Степени с нулевым и отрицательным показателем

{"questions":[{"content":"$$5^0$$[[fill_choice_big-1]]","widgets":{"fill_choice_big-1":{"type":"fill_choice_big","options":["$1$","$5$","$0$","$-5$"],"placeholder":0,"answer":0}},"step":1,"hints":["По правилу $a^0=1$, если $a\\neq0$.","Соответственно, $5\\neq0$ и $5^0=1$."]}]}

{"questions":[{"content":"$$(-3)^0$$[[fill_choice_big-1]]","widgets":{"fill_choice_big-1":{"type":"fill_choice_big","options":["$1$","$0$","$-3$","$3$"],"placeholder":0,"answer":0}},"step":1,"hints":["Вспомним правило: $a^0=1$, если $a\\neq 0$.","Основание равно $-3$, это число не ноль, значит $(-3)^0=1$."]},{"content":"$$5^0$$[[fill_choice_big-2]]","widgets":{"fill_choice_big-2":{"type":"fill_choice_big","options":["$0$","$1$","$5$","$-1$"],"placeholder":0,"answer":1}},"step":1,"hints":["По определению степени: $a^0=1$ при $a\\neq 0$.","Так как $5\\neq 0$, получаем $5^0=1$."]},{"content":"$$(-7)^0$$[[fill_choice_big-3]]","widgets":{"fill_choice_big-3":{"type":"fill_choice_big","options":["$-7$","$7$","$0$","$1$"],"placeholder":0,"answer":3}},"step":1,"hints":["Используем правило: $a^0=1$ при $a\\neq 0$.","Основание $-7\\neq 0$, значит $(-7)^0=1$."]},{"content":"$$12^0$$[[fill_choice_big-4]]","widgets":{"fill_choice_big-4":{"type":"fill_choice_big","options":["$12$","$1$","$0$","$-12$"],"placeholder":0,"answer":1}},"step":1,"hints":["В нулевой степени любое ненулевое число дает 1: $a^0=1$.","Так как $12\\neq 0$, то $12^0=1$."]},{"content":"$$\\left(-\\frac{2}{3}\\right)^0$$[[fill_choice_big-5]]","widgets":{"fill_choice_big-5":{"type":"fill_choice_big","options":["$\\frac{2}{3}$","$-\\frac{2}{3}$","$1$","$0$"],"placeholder":0,"answer":2}},"step":1,"hints":["Нулевой показатель: $a^0=1$, если $a\\neq 0$.","Основание $-\\frac{2}{3}\\neq 0$, значит результат равен $1$."]},{"content":"$$\\left(\\frac{9}{10}\\right)^0$$[[fill_choice_big-6]]","widgets":{"fill_choice_big-6":{"type":"fill_choice_big","options":["$0$","$1$","$\\frac{9}{10}$","$\\frac{10}{9}$"],"placeholder":0,"answer":1}},"step":1,"hints":["Степень с нулевым показателем: $a^0=1$ при $a\\neq 0$.","Основание $\\frac{9}{10}$ не равно нулю, значит значение равно $1$."]},{"content":"$$101^0$$[[fill_choice_big-7]]","widgets":{"fill_choice_big-7":{"type":"fill_choice_big","options":["$101$","$0$","$1$","$-101$"],"placeholder":0,"answer":2}},"step":1,"hints":["Любое ненулевое число в нулевой степени равно $1$: $a^0=1$.","Так как $101\\neq 0$, то $101^0=1$."]},{"content":"$$(-1)^0$$[[fill_choice_big-8]]","widgets":{"fill_choice_big-8":{"type":"fill_choice_big","options":["$-1$","$1$","$0$","$-0$"],"placeholder":0,"answer":1}},"step":1,"hints":["Применим правило: $a^0=1$, если $a\\neq 0$.","Основание $-1\\neq 0$, значит $(-1)^0=1$."]