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$a$ is the base
$n$ is the index (or power/exponent)

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

$a$ is the base
$n$ is the index (or power/exponent)

","correct":true,"explanation":"The base is the quantity repeatedly multiplied, while the index states how many times the base is used as a factor."},{"text":"

$a$ is the index
$n$ is the base (or power/exponent)

","correct":false,"explanation":"This reverses the two names. In $a^n$, $a$ is the base, while $n$ is the index, also called the power or exponent."},{"text":"

$a$ is the coefficient
$n$ is the index (or power/exponent)

","correct":false,"explanation":"A coefficient multiplies a term and is not the name for $a$ here. In $a^n$, $a$ is the base and $n$ is the index."},{"text":"

$a$ is the exponent
$n$ is the coefficient (or base number)

","correct":false,"explanation":"The letter $a$ is not called the exponent in this notation. It is the base, and the raised value $n$ is the index or exponent."}]},{"id":"053a31ea-79be-3c15-0b21-3366cebc92ed","cardType":"cloze","points":null,"front":"The law for multiplying powers with the same base is $\\text{[...]}$.","back":"The law for multiplying powers with the same base is $\\boxed{a^m \\cdot a^n = a^{m+n}}$.","mcqOptions":[{"text":"$$a^m \\cdot a^n = a^{m+n}$","correct":true,"explanation":"When powers with the same base are multiplied, their indices are added because the total number of factors of $a$ is $m+n$."},{"text":"$$a^m \\cdot a^n = a^{mn}$","correct":false,"explanation":"Multiplying powers does not multiply their indices. For a common base, count all factors by adding the indices, giving $a^m \\cdot a^n = a^{m+n}$."},{"text":"$$a^m \\cdot a^n = a^{m-n}$","correct":false,"explanation":"Subtracting indices is the rule for division, not multiplication. Multiplying powers with the same base requires adding the indices to obtain $a^{m+n}$."},{"text":"$$a^m \\cdot a^n = a^m$","correct":false,"explanation":"Keeping only one index ignores the factors contributed by the second power. The indices must be added, so the product is $a^{m+n}$."}]},{"id":"03cd7363-0139-dffb-9e14-0c92f9601b73","cardType":"cloze","points":null,"front":"The law for dividing powers with the same base is $\\text{[...]}$.","back":"The law for dividing powers with the same base is $\\boxed{\\frac{a^m}{a^n} = a^{m-n}}$.","mcqOptions":[{"text":"$$\\frac{a^m}{a^n} = a^{m-n}$","correct":true,"explanation":"Dividing equal bases cancels $n$ factors of $a$ from the numerator, leaving a power whose index is the difference $m-n$."},{"text":"$$\\frac{a^m}{a^n} = a^{m+n}$","correct":false,"explanation":"Adding indices applies when powers are multiplied. Division cancels factors, so the denominator’s index is subtracted and the correct result is $a^{m-n}$."},{"text":"$$\\frac{a^m}{a^n} = a^{n-m}$","correct":false,"explanation":"Reversing the subtraction changes the result to its reciprocal. Since the numerator has index $m$, the correct order is $m-n$, giving $a^{m-n}$."},{"text":"$$\\frac{a^m}{a^n} = a^{\\frac{m}{n}}$","correct":false,"explanation":"Dividing the indices is not an index law. For powers with the same base, division subtracts the denominator’s index, producing $a^{m-n}$."}]},{"id":"14b8b3ca-e0ec-79ea-d0b1-7b1711c36587","cardType":"cloze","points":null,"front":"The law for a power of a power is $\\text{[...]}$.","back":"The law for a power of a power is $\\boxed{(a^m)^n = a^{mn}}$.","mcqOptions":[{"text":"$$(a^m)^n = a^{mn}$","correct":true,"explanation":"Raising $a^m$ to the power $n$ repeats the factor group $n$ times, so the indices multiply to give $a^{mn}$."},{"text":"$$(a^m)^n = a^{m+n}$","correct":false,"explanation":"Adding indices is associated with multiplying separate powers of the same base. For a power raised to another power, multiply the indices to obtain $a^{mn}$."},{"text":"$$(a^m)^n = a^{m-n}$","correct":false,"explanation":"Subtracting indices applies to division of powers with a common base. A power of a power requires multiplication of the indices, giving $a^{mn}$."