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Therefore, the reciprocal of $8$ is $\\frac{1}{8}$."},{"text":"$$\\frac{8}{1}$","correct":false,"explanation":"Writing $8$ as a fraction over 1 does not invert it. Swapping the numerator and denominator gives the reciprocal $\\frac{1}{8}$."},{"text":"$$\\frac{1}{64}$","correct":false,"explanation":"This comes from squaring 8 rather than finding its reciprocal. A reciprocal multiplies by the original number to make 1, so it is $\\frac{1}{8}$."},{"text":"$$-\\frac{1}{8}$","correct":false,"explanation":"Finding a reciprocal does not change the sign of a positive number. Since $8 \\times \\frac{1}{8}=1$, the reciprocal is $\\frac{1}{8}$."}]},{"id":"e5c3fc56-89b2-4f17-9b4c-75a3bf1783a0","cardType":"basic","points":1,"front":"What is the reciprocal of $\\frac{4}{9}$?","back":"$$\\frac{9}{4}$","mcqOptions":[{"text":"$$\\frac{9}{4}$","correct":true,"explanation":"The reciprocal of a fraction is found by interchanging its numerator and denominator. Reversing $\\frac{4}{9}$ gives $\\frac{9}{4}$."},{"text":"$$\\frac{4}{9}$","correct":false,"explanation":"Leaving the fraction unchanged does not produce its reciprocal. The numerator and denominator must be interchanged, giving $\\frac{9}{4}$."},{"text":"$$-\\frac{9}{4}$","correct":false,"explanation":"Taking a reciprocal does not introduce a negative sign. Interchanging the numerator and denominator of $\\frac{4}{9}$ gives the positive fraction $\\frac{9}{4}$."},{"text":"$$\\frac{9}{8}$","correct":false,"explanation":"This incorrectly doubles the original numerator instead of reversing the fraction. A reciprocal swaps 4 and 9, so the correct result is $\\frac{9}{4}$."}]},{"id":"40dd996b-3691-4134-810e-ab58fc14a9f7","cardType":"basic","points":1,"front":"What operation does a negative index indicate?","back":"Taking the reciprocal of the base.","mcqOptions":[{"text":"Taking the reciprocal of the base.","correct":true,"explanation":"A negative index means the corresponding positive power is moved to the denominator. Equivalently, the reciprocal of the base is taken before applying the positive index."},{"text":"Taking the negative of the base.","correct":false,"explanation":"A negative index does not make the base negative; it indicates a reciprocal. For example, $a^{-n}=\\frac{1}{a^n}$ for any non-zero $a$."},{"text":"Multiplying the base by the index.","correct":false,"explanation":"Indices represent repeated multiplication or related roots, not multiplication by the exponent. A negative index specifically indicates taking the reciprocal of the corresponding positive power."},{"text":"Subtracting the index from the base.","correct":false,"explanation":"An exponent is not subtracted from its base. A negative index means taking the reciprocal, so $a^{-n}$ is equal to $\\frac{1}{a^n}$."}]},{"id":"74c91d25-76e4-4589-bd25-138bf10c5a98","cardType":"basic","points":1,"front":"Evaluate $5^{-2}$.","back":"$$\\frac{1}{25}$","mcqOptions":[{"text":"$$\\frac{1}{25}$","correct":true,"explanation":"The negative index gives the reciprocal of the positive power: $5^{-2}=\\frac{1}{5^2}$. Since $5^2=25$, the result is $\\frac{1}{25}$."},{"text":"$$\\frac{1}{10}$","correct":false,"explanation":"This treats the exponent as multiplication by 2 rather than a power. The correct calculation is $5^{-2}=\\frac{1}{5^2}=\\frac{1}{25}$."},{"text":"$$-\\frac{1}{25}$","correct":false,"explanation":"A negative exponent creates a reciprocal; it does not make the value negative. Since $5^2=25$, the correct value is the positive fraction $\\frac{1}{25}$."