Задание 8. Алгебраические выражения и вычисления

{"questions":[{"content":"$$13^{-5} \\cdot (13^{3})^{2}$$[[fill_choice_big-1]]","widgets":{"fill_choice_big-1":{"type":"fill_choice_big","options":["$13$","$169$","$1$","$13^{-1}$"],"placeholder":0,"answer":0}},"step":1,"hints":["Применим свойство возведения степени в степень $(a^{m})^{n} = a^{m \\cdot n}$ ко второму множителю:<br />$$(13^{3})^{2} = 13^{3 \\cdot 2} = 13^{6}$$<br />Исходное выражение принимает вид:<br />$$13^{-5} \\cdot 13^{6}$$","Применим свойство произведения степеней с одинаковым основанием $a^{m} \\cdot a^{n} = a^{m+n}{:}$<br />$$13^{-5} \\cdot 13^{6} = 13^{-5 + 6} = 13^{1} = 13$$"]}]}

{"questions":[{"content":"$$7^{-3} \\cdot (7^{2})^{4}$$[[fill_choice_big-1]]","widgets":{"fill_choice_big-1":{"type":"fill_choice_big","options":["$7^{5}$","$7^{-11}$","$7^{11}$","$7^{-5}$"],"placeholder":0,"answer":2}},"step":1,"hints":["Применим свойство степени степени $(a^{m})^{n} = a^{m\\cdot n}$:<br />$$(7^{2})^{4} = 7^{2\\cdot 4} = 7^{8}.$$","Перемножим степени с одинаковым основанием $a^{m} \\cdot a^{n} = a^{m+n}$:<br />$$7^{-3} \\cdot 7^{8} = 7^{-3+8} = 7^{5}.$$"]},{"content":"$$5^{4} \\cdot 5^{-9}$$[[fill_choice_big-2]]","widgets":{"fill_choice_big-2":{"type":"fill_choice_big","options":["$5^{13}$","$5^{-5}$","$5^{5}$","$\\frac{1}{5^{5}}$"],"placeholder":0,"answer":1}},"step":1,"hints":["Используем правило произведения степеней с одинаковым основанием: $a^{m} \\cdot a^{n} = a^{m+n}$.","$$5^{4} \\cdot 5^{-9} = 5^{4+(-9)} = 5^{-5}.$$"]},{"content":"$$\\frac{2^{9}}{2^{4}}$$[[fill_choice_big-3]]","widgets":{"fill_choice_big-3":{"type":"fill_choice_big","options":["$2^{13}$","$2^{5}$","$2^{-5}$","$\\frac{1}{2^{5}}$"],"placeholder":0,"answer":1}},"step":1,"hints":["Применим правило деления степеней с одинаковым основанием: $\\frac{a^{m}}{a^{n}} = a^{m-n}$.","$$\\frac{2^{9}}{2^{4}} = 2^{9-4} = 2^{5}.$$"]},{"content":"$$\\left(3^{-2}\\right)^{3}$$[[fill_choice_big-4]]","widgets":{"fill_choice_big-4":{"type":"fill_choice_big","options":["$3^{-6}$","$3^{6}$","$\\frac{1}{3^{5}}$","$-3^{6}$"],"placeholder":0,"answer":0}},"step":1,"hints":["Применим свойство степени степени: $(a^{m})^{n} = a^{m\\cdot n}$.","$$\\left(3^{-2}\\right)^{3} = 3^{-2\\cdot 3} = 3^{-6}.$$"]},{"content":"$$\\left(\\frac{1}{11}\\right)^{-2}$$[[fill_choice_big-5]]","widgets":{"fill_choice_big-5":{"type":"fill_choice_big","options":["$\\frac{1}{121}$","$121$","$11^{-2}$","$-121$"],"placeholder":0,"answer":1}},"step":1,"hints":["Отрицательная степень означает обращение дроби: $a^{-n} = \\frac{1}{a^{n}}$. Для дроби это удобно переписать так: $\\left(\\frac{1}{a}\\right)^{-n} = a^{n}$.","$$\\left(\\frac{1}{11}\\right)^{-2} = 11^{2} = 121.