Raise a power to a power

{"questions":[{"content":"$$(3^2)^4$$[[fill_choice_big-1]]","widgets":{"fill_choice_big-1":{"type":"fill_choice_big","options":["$3^6$","$3^2$","$3^8$","$3^{-2}$"],"placeholder":0,"answer":2}},"step":1,"hints":["A power is raised to a power, so we leave the base the same and multiply the exponents.","$(3^2)^4=3^{2\\cdot 4}=3^8$."]}]}

{"questions":[{"content":"$$(2^3)^4$$[[fill_choice_big-1]]","widgets":{"fill_choice_big-1":{"type":"fill_choice_big","options":["$2^{7}$","$2^{12}$","$8^{4}$","$2^{1}$"],"placeholder":0,"answer":1}},"step":1,"hints":["The power is raised to a power: we leave the base the same, we multiply the exponents.","$(2^3)^4=2^{3\\cdot 4}=2^{12}$."]},{"content":"$$(5^2)^3$$[[fill_choice_big-2]]","widgets":{"fill_choice_big-2":{"type":"fill_choice_big","options":["$5^{5}$","$25^{3}$","$5^{6}$","$5^{1}$"],"placeholder":0,"answer":2}},"step":1,"hints":["When raising a power to a power, we multiply the exponents: $(a^m)^n=a^{mn}$.","$(5^2)^3=5^{2\\cdot 3}=5^6$."]},{"content":"$$(7^3)^2$$[[fill_choice_big-3]]","widgets":{"fill_choice_big-3":{"type":"fill_choice_big","options":["$7^{6}$","$7^{9}$","$21^{2}$","$7^{1}$"],"placeholder":0,"answer":0}},"step":1,"hints":["We use the rule $(a^m)^n=a^{mn}$.","$(7^3)^2=7^{3\\cdot 2}=7^6$."]},{"content":"$$(3^4)^3$$[[fill_choice_big-4]]","widgets":{"fill_choice_big-4":{"type":"fill_choice_big","options":["$3^{7}$","$3^{12}$","$12^{3}$","$3^{1}$"],"placeholder":0,"answer":1}},"step":1,"hints":["The base is preserved, the exponents are multiplied.","$(3^4)^3=3^{4\\cdot 3}=3^{12}$."]},{"content":"$$(10^2)^5$$[[fill_choice_big-5]]","widgets":{"fill_choice_big-5":{"type":"fill_choice_big","options":["$10^{7}$","$10^{10}$","$100^{5}$","$10^{3}$"],"placeholder":0,"answer":1}},"step":1,"hints":["Let's apply the formula $(a^m)^n=a^{mn}$.","$(10^2)^5=10^{2\\cdot 5}=10^{10}$."]},{"content":"$$(4^3)^2$$[[fill_choice_big-6]]","widgets":{"fill_choice_big-6":{"type":"fill_choice_big","options":["$4^{5}$","$8^{2}$","$4^{6}$","$4^{1}$"],"placeholder":0,"answer":2}},"step":1,"hints":["When raising a power to a power, we multiply the exponents.","$(4^3)^2=4^{3\\cdot 2}=4^6$."]},{"content":"$$(6^2)^4$$[[fill_choice_big-7]]","widgets":{"fill_choice_big-7":{"type":"fill_choice_big","options":["$6^{6}$","$6^{8}$","$12^{4}$","$6^{1}$"],"placeholder":0,"answer":1}},"step":1,"hints":["We use the rule $(a^m)^n=a^{mn}$.","$(6^2)^4=6^{2\\cdot 4}=6^8$."]},{"content":"$$(8^2)^3$$[[fill_choice_big-8]]","widgets":{"fill_choice_big-8":{"type":"fill_choice_big","options":["$8^{6}$","$8^{5}$","$16^{3}$","$8^{1}$"],"placeholder":0,"answer":0}},"step":1,"hints":["We leave the base and multiply the exponents.","$(8^2)^3=8^{2\\cdot 3}=8^6$."]},{"content":"$$(9^3)^2$$[[fill_choice_big-9]]","widgets":{"fill_choice_big-9":{"type":"fill_choice_big","options":["$9^{9}$","$9^{6}$","$27^{2}$","$9^{1}$"],"placeholder":0,"answer":1}},"step":1,"hints":["Let's apply the power to power formula.","$(9^3)^2=9^{3\\cdot 2}=9^6$."]