},{"text":"$$25$","correct":false,"explanation":"This evaluates $5^2$ but ignores the negative sign on the index. The negative index requires the reciprocal, so $5^{-2}=\\frac{1}{25}$."}]},{"id":"b1308fe6-b390-4d3e-9c35-e4df45c0fc36","cardType":"basic","points":1,"front":"Evaluate $\\left(\\frac{3}{8}\\right)^{-1}$.","back":"$$\\frac{8}{3}$","mcqOptions":[{"text":"$$\\frac{8}{3}$","correct":true,"explanation":"An index of $-1$ means taking the reciprocal. Interchanging the numerator and denominator of $\\frac{3}{8}$ gives $\\frac{8}{3}$."},{"text":"$$\\frac{3}{8}$","correct":false,"explanation":"This leaves the original fraction unchanged and ignores the index of $-1$. Taking the reciprocal of $\\frac{3}{8}$ gives $\\frac{8}{3}$."},{"text":"$$-\\frac{8}{3}$","correct":false,"explanation":"A negative index does not introduce a negative sign into the value. It indicates a reciprocal, so the result is the positive fraction $\\frac{8}{3}$."},{"text":"$$\\frac{9}{64}$","correct":false,"explanation":"This squares the numerator and denominator, corresponding to an index of 2 rather than $-1$. The index $-1$ gives the reciprocal $\\frac{8}{3}$."}]},{"id":"a03adb1f-4311-48df-af3a-0e9a18724e47","cardType":"cloze","points":null,"front":"For any non-zero base $a$, the negative index rule is $a^{-n} = \\text{[...]}.","back":"For any non-zero base $a$, the negative index rule is $a^{-n} = \\boxed{\\frac{1}{a^n}}.","mcqOptions":[{"text":"$$\\frac{1}{a^n}$","correct":true,"explanation":"A negative index means taking the reciprocal of the corresponding positive power. Thus, for non-zero $a$, $a^{-n}$ is equal to $\\frac{1}{a^n}$."},{"text":"$$-a^n$","correct":false,"explanation":"A negative index affects the position of the power, not its sign. The base’s positive power belongs in the denominator, giving $a^{-n}=\\frac{1}{a^n}$."},{"text":"$$\\frac{1}{a^{-n}}$","correct":false,"explanation":"This takes the reciprocal while incorrectly keeping the exponent negative, producing $a^n$. The negative index rule instead gives $a^{-n}=\\frac{1}{a^n}$."},{"text":"$$\\frac{-1}{a^n}$","correct":false,"explanation":"The minus sign in the index does not make the fraction negative. It indicates a reciprocal, so the numerator is positive and the result is $\\frac{1}{a^n}$."}]},{"id":"c98ce278-23a1-472b-9ebf-3b7b5d2bffb0","cardType":"cloze","points":null,"front":"The fractional index $a^{\\frac{1}{n}}$ is equivalent to the radical expression $\\text{[...]}$.","back":"The fractional index $a^{\\frac{1}{n}}$ is equivalent to the radical expression $\\boxed{\\sqrt[n]{a}}$.","mcqOptions":[{"text":"$$\\sqrt[n]{a}$","correct":true,"explanation":"A denominator of $n$ in a fractional index indicates an $n$th root. Therefore, $a^{\\frac{1}{n}}$ is equivalent to $\\sqrt[n]{a}$."},{"text":"$$n\\sqrt{a}$","correct":false,"explanation":"The denominator $n$ specifies the order of the root; it is not a coefficient multiplying a square root. The equivalent expression is $\\sqrt[n]{a}$."},{"text":"$$\\sqrt{a^n}$","correct":false,"explanation":"This incorrectly treats $n$ as a power inside a square root. In $a^{\\frac{1}{n}}$, the denominator specifies an $n$th root, giving $\\sqrt[n]{a}$."},{"text":"$$\\frac{1}{\\sqrt[n]{a}}$","correct":false,"explanation":"A reciprocal would require a negative fractional index, such as $a^{-\\frac{1}{n}}$. Since the index is positive, the correct expression is $\\sqrt[n]{a}$."}]},{"id":"52cfa59f-e003-4544-8f42-7c3946f5426e","cardType":"basic","points":2,"front":"In the fractional index $a^{\\frac{m}{n}}$, what do the numerator and denominator represent?","back":"