},{"content":"$$2^0$$[[fill_choice_big-9]]","widgets":{"fill_choice_big-9":{"type":"fill_choice_big","options":["$0$","$2$","$1$","$-2$"],"placeholder":0,"answer":2}},"step":1,"hints":["Вспомним: $a^0=1$ при $a\\neq 0$.","Так как $2\\neq 0$, то $2^0=1$."]},{"content":"$$(-25)^0$$[[fill_choice_big-10]]","widgets":{"fill_choice_big-10":{"type":"fill_choice_big","options":["$-25$","$25$","$1$","$0$"],"placeholder":0,"answer":2}},"step":1,"hints":["Для любого ненулевого основания верно: $a^0=1$.","Основание $-25\\neq 0$, значит $(-25)^0=1$."]}],"mix":1}

{"questions":[{"content":"$$5^{-3}$$[[fill_choice_big-1]]","widgets":{"fill_choice_big-1":{"type":"fill_choice_big","options":["$\\frac{1}{125}$","$-\\frac{1}{125}$","$125$","$-125$"],"placeholder":0,"answer":0}},"step":1,"hints":["По правилу $a^{-n}=\\frac{1}{a^n}$ при $a\\neq0$.","Соответственно:<br />$5^{-3}=\\frac{1}{5^3}=\\frac{1}{125}$."]}]}

{"questions":[{"content":"$$2^{-5}$$[[fill_choice_big-1]]","widgets":{"fill_choice_big-1":{"type":"fill_choice_big","options":["$\\frac{1}{32}$","$-\\frac{1}{32}$","$32$","$-32$"],"placeholder":0,"answer":0}},"step":1,"hints":["По правилу $a^{-n}=\\frac{1}{a^n}$ при $a\\neq0$.","Тогда $2^{-5}=\\frac{1}{2^5}=\\frac{1}{32}$."]},{"content":"$$3^{-4}$$[[fill_choice_big-2]]","widgets":{"fill_choice_big-2":{"type":"fill_choice_big","options":["$\\frac{1}{81}$","$-\\frac{1}{81}$","$81$","$-81$"],"placeholder":0,"answer":0}},"step":1,"hints":["Используйте правило $a^{-n}=\\frac{1}{a^n}$ при $a\\neq0$.","Получаем $3^{-4}=\\frac{1}{3^4}=\\frac{1}{81}$."]},{"content":"$$4^{-3}$$[[fill_choice_big-3]]","widgets":{"fill_choice_big-3":{"type":"fill_choice_big","options":["$\\frac{1}{64}$","$-\\frac{1}{64}$","$64$","$-64$"],"placeholder":0,"answer":0}},"step":1,"hints":["По правилу отрицательной степени: $a^{-n}=\\frac{1}{a^n}$.","Имеем $4^{-3}=\\frac{1}{4^3}=\\frac{1}{64}$."]},{"content":"$$6^{-2}$$[[fill_choice_big-4]]","widgets":{"fill_choice_big-4":{"type":"fill_choice_big","options":["$\\frac{1}{36}$","$-\\frac{1}{36}$","$36$","$-36$"],"placeholder":0,"answer":0}},"step":1,"hints":["Примените формулу $a^{-n}=\\frac{1}{a^n}$.","Тогда $6^{-2}=\\frac{1}{6^2}=\\frac{1}{36}$."]},{"content":"$$10^{-3}$$[[fill_choice_big-5]]","widgets":{"fill_choice_big-5":{"type":"fill_choice_big","options":["$\\frac{1}{1000}$","$-\\frac{1}{1000}$","$1000$","$-1000$"],"placeholder":0,"answer":0}},"step":1,"hints":["Вспомните: $a^{-n}=\\frac{1}{a^n}$.","Получаем $10^{-3}=\\frac{1}{10^3}=\\frac{1}{1000}$."]