Умножение степеней с одинаковыми и разными основаниями

{"questions":[{"content":"$$3^4\\cdot 3^2$$[[fill_choice_big-1]]","widgets":{"fill_choice_big-1":{"type":"fill_choice_big","options":["$3^8$","$3^6$","$3^2$","$3$"],"placeholder":0,"answer":1}},"step":1,"hints":["У степеней основания одинаковые: $3$ и $3$. Значит, основание оставляем тем же, а показатели складываем.","$3^4\\cdot 3^2=3^{4+2}=3^6$."]}]}

{"questions":[{"content":"$$2^3\\cdot 2^5$$[[fill_choice_big-11]]","widgets":{"fill_choice_big-11":{"type":"fill_choice_big","options":["$2^8$","$2^{15}$","$4^8$","$2^2$"],"placeholder":0,"answer":0}},"step":1,"hints":["Основания одинаковые: $2$ и $2$. При умножении степеней с одинаковым основанием показатели складываются.","$2^3\\cdot 2^5=2^{3+5}=2^8$."]},{"content":"$$3^4\\cdot 3^2$$[[fill_choice_big-12]]","widgets":{"fill_choice_big-12":{"type":"fill_choice_big","options":["$3^6$","$3^8$","$6^3$","$9^6$"],"placeholder":0,"answer":0}},"step":1,"hints":["Основание одно и то же: $3$. Складываем показатели: $4+2$.","$3^4\\cdot 3^2=3^{4+2}=3^6$."]},{"content":"$$5^1\\cdot 5^7$$[[fill_choice_big-13]]","widgets":{"fill_choice_big-13":{"type":"fill_choice_big","options":["$5^8$","$5^6$","$25^7$","$10^8$"],"placeholder":0,"answer":0}},"step":1,"hints":["Одинаковое основание: $5$. При умножении складываем показатели: $1+7$.","$5^1\\cdot 5^7=5^{1+7}=5^8$."]},{"content":"$$7^3\\cdot 7^4$$[[fill_choice_big-14]]","widgets":{"fill_choice_big-14":{"type":"fill_choice_big","options":["$7^7$","$7^{12}$","$49^7$","$14^7$"],"placeholder":0,"answer":0}},"step":1,"hints":["Основание одинаковое: $7$. Складываем показатели: $3+4$.","$7^3\\cdot 7^4=7^{3+4}=7^7$."]},{"content":"$$10^2\\cdot 10^6$$[[fill_choice_big-15]]","widgets":{"fill_choice_big-15":{"type":"fill_choice_big","options":["$10^8$","$10^4$","$100^8$","$20^8$"],"placeholder":0,"answer":0}},"step":1,"hints":["Основания одинаковые: $10$. Складываем показатели: $2+6$.","$10^2\\cdot 10^6=10^{2+6}=10^8$."]},{"content":"$$4^5\\cdot 4^2$$[[fill_choice_big-16]]","widgets":{"fill_choice_big-16":{"type":"fill_choice_big","options":["$4^7$","$4^{10}$","$16^7$","$8^7$"],"placeholder":0,"answer":0}},"step":1,"hints":["Одинаковое основание: $4$. При умножении показатели складываются: $5+2$.","$4^5\\cdot 4^2=4^{5+2}=4^7$."]},{"content":"$$6^3\\cdot 6^3$$[[fill_choice_big-17]]","widgets":{"fill_choice_big-17":{"type":"fill_choice_big","options":["$6^6$","$6^9$","$12^6$","$36^3$"],"placeholder":0,"answer":0}},"step":1,"hints":["Основание одинаковое: $6$. Складываем показатели: $3+3$.","$6^3\\cdot 6^3=6^{3+3}=6^6$."]},{"content":"$$9^2\\cdot 9^5$$[[fill_choice_big-18]]","widgets":{"fill_choice_big-18":{"type":"fill_choice_big","options":["$9^7$","$9^{10}$","$81^7$","$18^7$"],"placeholder":0,"answer":0}},"step":1,"hints":["Основание одинаковое: $9$. Складываем показатели: $2+5$.","$9^2\\cdot 9^5=9^{2+5}=9^7$."]},{"content":"$$11^4\\cdot 11^1$$[[fill_choice_big-19]]","widgets":{"fill_choice_big-19":{"type":"fill_choice_big","options":["$11^5$","$11^4$","$121^5$","$22^5$"],"placeholder":0,"answer":0}},"step":1,"hints":["Основание одинаковое: $11$. При умножении степеней показатели складываются: $4+1$.","$11^4\\cdot 11^1=11^{4+1}=11^5$."]},{"content":"$$8^2\\cdot 8^4$$[[fill_choice_big-20]]","widgets":{"fill_choice_big-20":{"type":"fill_choice_big","options":["$8^6$","$8^8$","$64^6$","$16^6$"],"placeholder":0,"answer":0}},"step":1,"hints":["Основание одинаковое: $8$. Складываем показатели: $2+4$.","$8^2\\cdot 8^4=8^{2+4}=8^6$."]}]}