},{"content":"$$(2^5)^3$$[[fill_choice_big-10]]","widgets":{"fill_choice_big-10":{"type":"fill_choice_big","options":["$2^{8}$","$2^{15}$","$10^{3}$","$2^{2}$"],"placeholder":0,"answer":1}},"step":1,"hints":["When raising a power to a power, the exponents are multiplied.","$(2^5)^3=2^{5\\cdot 3}=2^{15}$."]},{"content":"$$(11^2)^2$$[[fill_choice_big-11]]","widgets":{"fill_choice_big-11":{"type":"fill_choice_big","options":["$11^{4}$","$11^{2}$","$22^{2}$","$11^{0}$"],"placeholder":0,"answer":0}},"step":1,"hints":["We use $(a^m)^n=a^{mn}$.","$(11^2)^2=11^{2\\cdot 2}=11^4$."]},{"content":"$$(3^5)^2$$[[fill_choice_big-12]]","widgets":{"fill_choice_big-12":{"type":"fill_choice_big","options":["$3^{7}$","$3^{10}$","$15^{2}$","$3^{3}$"],"placeholder":0,"answer":1}},"step":1,"hints":["When raising a power to a power, we multiply the exponents.","$(3^5)^2=3^{5\\cdot 2}=3^{10}$."]},{"content":"$$(12^1)^7$$[[fill_choice_big-13]]","widgets":{"fill_choice_big-13":{"type":"fill_choice_big","options":["$12^{8}$","$12^{7}$","$12^{1}$","$7^{12}$"],"placeholder":0,"answer":1}},"step":1,"hints":["We use the rule $(a^m)^n=a^{mn}$, even if $m=1$.","$(12^1)^7=12^{1\\cdot 7}=12^7$."]},{"content":"$$(4^2)^6$$[[fill_choice_big-14]]","widgets":{"fill_choice_big-14":{"type":"fill_choice_big","options":["$4^{8}$","$4^{12}$","$16^{6}$","$4^{4}$"],"placeholder":0,"answer":1}},"step":1,"hints":["When raising a power to a power, the exponents are multiplied.","$(4^2)^6=4^{2\\cdot 6}=4^{12}$."]},{"content":"$$(15^2)^3$$[[fill_choice_big-15]]","widgets":{"fill_choice_big-15":{"type":"fill_choice_big","options":["$15^{6}$","$15^{5}$","$30^{3}$","$225^{3}$"],"placeholder":0,"answer":0}},"step":1,"hints":["We use the formula $(a^m)^n=a^{mn}$.","$(15^2)^3=15^{2\\cdot 3}=15^6$."]}]}

{"questions":[{"content":"$$(2^4)^3$$[[fill_choice_big-16]]","widgets":{"fill_choice_big-16":{"type":"fill_choice_big","options":["$2^{7}$","$2^{12}$","$8^{3}$","$2^{4}$"],"placeholder":0,"answer":1}},"step":1,"hints":["We use the $(a^m)^n=a^{mn}$ rule: we leave the base the same, we multiply the exponents.","$(2^4)^3=2^{4\\cdot 3}=2^{12}$."]},{"content":"$$(5^3)^2$$[[fill_choice_big-17]]","widgets":{"fill_choice_big-17":{"type":"fill_choice_big","options":["$5^{6}$","$5^{5}$","$15^{2}$","$125^{2}$"],"placeholder":0,"answer":0}},"step":1,"hints":["When raising a power to a power, we multiply the exponents: $(a^m)^n=a^{mn}$.","$(5^3)^2=5^{3\\cdot 2}=5^6$."]},{"content":"$$(3^2)^5$$[[fill_choice_big-18]]","widgets":{"fill_choice_big-18":{"type":"fill_choice_big","options":["$3^{7}$","$3^{10}$","$9^{5}$","$3^{3}$"],"placeholder":0,"answer":1}},"step":1,"hints":["The base is preserved, the exponents are multiplied.","$(3^2)^5=3^{2\\cdot 5}=3^{10}$."]},{"content":"$$(7^2)^4$$[[fill_choice_big-19]]","widgets":{"fill_choice_big-19":{"type":"fill_choice_big","options":["$7^{6}$","$7^{8}$","$14^{4}$","$49^{4}$"],"placeholder":0,"answer":1}},"step":1,"hints":["We apply the formula $(a^m)^n=a^{mn}$.","$(7^2)^4=7^{2\\cdot 4}=7^8$."]