},{"content":"$$12^{-2}$$[[fill_choice_big-6]]","widgets":{"fill_choice_big-6":{"type":"fill_choice_big","options":["$\\frac{1}{144}$","$-\\frac{1}{144}$","$144$","$-144$"],"placeholder":0,"answer":0}},"step":1,"hints":["По правилу: $a^{-n}=\\frac{1}{a^n}$.","Тогда $12^{-2}=\\frac{1}{12^2}=\\frac{1}{144}$."]},{"content":"$$(-2)^{-4}$$[[fill_choice_big-7]]","widgets":{"fill_choice_big-7":{"type":"fill_choice_big","options":["$\\frac{1}{16}$","$-\\frac{1}{16}$","$16$","$-16$"],"placeholder":0,"answer":0}},"step":1,"hints":["Сначала примените $a^{-n}=\\frac{1}{a^n}$.","Далее $(-2)^{-4}=\\frac{1}{(-2)^4}=\\frac{1}{16}$, так как четная степень дает положительное число."]},{"content":"$$(-3)^{-3}$$[[fill_choice_big-8]]","widgets":{"fill_choice_big-8":{"type":"fill_choice_big","options":["$-\\frac{1}{27}$","$\\frac{1}{27}$","$-27$","$27$"],"placeholder":0,"answer":0}},"step":1,"hints":["Используйте $a^{-n}=\\frac{1}{a^n}$.","Имеем $(-3)^{-3}=\\frac{1}{(-3)^3}=\\frac{1}{-27}=-\\frac{1}{27}$, так как степень нечетная."]},{"content":"$$(-5)^{-2}$$[[fill_choice_big-9]]","widgets":{"fill_choice_big-9":{"type":"fill_choice_big","options":["$\\frac{1}{25}$","$-\\frac{1}{25}$","$25$","$-25$"],"placeholder":0,"answer":0}},"step":1,"hints":["Примените формулу $a^{-n}=\\frac{1}{a^n}$.","Тогда $(-5)^{-2}=\\frac{1}{(-5)^2}=\\frac{1}{25}$."]},{"content":"$$8^{-1}$$[[fill_choice_big-10]]","widgets":{"fill_choice_big-10":{"type":"fill_choice_big","options":["$\\frac{1}{8}$","$-\\frac{1}{8}$","$8$","$-8$"],"placeholder":0,"answer":0}},"step":1,"hints":["По правилу: $a^{-1}=\\frac{1}{a}$ при $a\\neq0$.","Значит, $8^{-1}=\\frac{1}{8}$."]},{"content":"$$7^{-2}$$[[fill_choice_big-11]]","widgets":{"fill_choice_big-11":{"type":"fill_choice_big","options":["$\\frac{1}{49}$","$-\\frac{1}{49}$","$49$","$-49$"],"placeholder":0,"answer":0}},"step":1,"hints":["Используйте правило $a^{-n}=\\frac{1}{a^n}$.","Получаем $7^{-2}=\\frac{1}{7^2}=\\frac{1}{49}$."]},{"content":"$$9^{-2}$$[[fill_choice_big-12]]","widgets":{"fill_choice_big-12":{"type":"fill_choice_big","options":["$\\frac{1}{81}$","$-\\frac{1}{81}$","$81$","$-81$"],"placeholder":0,"answer":0}},"step":1,"hints":["Примените формулу $a^{-n}=\\frac{1}{a^n}$.","Тогда $9^{-2}=\\frac{1}{9^2}=\\frac{1}{81}$."]},{"content":"$$25^{-2}$$[[fill_choice_big-13]]","widgets":{"fill_choice_big-13":{"type":"fill_choice_big","options":["$\\frac{1}{625}$","$-\\frac{1}{625}$","$625$","$-625$"],"placeholder":0,"answer":0}},"step":1,"hints":["Воспользуйтесь правилом отрицательной степени: $a^{-n}=\\frac{1}{a^n}$.","Имеем $25^{-2}=\\frac{1}{25^2}=\\frac{1}{625}$."]