{"questions":[{"content":"$$12^3\\cdot 12^2$$[[fill_choice_big-21]]","widgets":{"fill_choice_big-21":{"type":"fill_choice_big","options":["$12^5$","$12^6$","$24^5$","$144^5$"],"placeholder":0,"answer":0}},"step":1,"hints":["Основания одинаковые: $12$. При умножении степеней показатели складываются: $3+2$.","$12^3\\cdot 12^2=12^{3+2}=12^5$."]},{"content":"$$2^7\\cdot 2^4$$[[fill_choice_big-22]]","widgets":{"fill_choice_big-22":{"type":"fill_choice_big","options":["$2^{11}$","$2^{28}$","$4^{11}$","$2^3$"],"placeholder":0,"answer":0}},"step":1,"hints":["Основание одинаковое: $2$. Складываем показатели: $7+4$.","$2^7\\cdot 2^4=2^{7+4}=2^{11}$."]},{"content":"$$15^1\\cdot 15^9$$[[fill_choice_big-23]]","widgets":{"fill_choice_big-23":{"type":"fill_choice_big","options":["$15^{10}$","$15^8$","$30^{10}$","$225^{10}$"],"placeholder":0,"answer":0}},"step":1,"hints":["Основание одинаковое: $15$. При умножении показателей степени складываются: $1+9$.","$15^1\\cdot 15^9=15^{1+9}=15^{10}$."]},{"content":"$$3^8\\cdot 3^3$$[[fill_choice_big-24]]","widgets":{"fill_choice_big-24":{"type":"fill_choice_big","options":["$3^{11}$","$3^{24}$","$6^{11}$","$9^{11}$"],"placeholder":0,"answer":0}},"step":1,"hints":["Основание одинаковое: $3$. Складываем показатели: $8+3$.","$3^8\\cdot 3^3=3^{8+3}=3^{11}$."]},{"content":"$$14^4\\cdot 14^4$$[[fill_choice_big-25]]","widgets":{"fill_choice_big-25":{"type":"fill_choice_big","options":["$14^8$","$14^{16}$","$28^8$","$196^8$"],"placeholder":0,"answer":0}},"step":1,"hints":["Основание одинаковое: $14$. Складываем показатели: $4+4$.","$14^4\\cdot 14^4=14^{4+4}=14^8$."]},{"content":"$$5^9\\cdot 5^2$$[[fill_choice_big-26]]","widgets":{"fill_choice_big-26":{"type":"fill_choice_big","options":["$5^{11}$","$5^{18}$","$10^{11}$","$25^{11}$"],"placeholder":0,"answer":0}},"step":1,"hints":["Основание одинаковое: $5$. Складываем показатели: $9+2$.","$5^9\\cdot 5^2=5^{9+2}=5^{11}$."]},{"content":"$$9^6\\cdot 9^1$$[[fill_choice_big-27]]","widgets":{"fill_choice_big-27":{"type":"fill_choice_big","options":["$9^7$","$9^6$","$18^7$","$81^7$"],"placeholder":0,"answer":0}},"step":1,"hints":["Основание одинаковое: $9$. Складываем показатели: $6+1$.","$9^6\\cdot 9^1=9^{6+1}=9^7$."]},{"content":"$$4^7\\cdot 4^3$$[[fill_choice_big-28]]","widgets":{"fill_choice_big-28":{"type":"fill_choice_big","options":["$4^{10}$","$4^{21}$","$8^{10}$","$16^{10}$"],"placeholder":0,"answer":0}},"step":1,"hints":["Основание одинаковое: $4$. Складываем показатели: $7+3$.","$4^7\\cdot 4^3=4^{7+3}=4^{10}$."]},{"content":"$$13^2\\cdot 13^6$$[[fill_choice_big-29]]","widgets":{"fill_choice_big-29":{"type":"fill_choice_big","options":["$13^8$","$13^4$","$26^8$","$169^8$"],"placeholder":0,"answer":0}},"step":1,"hints":["Основание одинаковое: $13$. Складываем показатели: $2+6$.","$13^2\\cdot 13^6=13^{2+6}=13^8$."]},{"content":"$$7^5\\cdot 7^8$$[[fill_choice_big-30]]","widgets":{"fill_choice_big-30":{"type":"fill_choice_big","options":["$7^{13}$","$7^{40}$","$14^{13}$","$49^{13}$"],"placeholder":0,"answer":0}},"step":1,"hints":["Основание одинаковое: $7$. Складываем показатели: $5+8$.","$7^5\\cdot 7^8=7^{5+8}=7^{13}$."]}],"mix":1}