},{"text":"$$(a^m)^n = a^{m^n}$","correct":false,"explanation":"Applying the outer power only to the index misinterprets exponent notation. Repeating $a^m$ exactly $n$ times multiplies the indices, producing $a^{mn}$."}]},{"id":"934d7b90-ad5d-9527-f5ec-1afacffd2dcf","cardType":"basic","points":2,"front":"If $x = 2^p$ and $y = 2^q$, how is $2^{p+q}$ expressed in terms of $x$ and $y$?","back":"$$xy$","mcqOptions":[{"text":"$$xy$","correct":true,"explanation":"Since $2^{p+q}=2^p \\cdot 2^q$, substituting $x=2^p$ and $y=2^q$ gives the product $xy$."},{"text":"$$x+y$","correct":false,"explanation":"Adding $x$ and $y$ does not use the multiplication law for indices. Because adding indices corresponds to multiplying powers, $2^{p+q}=xy$."},{"text":"$$\\frac{x}{y}$","correct":false,"explanation":"The quotient $\\frac{x}{y}$ corresponds to $2^{p-q}$ because division subtracts indices. The sum $p+q$ instead gives the product $xy$."},{"text":"$$pq$","correct":false,"explanation":"The product $pq$ combines the indices themselves rather than the powers they define. Using $2^{p+q}=2^p \\cdot 2^q$ gives $xy$."}]},{"id":"a50858aa-b89d-dbcd-85cf-ec0801aaff4f","cardType":"basic","points":1,"front":"What does a negative index signify in mathematics?","back":"The reciprocal (1 divided by the base raised to the positive power).","mcqOptions":[{"text":"The reciprocal (1 divided by the base raised to the positive power).","correct":true,"explanation":"A negative index instructs you to take the reciprocal of the corresponding positive power, so $a^{-n}$ means $1$ divided by $a^n$."},{"text":"The negative value (the base raised to the positive power and multiplied by $-1$).","correct":false,"explanation":"A negative index does not make the value negative. It produces the reciprocal of the positive power, so $a^{-n}=1/a^n$ for a non-zero base."},{"text":"The opposite base (the negative of the base raised to the positive power).","correct":false,"explanation":"A negative index does not change the sign of the base. It means take the reciprocal of the positive power, giving $1$ divided by the base raised to that power."},{"text":"The root (the base rooted according to the positive value of the index).","correct":false,"explanation":"A negative index does not require taking a root; fractional indices perform that role. It signifies the reciprocal of the base raised to the corresponding positive power."}]},{"id":"099bd979-025d-58ce-7d57-d1d2bc7adb73","cardType":"cloze","points":null,"front":"The general rule for negative indices is $\\text{[...]}$.","back":"The general rule for negative indices is $\\boxed{a^{-n} = \\frac{1}{a^n}}$.","mcqOptions":[{"text":"$$a^{-n} = \\frac{1}{a^n}$","correct":true,"explanation":"Changing the sign of an index moves the power across the fraction line, so a negative power equals the reciprocal of the corresponding positive power."},{"text":"$$a^{-n} = -a^n$","correct":false,"explanation":"A negative index does not make the power’s value negative. It forms the reciprocal, so the correct general rule is $a^{-n}=\\frac{1}{a^n}$."},{"text":"$$a^{-n} = \\frac{1}{an}$","correct":false,"explanation":"The reciprocal must include the entire positive power in the denominator. Dividing only by the index confuses an exponent with an ordinary numerical factor."},{"text":"$$a^{-n} = (-a)^n$","correct":false,"explanation":"Negating the base changes its sign behaviour but does not represent a negative index. The required operation is reciprocation, giving $a^{-n}=\\frac{1}{a^n}$."}]},{"id":"56616684-0f73-2279-bda8-4d084767a0bc","cardType":"basic","points":1,"front":"What is the value of $\\left(\\frac{3}{7}\\right)^{-1}$?","back":"$$\\frac{7}{3}$","mcqOptions":[{"text":"$$\\frac{7}{3}$","correct":true,"explanation":"An index of $-1$ means take the reciprocal. Swapping the numerator and denominator of $\\frac{3}{7}$ gives $\\frac{7}{3}$."},{"text":"$$\\frac{3}{7}$","correct":false,"explanation":"Leaving the fraction unchanged treats the index as positive $1$. The index $-1$ requires the reciprocal, so the value is $\\frac{7}{3}$."},{"text":"$$-\\frac{7}{3}$","correct":false,"explanation":"A negative index does not make the fraction negative. It instructs you to take the reciprocal, so the positive result is $\\frac{7}{3}$."