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$x$ is the base
$5$ is the index

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

$x$ is the base
$5$ is the index

","correct":true,"explanation":"The repeated factor is the base, while the index states how many times that factor is multiplied by itself. Therefore, $x$ is the base and $5$ is the index."},{"text":"

$x$ is the index
$5$ is the base

","correct":false,"explanation":"The labels have been reversed. In $x^5$, $x$ is the quantity repeatedly multiplied, making it the base, while $5$ is the index."},{"text":"

$x$ is the coefficient
$5$ is the index

","correct":false,"explanation":"A coefficient is a numerical multiplier written before a variable, so $x$ is not the coefficient here. It is the base, and $5$ is the index."},{"text":"

$x$ is the base
$5$ is the coefficient

","correct":false,"explanation":"The number $5$ is not multiplying the power, so it is not a coefficient. It is the index, while $x$ is the base being raised to that power."}]},{"id":"f4c4b263-1244-b07e-e30f-3b683a216f14","cardType":"cloze","points":null,"front":"When multiplying powers with the same base, $a^m \\times a^n = a^{\\text{[...]}}$.","back":"When multiplying powers with the same base, $a^m \\times a^n = a^{\\boxed{m+n}}$.","mcqOptions":[{"text":"$$m+n$","correct":true,"explanation":"Multiplying powers with the same base combines all repeated factors, so their indices are added. Therefore, $a^m \\times a^n=a^{m+n}$."},{"text":"$$m-n$","correct":false,"explanation":"Subtracting the indices is the rule for dividing powers with the same base. Multiplication requires addition, giving the exponent $m+n$."},{"text":"$$mn$","correct":false,"explanation":"Multiplying the indices is used when raising an existing power to another power, as in $(a^m)^n$. Here the powers are multiplied, so the exponent is $m+n$."},{"text":"$$\\frac{m}{n}$","correct":false,"explanation":"Dividing the indices is not an index law for multiplying equal bases. The repeated factors combine, so the correct resulting exponent is $m+n$."}]},{"id":"fbe09637-18d8-595c-4f78-922cfd66d869","cardType":"basic","points":1,"front":"Simplify $x^6 \\times x^5$.","back":"$$x^{11}$","mcqOptions":[{"text":"$$x^{11}$","correct":true,"explanation":"The powers have the same base, so their indices are added when multiplying. Since $6+5=11$, the product simplifies to $x^{11}$."},{"text":"$$x^{30}$","correct":false,"explanation":"Multiplying the indices confuses this rule with neither valid multiplication nor the power-of-a-power law. Equal bases are multiplied by adding indices, giving $x^{11}$."},{"text":"$$x^1$","correct":false,"explanation":"Subtracting the indices is appropriate for division of equal bases, not multiplication. Adding them gives $6+5=11$, so the result is $x^{11}$."},{"text":"$$2x^{11}$","correct":false,"explanation":"The implicit coefficients are both $1$, whose product remains $1$, rather than $2$. The indices add to $11$, so the product is $x^{11}$."}]},{"id":"eee81316-2381-e11e-e078-9c714c73d454","cardType":"basic","points":1,"front":"Simplify $\\frac{w^9}{w^4}$.","back":"$$w^5$","mcqOptions":[{"text":"$$w^5$","correct":true,"explanation":"When dividing powers with the same base, subtract the denominator’s index from the numerator’s index. Since $9-4=5$, the expression simplifies to $w^5$."},{"text":"$$w^{13}$","correct":false,"explanation":"Adding indices is the rule for multiplying equal bases, not dividing them. Division requires subtraction, so $9-4=5$ and the result is $w^5$."},{"text":"$$w^{36}$","correct":false,"explanation":"Multiplying the indices is not the division rule for powers. The denominator’s index must be subtracted from the numerator’s, producing $w^{9-4}=w^5$."},{"text":"$$w^{\\frac{9}{4}}$","correct":false,"explanation":"The indices themselves are not divided when powers share a base. Instead, subtract the denominator’s index from the numerator’s, giving $w^{9-4}=w^5$."