{"questions":[{"content":"$$2^3\\cdot 5^3$$[[fill_choice_big-1]]","widgets":{"fill_choice_big-1":{"type":"fill_choice_big","options":["$10^3$","$7^3$","$10^6$","$2^6$"],"placeholder":0,"answer":0}},"step":1,"hints":["Показатели степеней одинаковые: $3$ и $3$. Можно перемножить основания и оставить показатель тем же.","$2^3\\cdot 5^3=(2\\cdot 5)^3=10^3$."]},{"content":"$$3^4\\cdot 2^4$$[[fill_choice_big-2]]","widgets":{"fill_choice_big-2":{"type":"fill_choice_big","options":["$6^4$","$5^4$","$6^8$","$3^8$"],"placeholder":0,"answer":0}},"step":1,"hints":["Показатели одинаковые: $4$. Используем $a^n\\cdot b^n=(ab)^n$.","$3^4\\cdot 2^4=(3\\cdot 2)^4=6^4$."]},{"content":"$$4^2\\cdot 7^2$$[[fill_choice_big-3]]","widgets":{"fill_choice_big-3":{"type":"fill_choice_big","options":["$28^2$","$11^2$","$28^4$","$4^4$"],"placeholder":0,"answer":0}},"step":1,"hints":["Одинаковый показатель: $2$. Перемножьте основания.","$4^2\\cdot 7^2=(4\\cdot 7)^2=28^2$."]},{"content":"$$6^5\\cdot 2^5$$[[fill_choice_big-4]]","widgets":{"fill_choice_big-4":{"type":"fill_choice_big","options":["$12^5$","$8^5$","$12^{10}$","$6^{10}$"],"placeholder":0,"answer":0}},"step":1,"hints":["Показатели одинаковые: $5$. Можно объединить в одну степень.","$6^5\\cdot 2^5=(6\\cdot 2)^5=12^5$."]},{"content":"$$9^3\\cdot 4^3$$[[fill_choice_big-5]]","widgets":{"fill_choice_big-5":{"type":"fill_choice_big","options":["$36^3$","$13^3$","$36^6$","$9^6$"],"placeholder":0,"answer":0}},"step":1,"hints":["Одинаковый показатель: $3$. Умножаем основания, показатель сохраняем.","$9^3\\cdot 4^3=(9\\cdot 4)^3=36^3$."]},{"content":"$$8^2\\cdot 3^2$$[[fill_choice_big-6]]","widgets":{"fill_choice_big-6":{"type":"fill_choice_big","options":["$24^2$","$11^2$","$24^4$","$8^4$"],"placeholder":0,"answer":0}},"step":1,"hints":["Показатель одинаковый: $2$. Используем формулу произведения степеней с одинаковым показателем.","$8^2\\cdot 3^2=(8\\cdot 3)^2=24^2$."]},{"content":"$$5^4\\cdot 10^4$$[[fill_choice_big-7]]","widgets":{"fill_choice_big-7":{"type":"fill_choice_big","options":["$50^4$","$15^4$","$50^8$","$5^8$"],"placeholder":0,"answer":0}},"step":1,"hints":["Одинаковый показатель: $4$. Перемножаем основания и оставляем $4$.","$5^4\\cdot 10^4=(5\\cdot 10)^4=50^4$."]},{"content":"$$7^3\\cdot 11^3$$[[fill_choice_big-8]]","widgets":{"fill_choice_big-8":{"type":"fill_choice_big","options":["$77^3$","$18^3$","$77^6$","$7^6$"],"placeholder":0,"answer":0}},"step":1,"hints":["Показатели одинаковые: $3$. Используйте $a^n\\cdot b^n=(ab)^n$.","$7^3\\cdot 11^3=(7\\cdot 11)^3=77^3$."]},{"content":"$$12^2\\cdot 5^2$$[[fill_choice_big-9]]","widgets":{"fill_choice_big-9":{"type":"fill_choice_big","options":["$60^2$","$17^2$","$60^4$","$12^4$"],"placeholder":0,"answer":0}},"step":1,"hints":["Одинаковый показатель: $2$. Можно объединить в одну степень.","$12^2\\cdot 5^2=(12\\cdot 5)^2=60^2$."]},{"content":"$$15^3\\cdot 2^3$$[[fill_choice_big-10]]","widgets":{"fill_choice_big-10":{"type":"fill_choice_big","options":["$30^3$","$17^3$","$30^6$","$15^6$"],"placeholder":0,"answer":0}},"step":1,"hints":["Показатели одинаковые: $3$. Умножьте основания и оставьте показатель $3$.","$15^3\\cdot 2^3=(15\\cdot 2)^3=30^3$."]}]}

