":{"fill_choice_big-617":{"type":"fill_choice_big","options":["$6$","$18$","$9$"],"placeholder":0,"answer":0}},"step":1,"hints":["Занесем множители под один знак корня: $$\\sqrt{\\textcolor{blue}{2}} \\cdot \\sqrt{\\textcolor{coral}{18}}=\\sqrt{\\textcolor{blue}{2} \\cdot \\textcolor{coral}{18}}=\\sqrt{36}$$","$$\\sqrt{36}=6$$"],"id":"1"},{"content":"Вычислите:$$\\sqrt{32} \\cdot \\sqrt{2}$$[[fill_choice_big-784]]","widgets":{"fill_choice_big-784":{"type":"fill_choice_big","options":["$8$","$6$","$32$"],"placeholder":0,"answer":0}},"step":1,"hints":["Занесем множители под один знак корня: $$\\sqrt{\\textcolor{blue}{32}} \\cdot \\sqrt{\\textcolor{coral}{2}}=\\sqrt{\\textcolor{blue}{32} \\cdot \\textcolor{coral}{2}}=\\sqrt{64}$$","$$\\sqrt{64}=8$$"],"id":"1"},{"content":"Вычислите:$$\\sqrt{40} \\cdot \\sqrt{10}$$[[fill_choice_big-913]]","widgets":{"fill_choice_big-913":{"type":"fill_choice_big","options":["$20$","$100$","$140$"],"placeholder":0,"answer":0}},"step":1,"hints":["Занесем множители под один знак корня: $$\\sqrt{\\textcolor{blue}{40}} \\cdot \\sqrt{\\textcolor{coral}{10}}=\\sqrt{\\textcolor{blue}{40} \\cdot \\textcolor{coral}{10}}=\\sqrt{400}$$","$$\\sqrt{400}=20$$"],"id":"1"},{"content":"Найдите значение корня:$$\\sqrt{\\frac{1}{64}}$$[[fill_choice_big-1294]]","widgets":{"fill_choice_big-1294":{"type":"fill_choice_big","options":["$\\frac{1}{8}$","$64$","$8$"],"placeholder":0,"answer":0}},"step":1,"hints":["Найдем корень числителя и знаменателя по отдельности: $$\\sqrt{\\frac{\\textcolor{blue}{1}}{\\textcolor{coral}{64}}}=\\frac{\\sqrt{\\textcolor{blue}{1}}}{\\sqrt{\\textcolor{coral}{64}}}$$","$$\\sqrt{1}=1$$ $$\\sqrt{64}=8$$ $$\\sqrt{\\frac{1}{64}}=\\frac{1}{8}$$"],"id":"2"},{"content":"Найдите значение корня:$$\\sqrt{\\frac{9}{49}}$$[[fill_choice_big-1469]]","widgets":{"fill_choice_big-1469":{"type":"fill_choice_big","options":["$\\frac{3}{7}$","$\\frac{49}{3}$","$14$"],"placeholder":0,"answer":0}},"step":1,"hints":["Найдем корень числителя и знаменателя по отдельности: $$\\sqrt{\\frac{\\textcolor{blue}{9}}{\\textcolor{coral}{49}}}=\\frac{\\sqrt{\\textcolor{blue}{9}}}{\\sqrt{\\textcolor{coral}{49}}}$$","$$\\sqrt{9}=3$$ $$\\sqrt{49}=7$$ $$\\sqrt{\\frac{9}{49}}=\\frac{3}{7}$$"],"id":"2"},{"content":"Найдите значение корня:$$\\sqrt{1\\frac{9}{16}}$$[[fill_choice_big-1661]]","widgets":{"fill_choice_big-1661":{"type":"fill_choice_big","options":["$1.25$","$0.8$","$\\frac{1}{4}$"],"placeholder":0,"answer":0}},"step":1,"hints":["Превратим смешанную дробь в неправильную: $$\\sqrt{1\\frac{9}{16}}=\\sqrt{\\frac{25}{16}}$$","Найдем корень числителя и знаменателя по отдельности: $$\\sqrt{\\frac{\\textcolor{blue}{25}}{\\textcolor{coral}{16}}}=\\frac{\\sqrt{\\textcolor{blue}{25}}}{\\sqrt{\\textcolor{coral}{16}}}$$","$$\\sqrt{25}=5$$ $$\\sqrt{16}=4$$ $$\\frac{5}{4}=1.25$$"],"id":"2"},{"content":"Найдите значение частного: $$\\frac{\\sqrt{23}}{\\sqrt{2300}}$$[[fill_choice_big-2033]]","widgets":{"fill_choice_big-2033":{"type":"fill_choice_big","options":["$0.1$","$0.01$","$23$"],"placeholder":0,"answer":0}},"step":1,"hints":["Занесем числитель и знаменатель дроби под один знак корня: $$\\frac{\\sqrt{\\textcolor{blue}{23}}}{\\sqrt{\\textcolor{coral}{2300}}}=\\sqrt{\\frac{\\textcolor{blue}{23}}{\\textcolor{coral}{2300}}}$$","Сократим дробь: $$\\sqrt{\\frac{\\cancel{23}}{{}^{100}\\cancel{2300}}}=\\sqrt{\\frac{1}{100}}=\\sqrt{0.01}$$","$$\\sqrt{0.01}=0.1$$"],"id":"3"},{"content":"найдите значение частного: $$\\frac{\\sqrt{52}}{\\sqrt{117}}$$[[fill_choice_big-2380]]","widgets":{"fill_choice_big-2380":{"type":"fill_choice_big","options":["$\\frac{2}{3}$","$11$","$3$"],"placeholder":0,"answer":0}},"step":1,"hints":["Занесем