},{"text":"$$-\\frac{3}{7}$","correct":false,"explanation":"This adds a negative sign without taking the reciprocal. An index of $-1$ swaps the numerator and denominator, producing the positive fraction $\\frac{7}{3}$."}]},{"id":"33cf7b6a-887c-dff2-fd71-30090649db58","cardType":"basic","points":2,"front":"Explain why $6^{-1}$ is not $-6$.","back":"A negative index represents a reciprocal, not a negative number; $6^{-1} = \\frac{1}{6}$.","mcqOptions":[{"text":"A negative index represents a reciprocal, not a negative number; $6^{-1} = \\frac{1}{6}$.","correct":true,"explanation":"A negative index means take the reciprocal of the base, so $a^{-1}=\\frac{1}{a}$. It does not change the sign of the number."},{"text":"A negative index represents a negative value, not a reciprocal; $6^{-1} = -6$.","correct":false,"explanation":"This confuses a negative exponent with a negative number. The exponent $-1$ means reciprocal, so the correct value is $6^{-1}=\\frac{1}{6}$."},{"text":"A negative index represents a subtraction, not a reciprocal; $6^{-1} = 6 - 1$.","correct":false,"explanation":"This treats the index as an instruction to subtract 1 from the base. Indices describe powers, and $6^{-1}$ means the reciprocal $\\frac{1}{6}$."},{"text":"A negative index represents a zero power, not a reciprocal; $6^{-1} = 1$.","correct":false,"explanation":"This mixes up the negative index rule with the zero index rule. $6^0=1$, but $6^{-1}$ means $\\frac{1}{6}$."}]},{"id":"4317594f-a32c-4ada-63a8-d152b6ab1902","cardType":"basic","points":1,"front":"Why is $0^{-1}$ undefined?","back":"It would require division by zero ($1/0$).","mcqOptions":[{"text":"It would require division by zero ($1/0$).","correct":true,"explanation":"A power of $-1$ means taking the reciprocal, so $0^{-1}$ would be $\\frac{1}{0}$. Division by zero is undefined."},{"text":"It would require multiplication by zero ($1 \\times 0$).","correct":false,"explanation":"The tempting slip is treating the reciprocal as multiplication by the base. A power of $-1$ means $\\frac{1}{0}$, which is division by zero and is undefined."},{"text":"It would require subtracting zero ($1 - 0$).","correct":false,"explanation":"This treats the negative index as a subtraction instruction. In fact, $0^{-1}$ means the reciprocal of 0, giving $\\frac{1}{0}$, which is undefined."},{"text":"It would require raising zero to zero ($0^0$).","correct":false,"explanation":"This confuses a negative index with a zero index. The expression $0^{-1}$ means $\\frac{1}{0}$, not $0^0$, so it is undefined because division by zero is not allowed."}]},{"id":"2fabcb67-6c41-4427-9dd7-e11aeb5b6d90","cardType":"basic","points":1,"front":"What root corresponds to the index $\\frac{1}{2}$?","back":"Square root ($\\sqrt{a}$)","mcqOptions":[{"text":"Square root ($\\sqrt{a}$)","correct":true,"explanation":"An index of $\\frac{1}{2}$ means the denominator 2 gives the root. Therefore $a^{\\frac{1}{2}}=\\sqrt{a}$."},{"text":"Cube root ($\\sqrt[3]{a}$)","correct":false,"explanation":"This uses denominator 3 instead of denominator 2. The index $\\frac{1}{2}$ corresponds to the square root, so $a^{\\frac{1}{2}}=\\sqrt{a}$."},{"text":"Fourth root ($\\sqrt[4]{a}$)","correct":false,"explanation":"This mistakes the denominator of the fractional index. A fourth root would come from $\\frac{1}{4}$, whereas $\\frac{1}{2}$ gives the square root."},{"text":"Reciprocal ($\\frac{1}{a}$)","correct":false,"explanation":"This confuses fractional indices with negative indices. A reciprocal comes from $a^{-1}$, while $a^{\\frac{1}{2}}$ means the square root of $a$."}]},{"id":"a69c1234-d814-3c0d-45f2-1a80d61e82a9","cardType":"basic","points":1,"front":"What root corresponds to the index $\\frac{1}{3}$?","back":"Cube root ($\\sqrt[3]{a}$)","mcqOptions":[{"text":"Cube root ($\\sqrt[3]{a}$)","correct":true,"explanation":"An index of $\\frac{1}{3}$ means the denominator 3 gives the root. Therefore $a^{\\frac{1}{3}}=\\sqrt[3]{a}$."},{"text":"Square root ($\\sqrt{a}$)","correct":false,"explanation":"This uses denominator 2 instead of denominator 3. The index $\\frac{1}{3}$ corresponds to the cube root, so $a^{\\frac{1}{3}}=\\sqrt[3]{a}$."