}]},{"id":"5cc21152-2afb-e124-8f44-16034118d64a","cardType":"cloze","points":null,"front":"A letter on its own, such as $x$, has an invisible index of [ ].","back":"A letter on its own, such as $x$, has an invisible index of 1.","mcqOptions":[{"text":"1","correct":true,"explanation":"A variable written without an explicit index represents one copy of itself. Thus $x$ means $x^1$, so its invisible index is 1."},{"text":"0","correct":false,"explanation":"An index of 0 would give $x^0=1$, not $x$. A lone variable represents $x^1$, so its invisible index is 1."},{"text":"2","correct":false,"explanation":"An index of 2 would mean $x^2=x\\times x$, which contains two factors of $x$. A lone $x$ is $x^1$, so the index is 1."},{"text":"-1","correct":false,"explanation":"An index of -1 represents a reciprocal, since $x^{-1}=\\frac{1}{x}$. A letter on its own is $x^1$, so its invisible index is 1."}]},{"id":"f45d9735-d36e-6a19-f462-a54e5a5016bd","cardType":"basic","points":2,"front":"Simplify $4m^3 \\times 6m$.","back":"$$24m^4$","mcqOptions":[{"text":"$$24m^4$","correct":true,"explanation":"Multiply the coefficients to get $4\\times6=24$, then add the indices of $m$, treating $m$ as $m^1$. This gives $24m^{3+1}=24m^4$."},{"text":"$$24m^3$","correct":false,"explanation":"Keeping the index at 3 overlooks the invisible index of 1 on $m$. The coefficients multiply to 24 and the indices add, giving $24m^4$."},{"text":"$$10m^4$","correct":false,"explanation":"The coefficients must be multiplied, not added: $4\\times6=24$. The indices also add as $3+1=4$, so the correct product is $24m^4$."},{"text":"$$24m^6$","correct":false,"explanation":"Multiplying the indices incorrectly treats the lone $m$ as though its index altered the 3 multiplicatively. Indices add here, so the result is $24m^4$."}]},{"id":"23a72787-f726-db2b-9a70-9264c48a489e","cardType":"basic","points":3,"front":"Simplify $\\frac{18p^5q^4}{6p^2q}$.","back":"$$3p^3q^3$","mcqOptions":[{"text":"$$3p^3q^3$","correct":true,"explanation":"Divide the coefficients to get $18\\div6=3$, then subtract corresponding indices: $5-2=3$ for $p$ and $4-1=3$ for $q$."},{"text":"$$3p^3q^4$","correct":false,"explanation":"Leaving the index of $q$ unchanged overlooks the denominator’s invisible index of 1. Subtracting it gives $q^{4-1}=q^3$, so the answer is $3p^3q^3$."},{"text":"$$12p^3q^3$","correct":false,"explanation":"Subtracting the coefficients is not how numerical factors are divided. Since $18\\div6=3$, and both variable indices reduce to 3, the answer is $3p^3q^3$."},{"text":"$$3p^7q^5$","correct":false,"explanation":"Adding indices applies when multiplying powers, but this expression is a quotient. Subtracting denominator indices gives $p^{5-2}q^{4-1}$, so the result is $3p^3q^3$."}]},{"id":"7a39700a-daa3-a4e5-3443-4d4f3e9f7d9b","cardType":"cloze","points":null,"front":"When raising a power to another power, $(a^m)^n = a^{\\text{[...]}}$.","back":"When raising a power to another power, $(a^m)^n = a^{\\boxed{mn}}$.","mcqOptions":[{"text":"$$mn$","correct":true,"explanation":"Raising a power to another power repeats the original power $n$ times, so the indices multiply. Therefore, $(a^m)^n=a^{mn}$."},{"text":"$$m+n$","correct":false,"explanation":"Adding indices applies when multiplying separate powers with the same base. For a power raised to another power, the indices multiply, giving $mn$."},{"text":"$$m^n$","correct":false,"explanation":"Raising one index to the other is not the power-of-a-power rule. The outer power repeats the inner power, so the resulting exponent is the product $mn$."},{"text":"$$\\frac{m}{n}$","correct":false,"explanation":"Dividing the indices does not describe repeatedly raising a power. A power raised to another power requires the indices to be multiplied, producing $mn$."}]},{"id":"9bb8bae2-231f-cdad-b84c-9e8cf7f405d4","cardType":"basic","points":1,"front":"Simplify $(t^4)^3$.","back":"$$t^{12}$","mcqOptions":[{"text":"$$t^{12}$","correct":true,"explanation":"When a power is raised to another power, multiply the indices. Multiplying $4$ by $3$ gives $12$, so the simplified expression is $t^{12}$."