Denominator ($n$): root
Numerator ($m$): power

","mcqOptions":[{"text":"

Denominator ($n$): root
Numerator ($m$): power

","correct":true,"explanation":"In $a^{\\frac{m}{n}}$, the denominator gives the order of the root and the numerator gives the power: $a^{\\frac{m}{n}}=\\sqrt[n]{a^m}$."},{"text":"

Denominator ($n$): power
Numerator ($m$): root

","correct":false,"explanation":"This reverses the roles of the numerator and denominator. In a fractional index, $n$ specifies the root and $m$ specifies the power."},{"text":"

Denominator ($n$): root
Numerator ($m$): coefficient

","correct":false,"explanation":"The numerator is not a multiplier placed before the radical. It specifies the power, while the denominator specifies the root, as in $\\sqrt[n]{a^m}$."},{"text":"

Denominator ($n$): reciprocal
Numerator ($m$): power

","correct":false,"explanation":"A denominator in a fractional index indicates a root, not a reciprocal; reciprocals are associated with negative indices. The numerator still represents the power."}]},{"id":"c26d0fef-43c7-45bc-9c89-1ba4a3baa30b","cardType":"basic","points":2,"front":"Evaluate $343^{\\frac{2}{3}}$.","back":"49","mcqOptions":[{"text":"49","correct":true,"explanation":"The denominator 3 means take the cube root, giving $\\sqrt[3]{343}=7$, and the numerator 2 means square the result: $7^2=49$."},{"text":"14","correct":false,"explanation":"This comes from doubling the cube root instead of squaring it. Since $343^{\\frac{2}{3}}=(\\sqrt[3]{343})^2=7^2$, the correct value is 49."},{"text":"98","correct":false,"explanation":"This incorrectly doubles 49 after evaluating the fractional power. The numerator 2 is already applied as a square, so $343^{\\frac{2}{3}}=7^2=49$."},{"text":"2401","correct":false,"explanation":"This uses $7^4$ rather than $7^2$, effectively applying the numerator twice. Taking the cube root and then squaring gives $343^{\\frac{2}{3}}=49$."}]},{"id":"cb119426-4675-4b39-9e93-bf27a382a174","cardType":"basic","points":1,"front":"Why is $144^{\\frac{1}{2}}$ not equal to 72?","back":"The index $\\frac{1}{2}$ means square root ($\\sqrt{144} = 12$), not dividing by 2.","mcqOptions":[{"text":"The index $\\frac{1}{2}$ means square root ($\\sqrt{144} = 12$), not dividing by 2.","correct":true,"explanation":"A denominator of 2 in a fractional index represents a square root. Therefore $144^{\\frac{1}{2}}=\\sqrt{144}=12$, whereas dividing 144 by 2 gives 72."},{"text":"The index $\\frac{1}{2}$ means divide the square root by 2 ($\\sqrt{144} \\div 2 = 6$), not dividing 144 by 2.","correct":false,"explanation":"The denominator of the index specifies the root; it is not an additional divisor. Thus $144^{\\frac{1}{2}}=\\sqrt{144}=12$, with no subsequent division by 2."},{"text":"The index $\\frac{1}{2}$ means square then divide ($144^2 \\div 2 = 10368$), not dividing 144 by 2.","correct":false,"explanation":"A fractional index does not instruct you to square the base and divide. The denominator 2 means take the square root, so $144^{\\frac{1}{2}}=12$."},{"text":"The index $\\frac{1}{2}$ means halve the square root ($\\frac{1}{2}\\sqrt{144} = 6$), not dividing 144 by 2.","correct":false,"explanation":"Treating the index as a coefficient incorrectly multiplies the square root by one-half. It represents the square-root operation itself, giving $144^{\\frac{1}{2}}=12$."}]},{"id":"3945a30f-a935-4e1a-9e80-1ce349596baf","cardType":"cloze","points":null,"front":"The combined rule for negative fractional indices is $a^{-\\frac{m}{n}} = \\text{[...]}.","back":"The combined rule for negative fractional indices is $a^{-\\frac{m}{n}} = \\boxed{\\frac{1}{(\\sqrt[n]{a})^m}}.","mcqOptions":[{"text":"$$\\frac{1}{(\\sqrt[n]{a})^m}$","correct":true,"explanation":"The fraction $\\frac{m}{n}$ means take the $n$th root and raise it to the power $m$, while the negative sign takes the reciprocal."},{"text":"$$\\frac{1}{(\\sqrt[m]{a})^n}$","correct":false,"explanation":"This swaps the numerator and denominator roles. In $a^{-\\frac{m}{n}}$, $n$ determines the root and $m$ determines the power, so the result is $\\frac{1}{(\\sqrt[n]{a})^m}$."},{"text":"$$(\\sqrt[n]{a})^m$","correct":false,"explanation":"This handles the fractional index but ignores the negative sign. A negative index requires the reciprocal, so the correct expression is $\\frac{1}{(\\sqrt[n]{a})^m}$."},{"text":"$$\\frac{1}{\\sqrt[n]{a^n}}$","correct":false,"explanation":"This replaces the numerator $m$ with $n$, causing the root and power to cancel. The correct rule is $a^{-\\frac{m}{n}}=\\frac{1}{(\\sqrt[n]{a})^m}$."}]},{"id":"349dc2f6-281e-4887-a9d1-e63ebc4c4192","cardType":"basic","points":2,"front":"Evaluate $32^{-\\frac{2}{5}}$.","back":"$$\\frac{1}{4}$","mcqOptions":[{"text":"$$\\frac{1}{4}$","correct":true,"explanation":"The fifth root of 32 is 2, and squaring gives 4. The negative index then takes the reciprocal, so $32^{-\\frac{2}{5}}=\\frac{1}{4}$."