},{"text":"Third power ($a^3$)","correct":false,"explanation":"This swaps the meaning of a fractional index with an integer power. The fraction $\\frac{1}{3}$ means cube root, not cubing the number."},{"text":"Reciprocal ($\\frac{1}{a}$)","correct":false,"explanation":"This confuses fractional indices with negative indices. A reciprocal comes from $a^{-1}$, while $a^{\\frac{1}{3}}$ means the cube root of $a$."}]},{"id":"1c7d4d0e-969c-7030-1cc3-4e7fc47dfca1","cardType":"cloze","points":null,"front":"For a fractional index $a^{\\frac{m}{n}}$, the denominator $n$ represents the [ ] and the numerator $m$ represents the power.","back":"For a fractional index $a^{\\frac{m}{n}}$, the denominator $n$ represents the root and the numerator $m$ represents the power.","mcqOptions":[{"text":"root","correct":true,"explanation":"In $a^{\\frac{m}{n}}$, the denominator tells you which root to take. For example, a denominator of 3 means cube root."},{"text":"power","correct":false,"explanation":"This reverses the roles of the numerator and denominator. In $a^{\\frac{m}{n}}$, $n$ gives the root, while $m$ gives the power."},{"text":"numerator","correct":false,"explanation":"This names the position in the fraction rather than its meaning. The denominator $n$ represents the root in a fractional index."},{"text":"base","correct":false,"explanation":"The base is the number being raised to the index, such as $a$. In $a^{\\frac{m}{n}}$, the denominator $n$ represents the root."}]},{"id":"6dc86dd2-a4c5-3251-a851-2152f6d3e58c","cardType":"cloze","points":null,"front":"For a fractional index $a^{\\frac{m}{n}}$, the denominator $n$ represents the root and the numerator $m$ represents the [ ].","back":"For a fractional index $a^{\\frac{m}{n}}$, the denominator $n$ represents the root and the numerator $m$ represents the power.","mcqOptions":[{"text":"power","correct":true,"explanation":"In $a^{\\frac{m}{n}}$, the numerator tells you the power to apply. For example, $a^{\\frac{2}{3}}$ means cube root, then square."},{"text":"root","correct":false,"explanation":"This reverses the roles of the numerator and denominator. In $a^{\\frac{m}{n}}$, the denominator $n$ gives the root, while the numerator $m$ gives the power."},{"text":"denominator","correct":false,"explanation":"This names the fraction position rather than its meaning. The numerator $m$ represents the power applied after taking the root."},{"text":"base","correct":false,"explanation":"The base is the quantity being raised to the index, such as $a$. In $a^{\\frac{m}{n}}$, the numerator $m$ represents the power."}]},{"id":"a06e790d-58dd-a809-a1cc-4f3681890fdd","cardType":"basic","points":2,"front":"When evaluating $a^{\\frac{m}{n}}$, why is it usually better to take the root first?","back":"It keeps the numbers smaller and easier to calculate before applying the power.","mcqOptions":[{"text":"It keeps the numbers smaller and easier to calculate before applying the power.","correct":true,"explanation":"Taking the root first usually reduces the size of the number before the power is applied. For example, $27^{\\frac{2}{3}}$ becomes $3^2$, not $27^2$ first."},{"text":"It keeps the numbers larger and easier to calculate before applying the power.","correct":false,"explanation":"The point of taking the root first is normally to make the numbers smaller, not larger. Smaller intermediate values reduce the chance of awkward arithmetic."},{"text":"It keeps the indices negative and easier to calculate before applying the power.","correct":false,"explanation":"This mixes up fractional indices with negative indices. Taking the root first does not make indices negative; it usually makes the base smaller before applying the power."},{"text":"It keeps the powers unchanged and easier to calculate before applying the root.","correct":false,"explanation":"This reverses the recommended order. For $a^{\\frac{m}{n}}$, taking the root first often gives smaller numbers before raising to the power $m$."}]},{"id":"b201b8d0-4cd8-c4c0-edbe-2e980b0368c7","cardType":"basic","points":2,"front":"Evaluate $27^{\\frac{2}{3}}$.","back":"