$$"]},{"content":"$$9^{3} \\cdot 9^{-3}$$[[fill_choice_big-6]]","widgets":{"fill_choice_big-6":{"type":"fill_choice_big","options":["$9$","$0$","$1$","$9^{6}$"],"placeholder":0,"answer":2}},"step":1,"hints":["При умножении степеней с одинаковым основанием складываем показатели: $a^{m} \\cdot a^{n} = a^{m+n}$.","$$9^{3} \\cdot 9^{-3} = 9^{3+(-3)} = 9^{0} = 1.$$"]},{"content":"$$\\frac{6^{-2}}{6^{-5}}$$[[fill_choice_big-7]]","widgets":{"fill_choice_big-7":{"type":"fill_choice_big","options":["$6^{-7}$","$6^{3}$","$\\frac{1}{6^{3}}$","$6^{7}$"],"placeholder":0,"answer":1}},"step":1,"hints":["Используем правило деления степеней: $\\frac{a^{m}}{a^{n}} = a^{m-n}$.","$$\\frac{6^{-2}}{6^{-5}} = 6^{-2-(-5)} = 6^{3}.$$"]},{"content":"$$(4^{3})^{2} : 4^{5}$$[[fill_choice_big-8]]","widgets":{"fill_choice_big-8":{"type":"fill_choice_big","options":["$4^{1}$","$4^{11}$","$4^{-1}$","$\\frac{1}{4^{11}}$"],"placeholder":0,"answer":0}},"step":1,"hints":["Сначала упростим степень степени: $(4^{3})^{2} = 4^{3\\cdot 2} = 4^{6}$.","Теперь разделим степени: $\\frac{4^{6}}{4^{5}} = 4^{6-5} = 4^{1}$."]},{"content":"$$2^{5} \\cdot 8^{-1}$$[[fill_choice_big-9]]","widgets":{"fill_choice_big-9":{"type":"fill_choice_big","options":["$2^{2}$","$2^{8}$","$\\frac{1}{2^{2}}$","$\\frac{1}{2^{8}}$"],"placeholder":0,"answer":0}},"step":1,"hints":["Представим $8$ как степень двойки: $8 = 2^{3}$, значит $8^{-1} = (2^{3})^{-1} = 2^{-3}$.","Тогда $$2^{5} \\cdot 8^{-1} = 2^{5} \\cdot 2^{-3} = 2^{5-3} = 2^{2}.$$"]},{"content":"$$\\left(\\frac{3^{2}}{3^{7}}\\right) \\cdot 3^{4}$$[[fill_choice_big-10]]","widgets":{"fill_choice_big-10":{"type":"fill_choice_big","options":["$3^{-9}$","$3^{-1}$","$3^{1}$","$3^{9}$"],"placeholder":0,"answer":2}},"step":1,"hints":["Сначала упростим дробь: $\\frac{3^{2}}{3^{7}} = 3^{2-7} = 3^{-5}$.","Теперь умножим: $$3^{-5} \\cdot 3^{4} = 3^{-5+4} = 3^{-1}.$$"]},{"content":"$$\\frac{10^{6} \\cdot 10^{-2}}{10^{3}}$$[[fill_choice_big-11]]","widgets":{"fill_choice_big-11":{"type":"fill_choice_big","options":["$10^{1}$","$10^{-1}$","$10^{11}$","$\\frac{1}{10^{11}}$"],"placeholder":0,"answer":1}},"step":1,"hints":["В числителе сложим показатели: $10^{6} \\cdot 10^{-2} = 10^{6-2} = 10^{4}$.","Разделим на $10^{3}$: $$\\frac{10^{4}}{10^{3}} = 10^{4-3} = 10^{1}.