},{"text":"$$4$","correct":false,"explanation":"This correctly evaluates $32^{\\frac{2}{5}}$ but ignores the negative sign. A negative index takes the reciprocal, so the correct value is $\\frac{1}{4}$."},{"text":"$$\\frac{1}{10}$","correct":false,"explanation":"This incorrectly multiplies the fifth root by the numerator, giving $2\\times5$. The numerator requires squaring: $(\\sqrt[5]{32})^2=4$, then reciprocating gives $\\frac{1}{4}$."},{"text":"$$\\frac{1}{64}$","correct":false,"explanation":"This squares 32 before taking account of the fifth root. The fifth root is 2, which squares to 4, and the negative index gives $\\frac{1}{4}$."}]},{"id":"451ddd34-f920-40d9-8036-8a49be808807","cardType":"basic","points":1,"front":"What is the most efficient first step to evaluate $\\left(\\frac{a}{b}\\right)^{-n}$?","back":"Invert the base to $\\left(\\frac{b}{a}\\right)^n$ to remove the negative index.","mcqOptions":[{"text":"Invert the base to $\\left(\\frac{b}{a}\\right)^n$ to remove the negative index.","correct":true,"explanation":"A negative index means take the reciprocal of the base. Inverting $\\frac{a}{b}$ changes the exponent from $-n$ to $n$, giving $\\left(\\frac{b}{a}\\right)^n$."},{"text":"Keep the base as $\\left(\\frac{a}{b}\\right)^n$ and simply remove the negative index.","correct":false,"explanation":"Removing a negative sign without taking a reciprocal changes the value. The base must be inverted, so $\\left(\\frac{a}{b}\\right)^{-n}=\\left(\\frac{b}{a}\\right)^n$."},{"text":"Invert the index to $\\left(\\frac{a}{b}\\right)^{\\frac{1}{n}}$ to remove the negative index.","correct":false,"explanation":"A negative exponent does not become a reciprocal exponent; that would introduce an $n$th root. Instead, invert the base to obtain $\\left(\\frac{b}{a}\\right)^n$."},{"text":"Negate the base to $\\left(-\\frac{a}{b}\\right)^n$ to remove the negative index.","correct":false,"explanation":"The negative sign belongs to the exponent and does not make the base negative. It requires taking the reciprocal, giving $\\left(\\frac{b}{a}\\right)^n$."}]},{"id":"03c189c5-e0c4-44ec-a8f3-6fcb3db8c2b8","cardType":"basic","points":2,"front":"Evaluate $\\left(\\frac{16}{25}\\right)^{-\\frac{3}{2}}$.","back":"$$\\frac{125}{64}$","mcqOptions":[{"text":"$$\\frac{125}{64}$","correct":true,"explanation":"Invert the fraction to get $\\left(\\frac{25}{16}\\right)^{\\frac{3}{2}}$. Taking square roots gives $\\frac{5}{4}$, and cubing produces $\\frac{125}{64}$."},{"text":"$$\\frac{64}{125}$","correct":false,"explanation":"This evaluates the positive fractional power but ignores the negative index. The base must first be inverted, leading to $\\left(\\frac{5}{4}\\right)^3=\\frac{125}{64}$."},{"text":"$$\\frac{25}{16}$","correct":false,"explanation":"This inverts the base but fails to apply the power $\\frac{3}{2}$. After inversion, take the square root and cube: $\\left(\\frac{5}{4}\\right)^3=\\frac{125}{64}$."},{"text":"$$\\frac{625}{256}$","correct":false,"explanation":"This effectively squares the inverted base instead of applying the exponent $\\frac{3}{2}$. Taking the square root first and then cubing gives $\\frac{125}{64}$."}]},{"id":"a3a45aec-feb8-4eab-a81f-36bde26ab90e","cardType":"basic","points":2,"front":"Simplify $(27x^6)^{\\frac{2}{3}}$.","back":"$$9x^4$","mcqOptions":[{"text":"$$9x^4$","correct":true,"explanation":"Taking the cube root gives $(27x^6)^{\\frac{1}{3}}=3x^2$, then squaring gives $(3x^2)^2=9x^4$."},{"text":"$$9x^3$","correct":false,"explanation":"The coefficient is handled correctly, but the variable exponent is multiplied incorrectly. Using the power rule gives $x^{6\\times\\frac{2}{3}}=x^4$, so the result is $9x^4$."},{"text":"$$6x^4$","correct":false,"explanation":"This doubles the cube root of 27 instead of squaring it. Since $27^{\\frac{2}{3}}=(\\sqrt[3]{27})^2=3^2=9$, the correct result is $9x^4$."},{"text":"$$9x^{12}$","correct":false,"explanation":"This multiplies the exponent 6 by 2 but ignores division by 3. The correct exponent is $6\\times\\frac{2}{3}=4$, giving $9x^4$."}]},{"id":"4c0b9776-68c4-4928-8933-39802c0e84b3","cardType":"cloze","points":null,"front":"The index law for a power of a power is $(a^p)^q = \\text{[...]}.","back":"The index law for a power of a power is $(a^p)^q = \\boxed{a^{pq}}.","mcqOptions":[{"text":"$$a^{pq}$","correct":true,"explanation":"When a power is raised to another power, the exponents multiply. Therefore $(a^p)^q=a^{p\\times q}=a^{pq}$."},{"text":"$$a^{p+q}$","correct":false,"explanation":"Adding exponents applies when multiplying powers with the same base, not when raising a power to another power. Here the exponents multiply, giving $a^{pq}$."},{"text":"$$a^{p-q}$","correct":false,"explanation":"Subtracting exponents applies when dividing powers with the same base. A power raised to another power requires multiplication, so $(a^p)^q=a^{pq}$."