},{"text":"$$t^7$","correct":false,"explanation":"Adding $4$ and $3$ confuses the power-of-a-power rule with multiplying powers of the same base. Here the indices must be multiplied, giving $t^{12}$."},{"text":"$$t^{64}$","correct":false,"explanation":"Calculating $4^3$ incorrectly treats the index itself as the base. The power-of-a-power rule multiplies the indices, so $4 \\times 3=12$ and the answer is $t^{12}$."},{"text":"$$t^4$","correct":false,"explanation":"Leaving the index unchanged ignores the outer power of $3$. A power raised to a power has its indices multiplied, so the correct result is $t^{12}$."}]},{"id":"877711c7-c78c-91e4-8507-151958df9b6b","cardType":"basic","points":2,"front":"Simplify $(4x)^2$.","back":"$$16x^2$","mcqOptions":[{"text":"$$16x^2$","correct":true,"explanation":"The power of $2$ applies to both factors inside the brackets: $4^2=16$ and $x^2=x^2$. Therefore, $(4x)^2=16x^2$."},{"text":"$$8x^2$","correct":false,"explanation":"Multiplying $4$ by $2$ instead of squaring $4$ gives the incorrect coefficient $8$. The coefficient should be $4^2=16$, so the answer is $16x^2$."},{"text":"$$16x$","correct":false,"explanation":"Squaring only the coefficient ignores that the power applies to every factor inside the brackets. Both $4$ and $x$ are squared, giving $16x^2$."},{"text":"$$4x^2$","correct":false,"explanation":"Squaring only the variable leaves the coefficient unchanged, but the outer power applies to the entire bracket. Since $4^2=16$, the correct answer is $16x^2$."}]},{"id":"9fea0ab8-1a28-1d90-cc9c-0b71e32f2c2b","cardType":"basic","points":3,"front":"Simplify $(3a^2b)^3$.","back":"$$27a^6b^3$","mcqOptions":[{"text":"$$27a^6b^3$","correct":true,"explanation":"Cube every factor inside the brackets: $3^3=27$, $(a^2)^3=a^6$, and $b^3=b^3$. Combining these gives $27a^6b^3$."},{"text":"$$9a^6b^3$","correct":false,"explanation":"Using $3^2=9$ applies the exponent on $a$ to the coefficient instead of cubing it. The coefficient is $3^3=27$, giving $27a^6b^3$."},{"text":"$$27a^5b^3$","correct":false,"explanation":"Adding the indices $2$ and $3$ gives $a^5$, but a power raised to a power requires multiplication. Since $2 \\times 3=6$, the answer is $27a^6b^3$."},{"text":"$$27a^6b$","correct":false,"explanation":"Leaving $b$ unchanged ignores that the outer power applies to every factor in the bracket. Cubing $b$ gives $b^3$, so the answer is $27a^6b^3$."}]},{"id":"38f73912-d9bb-2a94-f045-233856f9abfe","cardType":"cloze","points":null,"front":"Any non-zero base to the power of 0 equals [ ].","back":"Any non-zero base to the power of 0 equals 1.","mcqOptions":[{"text":"1","correct":true,"explanation":"For every non-zero number $a$, dividing $a^m$ by itself gives $a^{m-m}=a^0$ and also equals $1$. Therefore, $a^0=1$."},{"text":"0","correct":false,"explanation":"A zero index does not make the value zero; it makes the exponent difference zero when equal powers are divided. Any non-zero base to power $0$ equals $1$."},{"text":"-1","correct":false,"explanation":"A zero index does not introduce a negative sign. Using the index law $a^m \\div a^m=a^0$, the quotient is $1$, so any non-zero base gives $1$."},{"text":"undefined","correct":false,"explanation":"The zero-index rule is defined for every non-zero base. Since a non-zero quantity divided by itself is $1$, any non-zero base raised to power $0$ equals $1$."}]},{"id":"90e3b2bf-db7d-1882-1851-c12633d365a5","cardType":"basic","points":1,"front":"What is the reciprocal of $5$?","back":"$$\\frac{1}{5}$","mcqOptions":[{"text":"$$\\frac{1}{5}$","correct":true,"explanation":"The reciprocal of a non-zero number is found by writing $1$ divided by that number. Therefore, the reciprocal of $5$ is $\\frac{1}{5}$."},{"text":"$$-\\frac{1}{5}$","correct":false,"explanation":"Taking a reciprocal does not change a positive number into a negative number. The reciprocal must multiply by $5$ to make $1$, so it is $\\frac{1}{5}$."},{"text":"$$\\frac{5}{1}$","correct":false,"explanation":"Writing $5$ as $\\frac{5}{1}$ does not invert it. A reciprocal swaps the numerator and denominator, so the reciprocal is $\\frac{1}{5}$."