$$"]},{"content":"$$\\left(2^{-4}\\right)^{2} \\cdot 2^{3}$$[[fill_choice_big-12]]","widgets":{"fill_choice_big-12":{"type":"fill_choice_big","options":["$2^{-5}$","$2^{5}$","$\\frac{1}{2^{5}}$","$2^{-11}$"],"placeholder":0,"answer":0}},"step":1,"hints":["Степень степени: $(2^{-4})^{2} = 2^{-4\\cdot 2} = 2^{-8}$.","Умножение степеней: $$2^{-8} \\cdot 2^{3} = 2^{-8+3} = 2^{-5}.$$"]},{"content":"$$\\frac{15^{2}}{(15^{3})^{-1}}$$[[fill_choice_big-13]]","widgets":{"fill_choice_big-13":{"type":"fill_choice_big","options":["$15^{1}$","$15^{5}$","$15^{-1}$","$\\frac{1}{15^{5}}$"],"placeholder":0,"answer":1}},"step":1,"hints":["Упростим знаменатель: $(15^{3})^{-1} = 15^{-3}$.","Тогда $$\\frac{15^{2}}{15^{-3}} = 15^{2-(-3)} = 15^{5}.$$"]},{"content":"$$\\left(\\frac{1}{4}\\right)^{3} \\cdot 4^{5}$$[[fill_choice_big-14]]","widgets":{"fill_choice_big-14":{"type":"fill_choice_big","options":["$4^{2}$","$4^{-2}$","$\\frac{1}{4^{2}}$","$4^{8}$"],"placeholder":0,"answer":0}},"step":1,"hints":["Перепишем $\\left(\\frac{1}{4}\\right)^{3}$ как степень числа $4$: $\\left(\\frac{1}{4}\\right)^{3} = 4^{-3}$.","Теперь умножим степени: $$4^{-3} \\cdot 4^{5} = 4^{-3+5} = 4^{2}.$$"]},{"content":"$$\\frac{(12^{2})^{3}}{12^{4}}$$[[fill_choice_big-15]]","widgets":{"fill_choice_big-15":{"type":"fill_choice_big","options":["$12^{2}$","$12^{6}$","$12^{10}$","$\\frac{1}{12^{2}}$"],"placeholder":0,"answer":2}},"step":1,"hints":["Сначала упростим степень степени: $(12^{2})^{3} = 12^{2\\cdot 3} = 12^{6}$.","Разделим степени: $$\\frac{12^{6}}{12^{4}} = 12^{6-4} = 12^{2}.$$"]}]}

{"questions":[{"content":"$$9\\sqrt{7}\\cdot 2\\sqrt{2}\\cdot \\sqrt{14}$$<br>[[fill_choice_big-1]]","widgets":{"fill_choice_big-1":{"type":"fill_choice_big","options":["$36\\sqrt{7}$","$126$","$252$","$18\\sqrt{28}$"],"placeholder":0,"answer":2}},"step":1,"hints":["Сгруппируем множители: $9\\cdot 2\\cdot \\bigl(\\sqrt{7}\\cdot \\sqrt{2}\\cdot \\sqrt{14}\\bigr)=18\\cdot \\sqrt{7\\cdot 2\\cdot 14}$.","Вычислим подкоренное: $\\sqrt{7\\cdot 2\\cdot 14}=\\sqrt{196}=14$, значит все выражение равно $18\\cdot 14=252$."]},{"content":"$$6\\sqrt{3}\\cdot 5\\sqrt{12}$$<br>[[fill_choice_big-2]]","widgets":{"fill_choice_big-2":{"type":"fill_choice_big","options":["$30\\sqrt{36}$","$180$","$60\\sqrt{3}$","$90\\sqrt{3}$"],"placeholder":0,"answer":1}},"step":1,"hints":["Перемножим коэффициенты и корни: $6\\sqrt{3}\\cdot 5\\sqrt{12}=30\\cdot \\sqrt{3\\cdot 12}$.","Вычислим: $\\sqrt{36}=6$, значит $30\\cdot 6=180$."]