},{"text":"$$a^{\\frac{p}{q}}$","correct":false,"explanation":"Dividing exponents is associated with reversing a power operation, not applying another power. For $(a^p)^q$, multiply the exponents to obtain $a^{pq}$."}]},{"id":"c0c1b0be-4c0a-4464-a9cf-d7cc8210397f","cardType":"basic","points":1,"front":"What is the general strategy for finding an unknown index $n$ in an equation like $2 \\times \\sqrt{32} = 2^n$?","back":"Rewrite both sides using the same base and equate the indices.","mcqOptions":[{"text":"Rewrite both sides using the same base and equate the indices.","correct":true,"explanation":"Powers with the same base are equal only when their indices are equal, so expressing both sides as powers of 2 allows $n$ to be found directly."},{"text":"Rewrite both sides using the same index and equate the bases.","correct":false,"explanation":"Matching indices does not provide a valid general method for solving this equation. Both sides should be rewritten with the same base, after which their indices can be equated."},{"text":"Rewrite both sides using different bases and multiply the indices.","correct":false,"explanation":"Indices cannot be compared directly when the bases differ, and multiplication is not the relevant index operation. Use the same base on both sides and equate the indices."},{"text":"Evaluate both sides using decimal approximations and round the indices.","correct":false,"explanation":"Rounding decimal approximations can produce an inaccurate index and is unnecessary here. Rewriting both sides with the same base gives an exact result by equating the indices."}]},{"id":"4d5a0bd5-d435-468b-9029-b8fb52631742","cardType":"basic","points":2,"front":"Rewrite $\\sqrt{32}$ as a power of 2.","back":"$$2^{\\frac{5}{2}}$","mcqOptions":[{"text":"$$2^{\\frac{5}{2}}$","correct":true,"explanation":"Since $32=2^5$ and taking a square root is equivalent to raising to the power $\\frac{1}{2}$, $\\sqrt{32}=(2^5)^{1/2}=2^{5/2}$."},{"text":"$$2^5$","correct":false,"explanation":"This rewrites 32 itself but ignores the square root. A square root halves the index, so $\\sqrt{2^5}=2^{5/2}$."},{"text":"$$2^{\\frac{1}{5}}$","correct":false,"explanation":"This reverses the roles of the existing index and the root. Since $32=2^5$, applying a square root multiplies 5 by $\\frac{1}{2}$, giving $2^{5/2}$."},{"text":"$$2^{10}$","correct":false,"explanation":"This doubles the index as though 32 were being squared rather than square-rooted. Taking a square root halves the index, giving $2^{5/2}$."}]},{"id":"0db5a745-90c5-4f00-878f-37e015a49324","cardType":"basic","points":2,"front":"If $s = 5^q$, express $5^{q-1}$ in terms of $s$.","back":"$$\\frac{s}{5}$","mcqOptions":[{"text":"$$\\frac{s}{5}$","correct":true,"explanation":"Using $5^{q-1}=5^q\\div5$ and substituting $s=5^q$ gives $5^{q-1}=\\frac{s}{5}$."},{"text":"$$5s$","correct":false,"explanation":"Multiplying by 5 increases the index, since $5\\times5^q=5^{q+1}$. Decreasing the index by 1 requires division by 5, giving $\\frac{s}{5}$."},{"text":"$$\\frac{5}{s}$","correct":false,"explanation":"This incorrectly takes the reciprocal of $s$ and would correspond to a negative power involving $q$. Subtracting 1 from the index means dividing $s$ by 5."},{"text":"$$\\frac{s}{4}$","correct":false,"explanation":"Subtracting 1 from an index does not mean subtracting 1 from the base. The index law $5^{q-1}=5^q\\div5$ gives $\\frac{s}{5}$."}]},{"id":"9aca0684-6061-4c59-820c-0706daeac767","cardType":"basic","points":1,"front":"Why is $0^{-1}$ undefined?","back":"It requires division by zero ($\\frac{1}{0}$), which is impossible.","mcqOptions":[{"text":"It requires division by zero ($\\frac{1}{0}$), which is impossible.","correct":true,"explanation":"A negative index means taking a reciprocal, so $0^{-1}=\\frac{1}{0}$. Division by zero has no defined value, making the expression undefined."},{"text":"It requires multiplication by zero ($0 \\times 1$), which is impossible.","correct":false,"explanation":"Multiplication by zero is valid and always gives zero, so that is not the problem. A negative index creates the reciprocal $\\frac{1}{0}$, which is undefined."},{"text":"It requires taking a square root of zero ($\\sqrt{0}$), which is impossible.","correct":false,"explanation":"The square root of zero is defined and equals zero. The index $-1$ instead means taking a reciprocal, producing $\\frac{1}{0}$, which is undefined."},{"text":"It gives a negative result ($-1$), which is not permitted.","correct":false,"explanation":"A negative index does not make the value negative; it indicates a reciprocal. Thus $0^{-1}$ would be $\\frac{1}{0}$, and division by zero is undefined."}]},{"id":"0af5ad4d-c332-4c24-a6f9-d0fe0a3804b1","cardType":"basic","points":3,"front":"How can $\\sqrt[4]{5 \\times 125 \\times 10^8}$ be simplified without a calculator?","back":"