},{"content":"$$(10^3)^2$$[[fill_choice_big-20]]","widgets":{"fill_choice_big-20":{"type":"fill_choice_big","options":["$10^{5}$","$10^{6}$","$1000^{2}$","$30^{2}$"],"placeholder":0,"answer":1}},"step":1,"hints":["We use the power to power rule.","$(10^3)^2=10^{3\\cdot 2}=10^6$."]},{"content":"$$(4^4)^2$$[[fill_choice_big-21]]","widgets":{"fill_choice_big-21":{"type":"fill_choice_big","options":["$4^{8}$","$4^{6}$","$16^{2}$","$8^{4}$"],"placeholder":0,"answer":0}},"step":1,"hints":["Indicators are multiplied: $(a^m)^n=a^{mn}$.","$(4^4)^2=4^{4\\cdot 2}=4^8$."]},{"content":"$$(6^3)^3$$[[fill_choice_big-22]]","widgets":{"fill_choice_big-22":{"type":"fill_choice_big","options":["$6^{6}$","$6^{9}$","$18^{3}$","$216^{3}$"],"placeholder":0,"answer":1}},"step":1,"hints":["We use $(a^m)^n=a^{mn}$.","$(6^3)^3=6^{3\\cdot 3}=6^9$."]},{"content":"$$(8^3)^2$$[[fill_choice_big-23]]","widgets":{"fill_choice_big-23":{"type":"fill_choice_big","options":["$8^{5}$","$8^{6}$","$24^{2}$","$512^{2}$"],"placeholder":0,"answer":1}},"step":1,"hints":["We leave the base the same, we multiply the exponents.","$(8^3)^2=8^{3\\cdot 2}=8^6$."]},{"content":"$$(9^2)^4$$[[fill_choice_big-24]]","widgets":{"fill_choice_big-24":{"type":"fill_choice_big","options":["$9^{6}$","$9^{8}$","$18^{4}$","$81^{4}$"],"placeholder":0,"answer":1}},"step":1,"hints":["We apply the formula $(a^m)^n=a^{mn}$.","$(9^2)^4=9^{2\\cdot 4}=9^8$."]},{"content":"$$(12^2)^3$$[[fill_choice_big-25]]","widgets":{"fill_choice_big-25":{"type":"fill_choice_big","options":["$12^{5}$","$12^{6}$","$24^{3}$","$144^{3}$"],"placeholder":0,"answer":1}},"step":1,"hints":["When raising a power to a power, the exponents are multiplied.","$(12^2)^3=12^{2\\cdot 3}=12^6$."]},{"content":"$$(13^1)^5$$[[fill_choice_big-26]]","widgets":{"fill_choice_big-26":{"type":"fill_choice_big","options":["$13^{6}$","$13^{5}$","$13^{1}$","$5^{13}$"],"placeholder":0,"answer":1}},"step":1,"hints":["Even if the exponent is 1, the rule works: $(a^m)^n=a^{mn}$.","$(13^1)^5=13^{1\\cdot 5}=13^5$."]},{"content":"$$(2^6)^2$$[[fill_choice_big-27]]","widgets":{"fill_choice_big-27":{"type":"fill_choice_big","options":["$2^{8}$","$2^{12}$","$4^{6}$","$2^{36}$"],"placeholder":0,"answer":1}},"step":1,"hints":["We use $(a^m)^n=a^{mn}$.","$(2^6)^2=2^{6\\cdot 2}=2^{12}$."]},{"content":"$$(3^3)^4$$[[fill_choice_big-28]]","widgets":{"fill_choice_big-28":{"type":"fill_choice_big","options":["$3^{7}$","$3^{12}$","$9^{4}$","$12^{3}$"],"placeholder":0,"answer":1}},"step":1,"hints":["Multiply the exponents: $(a^m)^n=a^{mn}$.","$(3^3)^4=3^{3\\cdot 4}=3^{12}$."]},{"content":"$$(5^2)^4$$[[fill_choice_big-29]]","widgets":{"fill_choice_big-29":{"type":"fill_choice_big","options":["$5^{6}$","$5^{8}$","$25^{4}$","$10^{4}$"],"placeholder":0,"answer":1}},"step":1,"hints":["The base is preserved, the exponents are multiplied.","$(5^2)^4=5^{2\\cdot 4}=5^8$."]},{"content":"$$(16^2)^2$$[[fill_choice_big-30]]","widgets":{"fill_choice_big-30":{"type":"fill_choice_big","options":["$16^{3}$","$16^{4}$","$32^{2}$","$256^{2}$"],"placeholder":0,"answer":1}},"step":1,"hints":["We use the rule $(a^m)^n=a^{mn}$.","$(16^2)^2=16^{2\\cdot 2}=16^4$."]}]}