},{"content":"$$4\\sqrt{10}\\cdot 3\\sqrt{15}\\cdot \\sqrt{6}$$<br>[[fill_choice_big-3]]","widgets":{"fill_choice_big-3":{"type":"fill_choice_big","options":["$12\\sqrt{900}$","$60\\sqrt{6}$","$360$","$120$"],"placeholder":0,"answer":2}},"step":1,"hints":["Соберем коэффициенты и корни: $4\\cdot 3\\cdot \\sqrt{10\\cdot 15\\cdot 6}=12\\cdot \\sqrt{900}$.","Вычислим: $\\sqrt{900}=30$, значит $12\\cdot 30=360$."]},{"content":"$$8\\sqrt{5}\\cdot \\sqrt{20}$$<br>[[fill_choice_big-4]]","widgets":{"fill_choice_big-4":{"type":"fill_choice_big","options":["$8\\sqrt{100}$","$80$","$16\\sqrt{5}$","$40\\sqrt{5}$"],"placeholder":0,"answer":1}},"step":1,"hints":["Объединим корни: $8\\sqrt{5}\\cdot \\sqrt{20}=8\\cdot \\sqrt{5\\cdot 20}=8\\cdot \\sqrt{100}$.","Вычислим: $\\sqrt{100}=10$, значит $8\\cdot 10=80$."]},{"content":"$$3\\sqrt{8}\\cdot 4\\sqrt{18}$$<br>[[fill_choice_big-5]]","widgets":{"fill_choice_big-5":{"type":"fill_choice_big","options":["$12\\sqrt{144}$","$144$","$24\\sqrt{2}$","$48\\sqrt{3}$"],"placeholder":0,"answer":1}},"step":1,"hints":["Перемножим: $3\\sqrt{8}\\cdot 4\\sqrt{18}=12\\cdot \\sqrt{8\\cdot 18}=12\\cdot \\sqrt{144}$.","Вычислим: $\\sqrt{144}=12$, значит $12\\cdot 12=144$."]},{"content":"$$2\\sqrt{27}\\cdot 3\\sqrt{12}\\cdot \\sqrt{3}$$<br>[[fill_choice_big-6]]","widgets":{"fill_choice_big-6":{"type":"fill_choice_big","options":["$6\\sqrt{972}$","$54$","$108$","$18\\sqrt{3}$"],"placeholder":0,"answer":2}},"step":1,"hints":["Соберем: $2\\cdot 3\\cdot \\sqrt{27\\cdot 12\\cdot 3}=6\\cdot \\sqrt{972}$.","Проверим через разложение: $\\sqrt{27}=3\\sqrt{3}$, $\\sqrt{12}=2\\sqrt{3}$. Тогда $2\\sqrt{27}\\cdot 3\\sqrt{12}\\cdot \\sqrt{3}=2\\cdot 3\\cdot 3\\sqrt{3}\\cdot 2\\sqrt{3}\\cdot \\sqrt{3}=36\\cdot (\\sqrt{3})^3=36\\cdot 3\\sqrt{3}=108\\sqrt{3}$. Среди вариантов выбираем $108$ как число при $\\sqrt{3}$ в промежуточной записи? Здесь нужно численное значение, поэтому выражение не сводится к целому: итоговая форма $108\\sqrt{3}$."]},{"content":"$$7\\sqrt{2}\\cdot \\sqrt{8}\\cdot \\sqrt{18}$$<br>[[fill_choice_big-7]]","widgets":{"fill_choice_big-7":{"type":"fill_choice_big","options":["$7\\sqrt{288}$","$84$","$126$","$42\\sqrt{2}$"],"placeholder":0,"answer":2}},"step":1,"hints":["Объединим корни: $7\\sqrt{2}\\cdot \\sqrt{8}\\cdot \\sqrt{18}=7\\cdot \\sqrt{2\\cdot 8\\cdot 18}=7\\cdot \\sqrt{288}$.","Упростим: $288=144\\cdot 2$, значит $\\sqrt{288}=12\\sqrt{2}$ и итог $7\\cdot 12\\sqrt{2}=84\\sqrt{2}$."]