Combine $5 \\times 125$ into $5^4$
Rewrite $10^8$ as $(10^2)^4$
Take the fourth root: $5 \\times 10^2 = 500$

","mcqOptions":[{"text":"

Combine $5 \\times 125$ into $5^4$
Rewrite $10^8$ as $(10^2)^4$
Take the fourth root: $5 \\times 10^2 = 500$

","correct":true,"explanation":"Since $125=5^3$, the first product is $5^4$. Both factors are then fourth powers, so their fourth root is $5\\times10^2=500$."},{"text":"

Combine $5 \\times 125$ into $5^3$
Rewrite $10^8$ as $(10^2)^4$
Take the fourth root: $5^{\\frac{3}{4}} \\times 10^2$

","correct":false,"explanation":"This overlooks the extra factor of 5 because $125$ alone equals $5^3$. The full product is $5\\times5^3=5^4$, so the fourth root is 500."},{"text":"

Combine $5 \\times 125$ into $5^4$
Rewrite $10^8$ as $(10^4)^4$
Take the fourth root: $5 \\times 10^4 = 50000$

","correct":false,"explanation":"The power conversion is incorrect because $(10^4)^4=10^{16}$, not $10^8$. The correct fourth-power form is $(10^2)^4$, producing $5\\times10^2=500$."},{"text":"

Combine $5 \\times 125$ into $5^4$
Rewrite $10^8$ as $(10^2)^4$
Take the square root: $5^2 \\times 10^4 = 250000$

","correct":false,"explanation":"This applies a square root instead of the required fourth root, so each index is divided by 2 rather than 4. The fourth root gives $5\\times10^2=500$."}]}]}],"examCountdown":null,"heading":"Fractional and Negative Indices","title":"Fractional and Negative Indices Flashcards","topbar":{"backHref":"/igcse/maths/oxford-aqa/flashcards"},"topicSidebar":"$L38"}]}],"topicId":"22a88121-eac8-4909-b7c0-b67803598e97","walled":"$L39"}],"$L3a","$L3b"]