t":"What operation does a negative index indicate?","back":"Taking the reciprocal of the base.","points":1},{"id":"74c91d25-76e4-4589-bd25-138bf10c5a98","cardType":"basic","front":"Evaluate $5^{-2}$.","back":"$$\\frac{1}{25}$","points":1},{"id":"b1308fe6-b390-4d3e-9c35-e4df45c0fc36","cardType":"basic","front":"Evaluate $\\left(\\frac{3}{8}\\right)^{-1}$.","back":"$$\\frac{8}{3}$","points":1},{"id":"a03adb1f-4311-48df-af3a-0e9a18724e47","cardType":"cloze","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}}.","points":null},{"id":"c98ce278-23a1-472b-9ebf-3b7b5d2bffb0","cardType":"cloze","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 $\\boxed{\\sqrt[n]{a}}$.","points":null},{"id":"52cfa59f-e003-4544-8f42-7c3946f5426e","cardType":"basic","front":"In the fractional index $a^{\\frac{m}{n}}$, what do the numerator and denominator represent?","back":"

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

","points":2},{"id":"c26d0fef-43c7-45bc-9c89-1ba4a3baa30b","cardType":"basic","front":"Evaluate $343^{\\frac{2}{3}}$.","back":"49","points":2},{"id":"cb119426-4675-4b39-9e93-bf27a382a174","cardType":"basic","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.","points":1},{"id":"3945a30f-a935-4e1a-9e80-1ce349596baf","cardType":"cloze","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}}.","points":null},{"id":"349dc2f6-281e-4887-a9d1-e63ebc4c4192","cardType":"basic","front":"Evaluate $32^{-\\frac{2}{5}}$.","back":"$$\\frac{1}{4}$","points":2},{"id":"451ddd34-f920-40d9-8036-8a49be808807","cardType":"basic","front":"What is the most efficient first step to evaluate $\\left(\\frac{a}{b}\\right)^{-n}$?","back":"Invert the base to $\\left(\\frac{b}{a}\\right)^n$ to remove the negative index.","points":1},{"id":"03c189c5-e0c4-44ec-a8f3-6fcb3db8c2b8","cardType":"basic","front":"Evaluate $\\left(\\frac{16}{25}\\right)^{-\\frac{3}{2}}$.","back":"$$\\frac{125}{64}$","points":2},{"id":"a3a45aec-feb8-4eab-a81f-36bde26ab90e","cardType":"basic","front":"Simplify $(27x^6)^{\\frac{2}{3}}$.","back":"$$9x^4$","points":2},{"id":"4c0b9776-68c4-4928-8933-39802c0e84b3","cardType":"cloze","front":"The index law for a power of a power is $(a^p)^q = \\text{[...]}.","back":"The index law for a power of a power is $(a^p)^q = \\boxed{a^{pq}}.","points":null},{"id":"c0c1b0be-4c0a-4464-a9cf-d7cc8210397f","cardType":"basic","front":"What is the general strategy for finding an unknown index $n$ in an equation like $2 \\times \\sqrt{32} = 2^n$?","back":"Rewrite both sides using the same base and equate the indices.","points":1},{"id":"4d5a0bd5-d435-468b-9029-b8fb52631742","cardType":"basic","front":"Rewrite $\\sqrt{32}$ as a power of 2.","back":"$$2^{\\frac{5}{2}}$","points":2},{"id":"0db5a745-90c5-4f00-878f-37e015a49324","cardType":"basic","front":"If $s = 5^q$, express $5^{q-1}$ in terms of $s$.","back":"$$\\frac{s}{5}$","points":2},{"id":"9aca0684-6061-4c59-820c-0706daeac767","cardType":"basic","front":"Why is $0^{-1}$ undefined?","back":"It requires division by zero ($\\frac{1}{0}$), which is impossible.","points":1},{"id":"0af5ad4d-c332-4c24-a6f9-d0fe0a3804b1","cardType":"basic","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: $5 \\times 10^2 = 500$<

Fractional and Negative Indices Flashcards

Fractional and Negative Indices Flashcards

Fractional and Negative Indices Flashcards

Fractional and Negative Indices Flashcards