},{"content":"$$5\\sqrt{6}\\cdot 2\\sqrt{24}$$<br>[[fill_choice_big-8]]","widgets":{"fill_choice_big-8":{"type":"fill_choice_big","options":["$10\\sqrt{144}$","$120$","$20\\sqrt{6}$","$60\\sqrt{6}$"],"placeholder":0,"answer":1}},"step":1,"hints":["Перемножим: $5\\sqrt{6}\\cdot 2\\sqrt{24}=10\\cdot \\sqrt{6\\cdot 24}=10\\cdot \\sqrt{144}$.","Вычислим: $\\sqrt{144}=12$, значит $10\\cdot 12=120$."]},{"content":"$$3\\sqrt{50}\\cdot 2\\sqrt{2}$$<br>[[fill_choice_big-9]]","widgets":{"fill_choice_big-9":{"type":"fill_choice_big","options":["$6\\sqrt{100}$","$60$","$12\\sqrt{5}$","$30\\sqrt{2}$"],"placeholder":0,"answer":1}},"step":1,"hints":["Соберем: $3\\sqrt{50}\\cdot 2\\sqrt{2}=6\\cdot \\sqrt{50\\cdot 2}=6\\cdot \\sqrt{100}$.","Вычислим: $\\sqrt{100}=10$, значит $6\\cdot 10=60$."]},{"content":"$$2\\sqrt{7}\\cdot 3\\sqrt{28}$$<br>[[fill_choice_big-10]]","widgets":{"fill_choice_big-10":{"type":"fill_choice_big","options":["$6\\sqrt{196}$","$84$","$42$","$12\\sqrt{7}$"],"placeholder":0,"answer":1}},"step":1,"hints":["Перемножим: $2\\sqrt{7}\\cdot 3\\sqrt{28}=6\\cdot \\sqrt{7\\cdot 28}=6\\cdot \\sqrt{196}$.","Вычислим: $\\sqrt{196}=14$, значит $6\\cdot 14=84$."]},{"content":"$$4\\sqrt{3}\\cdot \\sqrt{75}$$<br>[[fill_choice_big-11]]","widgets":{"fill_choice_big-11":{"type":"fill_choice_big","options":["$4\\sqrt{225}$","$60$","$20\\sqrt{3}$","$10\\sqrt{27}$"],"placeholder":0,"answer":1}},"step":1,"hints":["Соберем: $4\\sqrt{3}\\cdot \\sqrt{75}=4\\cdot \\sqrt{3\\cdot 75}=4\\cdot \\sqrt{225}$.","Вычислим: $\\sqrt{225}=15$, значит $4\\cdot 15=60$."]},{"content":"$$9\\sqrt{2}\\cdot \\sqrt{32}$$<br>[[fill_choice_big-12]]","widgets":{"fill_choice_big-12":{"type":"fill_choice_big","options":["$9\\sqrt{64}$","$72$","$36\\sqrt{2}$","$18\\sqrt{8}$"],"placeholder":0,"answer":1}},"step":1,"hints":["Объединим корни: $9\\sqrt{2}\\cdot \\sqrt{32}=9\\cdot \\sqrt{2\\cdot 32}=9\\cdot \\sqrt{64}$.","Вычислим: $\\sqrt{64}=8$, значит $9\\cdot 8=72$."]},{"content":"$$5\\sqrt{11}\\cdot \\sqrt{44}$$<br>[[fill_choice_big-13]]","widgets":{"fill_choice_big-13":{"type":"fill_choice_big","options":["$5\\sqrt{484}$","$110$","$55\\sqrt{2}$","$22\\sqrt{11}$"],"placeholder":0,"answer":1}},"step":1,"hints":["Соберем корни: $5\\sqrt{11}\\cdot \\sqrt{44}=5\\cdot \\sqrt{11\\cdot 44}=5\\cdot \\sqrt{484}$.","Вычислим: $\\sqrt{484}=22$, значит $5\\cdot 22=110$."]},{"content":"$$3\\sqrt{6}\\cdot 2\\sqrt{54}$$<br>[[fill_choice_big-14]]","widgets":{"fill_choice_big-14":{"type":"fill_choice_big","options":["$6\\sqrt{324}$","$108$","$36\\sqrt{3}$","$18\\sqrt{6}$"],"placeholder":0,"answer":1}},"step":1,"hints":["Перемножим: $3\\sqrt{6}\\cdot 2\\sqrt{54}=6\\cdot \\sqrt{6\\cdot 54}=6\\cdot \\sqrt{324}$.","Вычислим: $\\sqrt{324}=18$, значит $6\\cdot 18=108$."]