},{"text":"$$\\frac{1}{25}$","correct":false,"explanation":"Using $25$ in the denominator squares $5$ instead of taking its reciprocal. The number that multiplies by $5$ to give $1$ is $\\frac{1}{5}$."}]},{"id":"c1a5d38b-3409-1d68-12be-7f6fa27d4bd6","cardType":"cloze","points":null,"front":"A negative index means you must take the [ ].","back":"A negative index means you must take the reciprocal.","mcqOptions":[{"text":"reciprocal","correct":true,"explanation":"A negative index reverses the corresponding positive power: $a^{-n}=\\frac{1}{a^n}$ for $a \\ne 0$. This means taking the reciprocal."},{"text":"opposite","correct":false,"explanation":"Taking the opposite changes the sign of a value, whereas a negative index changes its position in a fraction. The required operation is taking the reciprocal."},{"text":"square root","correct":false,"explanation":"A square root corresponds to an index of $\\frac{1}{2}$, not a negative index. A negative index instead means taking the reciprocal of the positive-index power."},{"text":"absolute value","correct":false,"explanation":"Absolute value removes a negative sign from a number but does not describe negative indices. A negative index means taking the reciprocal, as in $a^{-n}=\\frac{1}{a^n}$."}]},{"id":"09cdde04-b7d8-3915-fe91-3b2960f96331","cardType":"cloze","points":null,"front":"$$a^{-n} = \\text{[...]}$","back":"$$a^{-n} = \\boxed{\\frac{1}{a^n}}$","mcqOptions":[{"text":"$$\\frac{1}{a^n}$","correct":true,"explanation":"A negative index means taking the reciprocal of the corresponding positive power. Therefore, for $a \\ne 0$, $a^{-n}$ equals $\\frac{1}{a^n}$."},{"text":"$$-a^n$","correct":false,"explanation":"A negative index does not place a negative sign before the expression. It indicates a reciprocal, so $a^{-n}$ equals $\\frac{1}{a^n}$ for $a \\ne 0$."},{"text":"$$\\frac{1}{a^{-n}}$","correct":false,"explanation":"This takes the reciprocal of an already negative power, which gives $a^n$ rather than $a^{-n}$. The correct positive-index form is $\\frac{1}{a^n}$."},{"text":"$$-\\frac{1}{a^n}$","correct":false,"explanation":"Taking the reciprocal is correct, but adding a negative sign is not. The negative index changes the power’s position, not its sign, so the result is $\\frac{1}{a^n}$."}]},{"id":"f48cdef1-95d8-e9c8-52da-15c68028b31e","cardType":"basic","points":2,"front":"Calculate the value of $2^{-4}$.","back":"$$\\frac{1}{16}$","mcqOptions":[{"text":"$$\\frac{1}{16}$","correct":true,"explanation":"The negative index means take the reciprocal: $2^{-4}=\\frac{1}{2^4}$. Since $2^4=16$, the value is $\\frac{1}{16}$."},{"text":"$$-16$","correct":false,"explanation":"A negative index does not make the final value negative. It means take the reciprocal of $2^4=16$, so $2^{-4}=\\frac{1}{16}$."},{"text":"$$\\frac{1}{8}$","correct":false,"explanation":"The denominator $8$ comes from calculating $2^3$, so the exponent has effectively been reduced by one. Since $2^4=16$, the correct value is $\\frac{1}{16}$."},{"text":"$$16$","correct":false,"explanation":"Calculating $2^4=16$ ignores the negative sign on the index. A negative index requires the reciprocal, making the correct value $\\frac{1}{16}$."}]},{"id":"18144424-69d5-a965-f0bb-9fd08f17a646","cardType":"basic","points":2,"front":"If $y^3 \\times y^a = y^8$, what is the value of $a$?","back":"$$5$","mcqOptions":[{"text":"$$5$","correct":true,"explanation":"When multiplying powers with the same base, add the indices. Therefore $3+a=8$, and subtracting $3$ from both sides gives $a=5$."},{"text":"$$11$","correct":false,"explanation":"Adding $3$ to $8$ reverses the required calculation. The index law gives $3+a=8$, so subtracting $3$ shows that the correct value is $a=5$."},{"text":"$$24$","correct":false,"explanation":"Multiplying the indices is not the rule for multiplying powers with the same base. The indices must be added, giving $3+a=8$ and hence $a=5$."},{"text":"$$6$","correct":false,"explanation":"This is an off-by-one subtraction error. Since multiplication of equal bases adds indices, $3+a=8$, and calculating $8-3$ correctly gives $a=5$."