$\\sqrt[3]{27} = 3$
$3^2 = 9$

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

$\\sqrt[3]{27} = 3$
$3^2 = 9$

","correct":true,"explanation":"The denominator 3 means take the cube root first, giving 3. The numerator 2 then means square it, so the result is 9."},{"text":"

$\\sqrt[2]{27} = 3$
$3^3 = 27$

","correct":false,"explanation":"This swaps the numerator and denominator and also treats $\\sqrt{27}$ as 3. The denominator is 3, so take $\\sqrt[3]{27}=3$, then square to get 9."},{"text":"

$\\sqrt[3]{27} = 3$
$3^3 = 27$

","correct":false,"explanation":"The cube root step is right, but the numerator is 2, not 3. After $\\sqrt[3]{27}=3$, you must calculate $3^2=9$."},{"text":"

$\\sqrt[3]{27} = 9$
$9^2 = 81$

","correct":false,"explanation":"This confuses the cube root of 27 with another factor. Since $3^3=27$, $\\sqrt[3]{27}=3$, and then $3^2=9$."}]},{"id":"1296f6b8-055c-180b-b597-8b1137246f05","cardType":"basic","points":3,"front":"Outline the 3 steps to evaluate $16^{-\\frac{3}{2}}$.","back":"

Square root: $\\sqrt{16} = 4$
Power: $4^3 = 64$
Reciprocal: $\\frac{1}{64}$

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

Square root: $\\sqrt{16} = 4$
Power: $4^3 = 64$
Reciprocal: $\\frac{1}{64}$

","correct":true,"explanation":"The denominator 2 gives the square root, the numerator 3 gives the cube, and the negative sign means take the reciprocal at the end."},{"text":"

Square root: $\\sqrt{16} = 4$
Power: $4^3 = 64$
Negative: $-64$

","correct":false,"explanation":"This treats the negative index as making the answer negative. A negative index means reciprocal, so after getting 64 the correct result is $\\frac{1}{64}$."},{"text":"

Cube root: $\\sqrt[3]{16} = 4$
Power: $4^2 = 16$
Reciprocal: $\\frac{1}{16}$

","correct":false,"explanation":"This swaps the numerator and denominator, and $\\sqrt[3]{16}$ is not 4. The denominator 2 gives $\\sqrt{16}=4$, then $4^3=64$, then reciprocal."},{"text":"