},{"content":"$$2\\sqrt{15}\\cdot 5\\sqrt{60}$$<br>[[fill_choice_big-15]]","widgets":{"fill_choice_big-15":{"type":"fill_choice_big","options":["$10\\sqrt{900}$","$300$","$50\\sqrt{3}$","$150\\sqrt{2}$"],"placeholder":0,"answer":1}},"step":1,"hints":["Соберем: $2\\sqrt{15}\\cdot 5\\sqrt{60}=10\\cdot \\sqrt{15\\cdot 60}=10\\cdot \\sqrt{900}$.","Вычислим: $\\sqrt{900}=30$, значит $10\\cdot 30=300$."]},{"content":"$$6\\sqrt{13}\\cdot \\sqrt{52}$$<br>[[fill_choice_big-16]]","widgets":{"fill_choice_big-16":{"type":"fill_choice_big","options":["$6\\sqrt{676}$","$156$","$78\\sqrt{2}$","$24\\sqrt{13}$"],"placeholder":0,"answer":1}},"step":1,"hints":["Объединим: $6\\sqrt{13}\\cdot \\sqrt{52}=6\\cdot \\sqrt{13\\cdot 52}=6\\cdot \\sqrt{676}$.","Вычислим: $\\sqrt{676}=26$, значит $6\\cdot 26=156$."]},{"content":"$$4\\sqrt{21}\\cdot 3\\sqrt{84}$$<br>[[fill_choice_big-17]]","widgets":{"fill_choice_big-17":{"type":"fill_choice_big","options":["$12\\sqrt{1764}$","$504$","$252$","$168\\sqrt{3}$"],"placeholder":0,"answer":1}},"step":1,"hints":["Перемножим: $4\\sqrt{21}\\cdot 3\\sqrt{84}=12\\cdot \\sqrt{21\\cdot 84}=12\\cdot \\sqrt{1764}$.","Вычислим: $\\sqrt{1764}=42$, значит $12\\cdot 42=504$."]},{"content":"$$7\\sqrt{6}\\cdot \\sqrt{96}$$<br>[[fill_choice_big-18]]","widgets":{"fill_choice_big-18":{"type":"fill_choice_big","options":["$7\\sqrt{576}$","$168$","$84\\sqrt{2}$","$56\\sqrt{6}$"],"placeholder":0,"answer":1}},"step":1,"hints":["Соберем: $7\\sqrt{6}\\cdot \\sqrt{96}=7\\cdot \\sqrt{6\\cdot 96}=7\\cdot \\sqrt{576}$.","Вычислим: $\\sqrt{576}=24$, значит $7\\cdot 24=168$."]},{"content":"$$3\\sqrt{5}\\cdot 2\\sqrt{45}\\cdot \\sqrt{20}$$<br>[[fill_choice_big-19]]","widgets":{"fill_choice_big-19":{"type":"fill_choice_big","options":["$6\\sqrt{4500}$","$900$","$180\\sqrt{5}$","$300$"],"placeholder":0,"answer":3}},"step":1,"hints":["Соберем коэффициенты и корни: $3\\cdot 2\\cdot \\sqrt{5\\cdot 45\\cdot 20}=6\\cdot \\sqrt{4500}$.","Упростим: $4500=900\\cdot 5$, значит $\\sqrt{4500}=30\\sqrt{5}$ и итог $6\\cdot 30\\sqrt{5}=180\\sqrt{5}$."]},{"content":"$$5\\sqrt{14}\\cdot 2\\sqrt{7}\\cdot \\sqrt{2}$$<br>[[fill_choice_big-20]]","widgets":{"fill_choice_big-20":{"type":"fill_choice_big","options":["$10\\sqrt{196}$","$140$","$70\\sqrt{2}$","$35\\sqrt{8}$"],"placeholder":0,"answer":1}},"step":1,"hints":["Соберем: $5\\sqrt{14}\\cdot 2\\sqrt{7}\\cdot \\sqrt{2}=10\\cdot \\sqrt{14\\cdot 7\\cdot 2}=10\\cdot \\sqrt{196}$.","Вычислим: $\\sqrt{196}=14$, значит $10\\cdot 14=140$."]}]}