},{"content":"$$0.2^{-2}$$[[fill_choice_big-14]]","widgets":{"fill_choice_big-14":{"type":"fill_choice_big","options":["$25$","$-25$","$\\frac{1}{25}$","$-\\frac{1}{25}$"],"placeholder":0,"answer":0}},"step":1,"hints":["По правилу: $a^{-n}=\\frac{1}{a^n}$.","Тогда $0.2^{-2}=\\frac{1}{(0.2)^2}$. Так как $0.2=\\frac{1}{5}$, получаем $\\frac{1}{(\\frac{1}{5})^2}=\\frac{1}{\\frac{1}{25}}=25$."]},{"content":"$$\\left(\\frac{2}{3}\\right)^{-3}$$[[fill_choice_big-15]]","widgets":{"fill_choice_big-15":{"type":"fill_choice_big","options":["$\\frac{27}{8}$","$\\frac{8}{27}$","$-\\frac{27}{8}$","$-\\frac{8}{27}$"],"placeholder":0,"answer":0}},"step":1,"hints":["Используйте правило: $a^{-n}=\\frac{1}{a^n}$ или переворот дроби: $\\left(\\frac{m}{n}\\right)^{-k}=\\left(\\frac{n}{m}\\right)^k$.","Имеем $\\left(\\frac{2}{3}\\right)^{-3}=\\left(\\frac{3}{2}\\right)^3=\\frac{27}{8}$."]}],"mix":1}

{"questions":[{"content":"$$\\left(\\frac{3}{5}\\right)^{-2}$$[[fill_choice_big-16]]","widgets":{"fill_choice_big-16":{"type":"fill_choice_big","options":["$\\frac{25}{9}$","$\\frac{9}{25}$","$-\\frac{25}{9}$","$-\\frac{9}{25}$"],"placeholder":0,"answer":0}},"step":1,"hints":["Используйте правило: $\\left(\\frac{m}{n}\\right)^{-k}=\\left(\\frac{n}{m}\\right)^k$.","Тогда $\\left(\\frac{3}{5}\\right)^{-2}=\\left(\\frac{5}{3}\\right)^2=\\frac{25}{9}$."]},{"content":"$$\\left(\\frac{4}{7}\\right)^{-3}$$[[fill_choice_big-17]]","widgets":{"fill_choice_big-17":{"type":"fill_choice_big","options":["$\\frac{343}{64}$","$\\frac{64}{343}$","$-\\frac{343}{64}$","$-\\frac{64}{343}$"],"placeholder":0,"answer":0}},"step":1,"hints":["Отрицательная степень у дроби означает переворот: $\\left(\\frac{m}{n}\\right)^{-k}=\\left(\\frac{n}{m}\\right)^k$.","Получаем $\\left(\\frac{4}{7}\\right)^{-3}=\\left(\\frac{7}{4}\\right)^3=\\frac{343}{64}$."]},{"content":"$$\\left(\\frac{5}{2}\\right)^{-4}$$[[fill_choice_big-18]]","widgets":{"fill_choice_big-18":{"type":"fill_choice_big","options":["$\\frac{16}{625}$","$\\frac{625}{16}$","$-\\frac{16}{625}$","$-\\frac{625}{16}$"],"placeholder":0,"answer":0}},"step":1,"hints":["Примените правило: $\\left(\\frac{m}{n}\\right)^{-k}=\\left(\\frac{n}{m}\\right)^k$.","Тогда $\\left(\\frac{5}{2}\\right)^{-4}=\\left(\\frac{2}{5}\\right)^4=\\frac{16}{625}$."]},{"content":"$$\\left(\\frac{7}{9}\\right)^{-2}$$[[fill_choice_big-19]]","widgets":{"fill_choice_big-19":{"type":"fill_choice_big","options":["$\\frac{81}{49}$","$\\frac{49}{81}$","$-\\frac{81}{49}$","$-\\frac{49}{81}$"],"placeholder":0,"answer":0}},"step":1,"hints":["Отрицательная степень у дроби: переверните дробь и возведите в положительную степень.","Имеем $\\left(\\frac{7}{9}\\right)^{-2}=\\left(\\frac{9}{7}\\right)^2=\\frac{81}{49}$."]