Square root: $\\sqrt{16} = 4$
Power: $4^3 = 64$
Reciprocal: $64$

","correct":false,"explanation":"The root and power steps are right, but the reciprocal has not actually been taken. Because the index is negative, the final answer must be $\\frac{1}{64}$."}]},{"id":"655197b1-6c74-58cd-856f-77e79527b154","cardType":"basic","points":2,"front":"How do you evaluate a negative index applied to a fraction, such as $\\left(\\frac{27}{64}\\right)^{-\\frac{2}{3}}$?","back":"Apply the fractional power to both parts, then flip the fraction (take the reciprocal).","mcqOptions":[{"text":"Apply the fractional power to both parts, then flip the fraction (take the reciprocal).","correct":true,"explanation":"A negative index means take the reciprocal, and the fractional power applies to the numerator and denominator. You can evaluate the fractional power, then flip the fraction."},{"text":"Apply the fractional power to both parts, then change the sign (make it negative).","correct":false,"explanation":"This confuses a negative exponent with a negative value. The negative index means take the reciprocal, so you flip the fraction rather than making the result negative."},{"text":"Apply the fractional power to the top only, then flip the fraction (take the reciprocal).","correct":false,"explanation":"A power outside brackets applies to both numerator and denominator. For a fraction in brackets, the fractional power must be applied to both parts before taking the reciprocal."},{"text":"Apply the fractional power to both parts, then square the fraction (take the power).","correct":false,"explanation":"The extra squaring comes from misreading the negative sign as another power. The negative index tells you to take the reciprocal, so the fraction should be flipped."}]},{"id":"c5fc7405-0cfe-e43b-9b70-6185b0d7ed9d","cardType":"cloze","points":null,"front":"When a power is outside brackets containing a product, the rule is $\\text{[...]}$.","back":"When a power is outside brackets containing a product, the rule is $\\boxed{(ab)^n = a^n b^n}$.","mcqOptions":[{"text":"$$(ab)^n = a^n b^n$","correct":true,"explanation":"A power outside brackets containing a product applies to every factor inside. Therefore both $a$ and $b$ are raised to the power $n$."},{"text":"$$(ab)^n = a^n + b^n$","correct":false,"explanation":"This incorrectly changes a product into a sum. Powers distribute over multiplication here, so the correct rule is $(ab)^n=a^n b^n$."},{"text":"$$(ab)^n = ab^n$","correct":false,"explanation":"This applies the outside power only to $b$ and forgets $a$. The power applies to every factor inside the brackets, so $(ab)^n=a^n b^n$."},{"text":"$$(ab)^n = a^{n}b$","correct":false,"explanation":"This applies the outside power only to $a$ and forgets $b$. For a product in brackets, both factors are raised to the power: $(ab)^n=a^n b^n$."}]},{"id":"0afba4c7-da9d-1f2d-9723-fb041279f9b2","cardType":"basic","points":2,"front":"Evaluate $(27x^6)^{\\frac{2}{3}}$.","back":"

$\\sqrt[3]{27x^6} = 3x^2$
$(3x^2)^2 = 9x^4$

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

$\\sqrt[3]{27x^6} = 3x^2$
$(3x^2)^2 = 9x^4$

","correct":true,"explanation":"The denominator 3 means take the cube root: $27$ becomes 3 and $x^6$ becomes $x^2$. Then squaring gives $9x^4$."},{"text":"

$\\sqrt[3]{27x^6} = 3x^2$
$(3x^2)^2 = 6x^4$

","correct":false,"explanation":"The cube root step is right, but squaring $3x^2$ means squaring both factors. $3^2=9$, not 6, so the result is $9x^4$."},{"text":"

$\\sqrt[2]{27x^6} = 3x^3$
$(3x^3)^3 = 27x^9$

","correct":false,"explanation":"This swaps the numerator and denominator. The denominator 3 gives a cube root first, so $\\sqrt[3]{27x^6}=3x^2$, then squaring gives $9x^4$."},{"text":"

$\\sqrt[3]{27x^6} = 9x^2$
$(9x^2)^2 = 81x^4$

","correct":false,"explanation":"This takes the cube root of 27 incorrectly. Since $3^3=27$, the cube root is 3, and the final answer is $(3x^2)^2=9x^4$."}]},{"id":"3f8f1a67-cffc-a534-7a0e-5fdd4cb7df38","cardType":"basic","points":2,"front":"Evaluate $\\sqrt[4]{81 \\cdot 10^8}$.","back":"

$\\sqrt[4]{81} = 3$
$\\sqrt[4]{10^8} = 10^2$
Result: $3 \\cdot 100 = 300$

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

$\\sqrt[4]{81} = 3$
$\\sqrt[4]{10^8} = 10^2$
Result: $3 \\cdot 100 = 300$

","correct":true,"explanation":"The fourth root can be taken for each factor. Since $3^4=81$ and $(10^2)^4=10^8$, the result is $3 \\cdot 100=300$."},{"text":"

$\\sqrt[4]{81} = 9$
$\\sqrt[4]{10^8} = 10^2$
Result: $9 \\cdot 100 = 900$

","correct":false,"explanation":"This uses the square root of 81 instead of the fourth root. The fourth root of 81 is 3, so the correct result is $3 \\cdot 100=300$."},{"text":"