числитель и знаменатель дроби под один знак корня: $$\\frac{\\sqrt{\\textcolor{blue}{52}}}{\\sqrt{\\textcolor{coral}{117}}}=\\sqrt{\\frac{\\textcolor{blue}{52}}{\\textcolor{coral}{117}}}$$","Сократим дробь на $13$: $$\\sqrt{\\frac{{}^{4}\\cancel{52}}{{}^{9}\\cancel{117}}}=\\sqrt{\\frac{4}{9}}$$","Найдем корень отдельно из числителя и знаменателя: $$\\sqrt{\\frac{\\textcolor{blue}{4}}{\\textcolor{coral}{9}}}=\\frac{\\sqrt{\\textcolor{blue}{4}}}{\\sqrt{\\textcolor{coral}{9}}}$$ $$\\sqrt{4}=2$$ $$\\sqrt{9}=3$$ $$\\frac{\\sqrt{52}}{\\sqrt{117}}=\\frac{2}{3}$$"],"id":"3"},{"content":"найдите значение частного: $$\\frac{\\sqrt{12500}}{\\sqrt{500}}$$[[fill_choice_big-2682]]","widgets":{"fill_choice_big-2682":{"type":"fill_choice_big","options":["$5$","$25$","$125$"],"placeholder":0,"answer":0}},"step":1,"hints":["Занесем числитель и знаменатель дроби под один знак корня: $$\\frac{\\sqrt{\\textcolor{blue}{12500}}}{\\sqrt{\\textcolor{coral}{500}}}=\\sqrt{\\frac{\\textcolor{blue}{12500}}{\\textcolor{coral}{500}}}$$","Сократим дробь: $$\\sqrt{\\frac{12500}{500}}=\\sqrt{25}=5$$"],"id":"3"},{"content":"Вычислите значение выражения: $$\\sqrt{8^2+6^2}$$[[image-114]][[fill_choice_big-3056]]","widgets":{"image-114":{"type":"image","url":"https://obrazavr.ru/wp-content/uploads/2022/10/uchenik-uchitel-uchitelnitsa-papka-1.svg","width":"400"},"fill_choice_big-3056":{"type":"fill_choice_big","options":["$10$","$14$","$2$"],"placeholder":0,"answer":0}},"step":1,"hints":["Найдем значение выражения под корнем: $$\\sqrt{\\textcolor{blue}{8^2}+\\textcolor{coral}{6^2}}=\\sqrt{\\textcolor{blue}{64}+\\textcolor{coral}{36}}=\\sqrt{100}$$","$$\\sqrt{100}=10$$"],"id":"5"},{"content":"Вычислите значение выражения: $$\\sqrt{122^2-22^2}$$[[image-123]][[fill_choice_big-3566]]","widgets":{"image-123":{"type":"image","url":"https://obrazavr.ru/wp-content/uploads/2022/12/phase-3-final-v2.svg","width":"400"},"fill_choice_big-3566":{"type":"fill_choice_big","options":["$120$","$100$","$22$"],"placeholder":0,"answer":0}},"step":1,"hints":["Распишем подкоренное выражение по формуле «разность квадратов»: $$\\textcolor{blue}{a}^\\textcolor{coral}{2}-\\textcolor{darkgreen}{b}^\\textcolor{coral}{2}=(\\textcolor{blue}{a}+\\textcolor{darkgreen}{b})(\\textcolor{blue}{a}-\\textcolor{darkgreen}{b})$$","$$\\sqrt{\\textcolor{blue}{122}^\\textcolor{coral}{2}-\\textcolor{darkgreen}{22}^\\textcolor{coral}{2}}=\\sqrt{(\\textcolor{blue}{122}+\\textcolor{darkgreen}{22})(\\textcolor{blue}{122}-\\textcolor{darkgreen}{22})}$$ $$\\sqrt{144 \\cdot 100}=\\sqrt{144} \\cdot \\sqrt{100}$$","$$\\sqrt{144}=12$$ $$\\sqrt{100}=10$$ $$12 \\cdot 10 =120$$"],"id":"5"},{"content":"Вычислите значение выражения: $$\\sqrt{82^2-18^2}$$[[image-134]][[fill_choice_big-3863]]","widgets":{"image-134":{"type":"image","url":"https://obrazavr.ru/wp-content/uploads/2023/10/obrazavr_ustal.svg","width":"400"},"fill_choice_big-3863":{"type":"fill_choice_big","options":["$80$","$64$","$18$"],"placeholder":0,"answer":0}},"step":1,"hints":["Распишем подкоренное выражение по формуле «разность квадратов»: $$\\textcolor{blue}{a}^\\textcolor{coral}{2}-\\textcolor{darkgreen}{b}^\\textcolor{coral}{2}=(\\textcolor{blue}{a}+\\textcolor{darkgreen}{b})(\\textcolor{blue}{a}-\\textcolor{darkgreen}{b})$$","$$\\sqrt{\\textcolor{blue}{82}^\\textcolor{coral}{2}-\\textcolor{darkgreen}{18}^\\textcolor{coral}{2}}=\\sqrt{(\\textcolor{blue}{82}+\\textcolor{darkgreen}{18})(\\textcolor{blue}{82}-\\textcolor{darkgreen}{18})}$$ $$\\sqrt{100\\cdot 64}=\\sqrt{100} \\cdot \\sqrt{64}$$","$$\\sqrt{100}=10$$ $$\\sqrt{64}=8$$ $$10\\cdot 8=80$$"],"id":"5"}]}</textarea></pre></div>
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