},{"content":"$$\\left(\\frac{11}{3}\\right)^{-2}$$[[fill_choice_big-20]]","widgets":{"fill_choice_big-20":{"type":"fill_choice_big","options":["$\\frac{9}{121}$","$\\frac{121}{9}$","$-\\frac{9}{121}$","$-\\frac{121}{9}$"],"placeholder":0,"answer":0}},"step":1,"hints":["Воспользуйтесь правилом: $\\left(\\frac{m}{n}\\right)^{-k}=\\left(\\frac{n}{m}\\right)^k$.","Тогда $\\left(\\frac{11}{3}\\right)^{-2}=\\left(\\frac{3}{11}\\right)^2=\\frac{9}{121}$."]},{"content":"$$\\left(\\frac{8}{5}\\right)^{-3}$$[[fill_choice_big-21]]","widgets":{"fill_choice_big-21":{"type":"fill_choice_big","options":["$\\frac{125}{512}$","$\\frac{512}{125}$","$-\\frac{125}{512}$","$-\\frac{512}{125}$"],"placeholder":0,"answer":0}},"step":1,"hints":["Сначала переверните дробь: $\\left(\\frac{8}{5}\\right)^{-3}=\\left(\\frac{5}{8}\\right)^3$.","Далее $\\left(\\frac{5}{8}\\right)^3=\\frac{125}{512}$."]},{"content":"$$\\left(\\frac{13}{4}\\right)^{-2}$$[[fill_choice_big-22]]","widgets":{"fill_choice_big-22":{"type":"fill_choice_big","options":["$\\frac{16}{169}$","$\\frac{169}{16}$","$-\\frac{16}{169}$","$-\\frac{169}{16}$"],"placeholder":0,"answer":0}},"step":1,"hints":["Используйте: $\\left(\\frac{m}{n}\\right)^{-k}=\\left(\\frac{n}{m}\\right)^k$.","Тогда $\\left(\\frac{13}{4}\\right)^{-2}=\\left(\\frac{4}{13}\\right)^2=\\frac{16}{169}$."]},{"content":"$$\\left(\\frac{2}{9}\\right)^{-3}$$[[fill_choice_big-23]]","widgets":{"fill_choice_big-23":{"type":"fill_choice_big","options":["$\\frac{729}{8}$","$\\frac{8}{729}$","$-\\frac{729}{8}$","$-\\frac{8}{729}$"],"placeholder":0,"answer":0}},"step":1,"hints":["При отрицательной степени переворачиваем дробь: $\\left(\\frac{2}{9}\\right)^{-3}=\\left(\\frac{9}{2}\\right)^3$.","Затем $\\left(\\frac{9}{2}\\right)^3=\\frac{9^3}{2^3}=\\frac{729}{8}$."]},{"content":"$$\\left(\\frac{6}{11}\\right)^{-2}$$[[fill_choice_big-24]]","widgets":{"fill_choice_big-24":{"type":"fill_choice_big","options":["$\\frac{121}{36}$","$\\frac{36}{121}$","$-\\frac{121}{36}$","$-\\frac{36}{121}$"],"placeholder":0,"answer":0}},"step":1,"hints":["Используйте правило: $\\left(\\frac{m}{n}\\right)^{-k}=\\left(\\frac{n}{m}\\right)^k$.","Тогда $\\left(\\frac{6}{11}\\right)^{-2}=\\left(\\frac{11}{6}\\right)^2=\\frac{121}{36}$."]},{"content":"$$\\left(\\frac{15}{8}\\right)^{-2}$$[[fill_choice_big-25]]","widgets":{"fill_choice_big-25":{"type":"fill_choice_big","options":["$\\frac{64}{225}$","$\\frac{225}{64}$","$-\\frac{64}{225}$","$-\\frac{225}{64}$"],"placeholder":0,"answer":0}},"step":1,"hints":["Переворачиваем дробь и убираем минус у показателя: $\\left(\\frac{15}{8}\\right)^{-2}=\\left(\\frac{8}{15}\\right)^2$.","Получаем $\\left(\\frac{8}{15}\\right)^2=\\frac{64}{225}$."]}],"mix":1}