$\\sqrt[4]{81} = 3$
$\\sqrt[4]{10^8} = 10^4$
Result: $3 \\cdot 10000 = 30000$

","correct":false,"explanation":"This halves the exponent instead of dividing it by 4. The fourth root of $10^8$ is $10^{8/4}=10^2$, giving 300."},{"text":"

$\\sqrt[4]{81} = 3$
$\\sqrt[4]{10^8} = 10^2$
Result: $3 + 100 = 103$

","correct":false,"explanation":"This adds the two factor results instead of multiplying them. The fourth root of a product gives the product of the fourth roots, so the result is $3 \\cdot 100=300$."}]},{"id":"4686f37c-2de3-ba8a-a6f7-7b33c5823d63","cardType":"basic","points":2,"front":"What is the general strategy for solving an equation with an unknown index, such as $9^n = 27$?","back":"Rewrite both sides using the same base (e.g., base 3).","mcqOptions":[{"text":"Rewrite both sides using the same base (e.g., base 3).","correct":true,"explanation":"Writing both sides with the same base lets you compare the indices directly. For example, $9^n=27$ becomes $(3^2)^n=3^3$."},{"text":"Rewrite both sides using the same index (e.g., index 3).","correct":false,"explanation":"The useful comparison comes from matching the bases, not forcing the indices to match. Once both sides have base 3, the powers can be equated."},{"text":"Rewrite both sides using the same coefficient (e.g., coefficient 3).","correct":false,"explanation":"Coefficients are not the key feature in an exponential equation like $9^n=27$. The standard strategy is to rewrite both sides using the same base."},{"text":"Rewrite both sides using the same fraction (e.g., fraction $\\frac{1}{3}$).","correct":false,"explanation":"This confuses index equations with fractional-index calculations. For unknown indices, rewrite both sides with a common base, such as base 3."}]},{"id":"ca07c3bc-9bc7-0a06-7949-ba6ef5fad269","cardType":"basic","points":3,"front":"Write $2\\sqrt{8}$ as a single power of 2.","back":"

$8 = 2^3$, so $\\sqrt{8} = 2^{\\frac{3}{2}}$
$2^1 \\cdot 2^{\\frac{3}{2}} = 2^{\\frac{5}{2}}$

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

$8 = 2^3$, so $\\sqrt{8} = 2^{\\frac{3}{2}}$
$2^1 \\cdot 2^{\\frac{3}{2}} = 2^{\\frac{5}{2}}$

","correct":true,"explanation":"A square root is equivalent to a power of one half, so $sqrt{2^3}=2^{rac{3}{2}}$. Multiplying powers with the same base adds their indices, giving $2^{1+rac{3}{2}}=2^{rac{5}{2}}$."},{"text":"

$8 = 2^3$, so $sqrt{8} = 2^3$
$2^1 cdot 2^3 = 2^4$

","correct":false,"explanation":"This fails to apply the square-root power of $rac{1}{2}$, leaving the index unchanged. In fact, $sqrt{2^3}=2^{rac{3}{2}}$, so adding indices gives $2^{rac{5}{2}}$."},{"text":"

$8 = 2^3$, so $sqrt{8} = 2^{rac{1}{3}}$
$2^1 cdot 2^{rac{1}{3}} = 2^{rac{4}{3}}$

","correct":false,"explanation":"This confuses a square root with a cube root by dividing the index by 3. A square root halves the index, so $sqrt{8}=2^{rac{3}{2}}$ and the product is $2^{rac{5}{2}}$."},{"text":"

$8 = 2^3$, so $sqrt{8} = 2^{rac{3}{2}}$
$2^1 cdot 2^{rac{3}{2}} = 2^{rac{3}{2}}$

","correct":false,"explanation":"This multiplies the indices when multiplying powers with the same base. The indices must be added instead, so $1+rac{3}{2}=rac{5}{2}$ and the result is $2^{rac{5}{2}}$."}]}]}],"examCountdown":{"examDate":"2027-05-14","paperLabel":"Component 1"},"heading":"Fractional and Negative Indices","title":"Fractional and Negative Indices Flashcards","topbar":{"backHref":"/gcse/maths/eduqas/flashcards"},"topicSidebar":"$L37"}]}],"topicId":"b9b74c4c-ee62-430c-9931-4a827d9dff24","walled":"$L38"}],"$L39","$L3a"]