}]},{"id":"9b49c065-871b-e933-22cc-ebe90361e54e","cardType":"basic","points":2,"front":"If $(x^5)^b = x^{20}$, what is the value of $b$?","back":"$$4$","mcqOptions":[{"text":"$$4$","correct":true,"explanation":"When raising a power to another power, multiply the indices. Thus $(x^5)^b=x^{5b}$, so $5b=20$ and dividing by $5$ gives $b=4$."},{"text":"$$15$","correct":false,"explanation":"Subtracting the indices confuses the power-of-a-power rule with rearranging an addition equation. Here the indices multiply, so $5b=20$ and the correct value is $b=4$."},{"text":"$$25$","correct":false,"explanation":"Adding the indices is used when multiplying powers with the same base, not when raising a power to a power. Since $5b=20$, the correct value is $b=4$."},{"text":"$$100$","correct":false,"explanation":"Multiplying $5$ by $20$ does not solve for the unknown index. The power-of-a-power rule gives $5b=20$, so division by $5$ gives $b=4$."}]},{"id":"79923dd0-dc13-28ef-f5a5-4e49fd026df8","cardType":"basic","points":2,"front":"If $\\frac{x^5}{x^a} = x^7$, what is the value of $a$?","back":"$$-2$","mcqOptions":[{"text":"$$-2$","correct":true,"explanation":"When dividing powers with the same base, subtract the denominator’s index: $x^5/x^a=x^{5-a}$. Therefore $5-a=7$, which rearranges to $a=-2$."},{"text":"$$2$","correct":false,"explanation":"The negative sign before $a$ must be preserved when solving $5-a=7$. Subtracting $5$ gives $-a=2$, so multiplying by $-1$ gives $a=-2$."},{"text":"$$12$","correct":false,"explanation":"Adding $5$ and $7$ ignores the quotient law and does not solve the exponent equation. Division gives $5-a=7$, which rearranges to the correct value $a=-2$."},{"text":"$$-12$","correct":false,"explanation":"Combining $5$ and $7$ with the wrong operation produces an excessive negative value. From $5-a=7$, subtracting $5$ gives $-a=2$, so $a=-2$."}]},{"id":"c169a130-517d-8412-fd2f-b68a6705cc34","cardType":"basic","points":2,"front":"Write $16 \\times 4^6$ as a single power of 4.","back":"$$4^8$","mcqOptions":[{"text":"$$4^8$","correct":true,"explanation":"Since $16=4^2$, the expression is $4^2\\times4^6$. Multiplying powers with the same base means adding the indices, so the result is $4^{2+6}=4^8$."},{"text":"$$4^7$","correct":false,"explanation":"This treats $16$ as one additional factor of $4$, but $16=4^2$. Therefore the indices are $2$ and $6$, which add to give the correct result $4^8$."},{"text":"$$4^{12}$","correct":false,"explanation":"Multiplying the indices confuses multiplication of equal-base powers with the power-of-a-power rule. Since $16=4^2$, the calculation is $4^2\\times4^6=4^{2+6}=4^8$."},{"text":"$$4^6$","correct":false,"explanation":"This effectively ignores the factor of $16$. Rewriting $16$ as $4^2$ gives $4^2\\times4^6$, and adding the indices produces the correct single power $4^8$."}]},{"id":"3be9ba4a-c860-b639-c3d7-20ace46e88ef","cardType":"basic","points":2,"front":"If $1000^3 = 10^x$, what is the value of $x$?","back":"$$9$","mcqOptions":[{"text":"$$9$","correct":true,"explanation":"Since $1000=10^3$, cubing gives $(10^3)^3=10^{3\\times3}=10^9$. Therefore the exponent satisfying $1000^3=10^x$ is $x=9$."},{"text":"$$6$","correct":false,"explanation":"Adding the two $3$s confuses a power of a power with multiplication of equal-base powers. The indices multiply, so $(10^3)^3=10^9$ and $x=9$."},{"text":"$$27$","correct":false,"explanation":"Cubing the index itself is not the power-of-a-power rule. Since $1000=10^3$, the indices multiply as $3\\times3$, giving $10^9$ and therefore $x=9$."},{"text":"$$3000$","correct":false,"explanation":"Multiplying the base value by the outer exponent does not express the result as a power of $10$. Writing $1000=10^3$ gives $(10^3)^3=10^9$, so $x=9$."}]}]}],"examCountdown":null,"heading":"Indices","title":"Indices Flashcards","topbar":{"backHref":"/gcse/maths/ccea/flashcards"},"topicSidebar":"$L39"}]}],"topicId":"af59b5c8-149c-3f24-05a7-74b683c2b970","walled":"$L3a"}],"$L3b","$L3c"]