l name for $x$ and $5$?","back":"

$x$ 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":"d4037d6f-9e4c-19a9-0811-f221d07e3bfa","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":"2b1b0c2f-1aa1-bd7e-11cb-d990a47936f6","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":"$$2x^{11}$","correct":false,"explanation":"The implicit coefficients are both $1$, whose product remains $1$, rather than $2$. The indices add to $11$, so the product is $x^{11}$."}]},{"id":"e5a8222b-aafb-c3b0-447c-e3420803e7ad","cardType":"basic","points":1,"front":"Simplify $\\frac{w^9}{w^4}$.","back":"$$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 ar

Indices Flashcards

Indices Flashcards