{"questions":[{"content":"$$14^2\\cdot 3^2$$[[fill_choice_big-11]]","widgets":{"fill_choice_big-11":{"type":"fill_choice_big","options":["$42^2$","$17^2$","$42^4$","$14^4$"],"placeholder":0,"answer":0}},"step":1,"hints":["Показатели степеней одинаковые: $2$. Используйте $a^n\\cdot b^n=(ab)^n$.","$14^2\\cdot 3^2=(14\\cdot 3)^2=42^2$."]},{"content":"$$5^3\\cdot 6^3$$[[fill_choice_big-12]]","widgets":{"fill_choice_big-12":{"type":"fill_choice_big","options":["$30^3$","$11^3$","$30^6$","$5^6$"],"placeholder":0,"answer":0}},"step":1,"hints":["Показатели одинаковые: $3$. Можно перемножить основания и сохранить показатель.","$5^3\\cdot 6^3=(5\\cdot 6)^3=30^3$."]},{"content":"$$8^4\\cdot 5^4$$[[fill_choice_big-13]]","widgets":{"fill_choice_big-13":{"type":"fill_choice_big","options":["$40^4$","$13^4$","$40^8$","$8^8$"],"placeholder":0,"answer":0}},"step":1,"hints":["Одинаковый показатель: $4$. Используем $a^n\\cdot b^n=(ab)^n$.","$8^4\\cdot 5^4=(8\\cdot 5)^4=40^4$."]},{"content":"$$9^2\\cdot 2^2$$[[fill_choice_big-14]]","widgets":{"fill_choice_big-14":{"type":"fill_choice_big","options":["$18^2$","$11^2$","$18^4$","$9^4$"],"placeholder":0,"answer":0}},"step":1,"hints":["Показатели одинаковые: $2$. Перемножьте основания.","$9^2\\cdot 2^2=(9\\cdot 2)^2=18^2$."]},{"content":"$$11^3\\cdot 4^3$$[[fill_choice_big-15]]","widgets":{"fill_choice_big-15":{"type":"fill_choice_big","options":["$44^3$","$15^3$","$44^6$","$11^6$"],"placeholder":0,"answer":0}},"step":1,"hints":["Показатели одинаковые: $3$. Умножаем основания, показатель сохраняем.","$11^3\\cdot 4^3=(11\\cdot 4)^3=44^3$."]},{"content":"$$13^2\\cdot 6^2$$[[fill_choice_big-16]]","widgets":{"fill_choice_big-16":{"type":"fill_choice_big","options":["$78^2$","$19^2$","$78^4$","$13^4$"],"placeholder":0,"answer":0}},"step":1,"hints":["Одинаковый показатель: $2$. Используем формулу произведения степеней с одинаковым показателем.","$13^2\\cdot 6^2=(13\\cdot 6)^2=78^2$."]},{"content":"$$16^3\\cdot 5^3$$[[fill_choice_big-17]]","widgets":{"fill_choice_big-17":{"type":"fill_choice_big","options":["$80^3$","$21^3$","$80^6$","$16^6$"],"placeholder":0,"answer":0}},"step":1,"hints":["Показатели степеней одинаковые: $3$. Перемножьте основания и оставьте показатель.","$16^3\\cdot 5^3=(16\\cdot 5)^3=80^3$."]},{"content":"$$7^4\\cdot 3^4$$[[fill_choice_big-18]]","widgets":{"fill_choice_big-18":{"type":"fill_choice_big","options":["$21^4$","$10^4$","$21^8$","$7^8$"],"placeholder":0,"answer":0}},"step":1,"hints":["Показатели одинаковые: $4$. Используйте $a^n\\cdot b^n=(ab)^n$.","$7^4\\cdot 3^4=(7\\cdot 3)^4=21^4$."]},{"content":"$$12^4\\cdot 2^4$$[[fill_choice_big-19]]","widgets":{"fill_choice_big-19":{"type":"fill_choice_big","options":["$24^4$","$14^4$","$24^8$","$12^8$"],"placeholder":0,"answer":0}},"step":1,"hints":["Одинаковый показатель: $4$. Можно объединить в одну степень.","$12^4\\cdot 2^4=(12\\cdot 2)^4=24^4$."]},{"content":"$$25^2\\cdot 4^2$$[[fill_choice_big-20]]","widgets":{"fill_choice_big-20":{"type":"fill_choice_big","options":["$100^2$","$29^2$","$100^4$","$25^4$"],"placeholder":0,"answer":0}},"step":1,"hints":["Показатели одинаковые: $2$. Перемножьте основания и сохраните показатель.","$25^2\\cdot 4^2=(25\\cdot 4)^2=100^2$."]}]}