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$\\frac{18p^{7/4}q^5}{6p^{1/2}q^8}$ using positive indices.","back":"$$\\boldsymbol{\\frac{3p^{5/4}}{q^3}}$","mcqOptions":null},{"id":"22491259-6125-4b01-bf17-94c795eb5d0f","cardType":"basic","points":2,"front":"Simplify $(8x^6)^{2/3}$ for $x\\geq 0$.","back":"$$\\boldsymbol{4x^4}$","mcqOptions":null},{"id":"80caf505-5596-49ec-a110-c289240703a4","cardType":"basic","points":1,"front":"Apply the index $n$ to the product $(ab)^n$.","back":"$$\\boldsymbol{a^nb^n}$","mcqOptions":null},{"id":"9cbb696c-9927-4ab9-86bd-a0010c3e3fa8","cardType":"basic","points":1,"front":"Apply the index $n$ to $\\left(\\frac{a}{b}\\right)^n$, where $b\\neq 0$.","back":"$$\\boldsymbol{\\frac{a^n}{b^n}}$","mcqOptions":null},{"id":"555cefb1-2be6-40b8-86c4-eb18d54986f8","cardType":"basic","points":1,"front":"Does an index generally distribute over addition: $(a+b)^n=a^n+b^n$?","back":"No - index distribution applies to products and quotients.","mcqOptions":null},{"id":"34d77807-a881-4812-9485-969add518601","cardType":"basic","points":1,"front":"Evaluate $a^0$ when $a\\neq 0$.","back":"$$\\boldsymbol{1}$","mcqOptions":null},{"id":"9541d6f7-aba3-42a0-8008-7282cefcfc01","cardType":"basic","points":2,"front":"Why must $a\\neq 0$ in the zero-index law?","back":"Its derivation uses division by $a^m$, which is undefined for $a=0$.","mcqOptions":null},{"id":"dfa6ec0f-56f7-4363-9140-765cb223b08d","cardType":"basic","points":1,"front":"Rewrite $\\frac{6x^{-2}}{y^{-3}}$ using positive indices.","back":"$$\\boldsymbol{\\frac{6y^3}{x^2}}$","mcqOptions":null},{"id":"879df514-0288-410b-937e-7ea73063cac9","cardType":"cloze","points":null,"front":"A negative index gives a reciprocal: $a^{-n}=$ $\\text{[...]}$, for $a\\neq 0$.","back":"A negative index gives a reciprocal: $a^{-n}=$ $\\boxed{\\frac{1}{a^n}}$, for $a\\neq 0$.","mcqOptions":null},{"id":"16a2f598-1ad3-4274-a663-1cf910968087","cardType":"cloze","points":null,"front":"In $a^{m/n}$, the denominator $n$ gives the [ ] and the numerator $m$ gives the power.","back":"In $a^{m/n}$, the denominator $n$ gives the root and the numerator $m$ gives the power.","mcqOptions":null},{"id":"03ea6d24-b579-4911-b9f0-1d405f6dd246","cardType":"basic","points":1,"front":"Evaluate $81^{3/4}$.","back":"$$\\boldsymbol{27}$","mcqOptions":null},{"id":"ad21d4b8-40a8-4b2f-be18-e3b99df4bb46","cardType":"basic","points":2,"front":"Evaluate $32^{-2/5}$.","back":"$$\\boldsymbol{\\frac{1}{4}}$","mcqOptions":null},{"id":"b0c210df-3ebf-4a4d-89a9-3f91d4b9f236","cardType":"cloze","points":null,"front":"Over the real numbers, $a^{1/n}$ requires $a\\geq 0$ when $n$ is [ ].","back":"Over the real numbers, $a^{1/n}$ requires $a\\geq 0$ when $n$ is even.","mcqOptions":null},{"id":"eb6b304a-3488-4766-8a36-f9c76e910d31","cardType":"basic","points":2,"front":"Simplify $\\frac{(2x^{3/2}y^{-1})^2}{4x^{-1}y^3}$ using positive indices.","back":"$$\\boldsymbol{\\frac{x^4}{y^5}}$","mcqOptions":null},{"id":"61149526-2d86-4c5e-85a6-f7af61d2d4f2","cardType":"basic","points":2,"front":"Solve $x^{3/2}=64$ for real $x$.","back":"$$\\boldsymbol{x=16}$","mcqOptions":null}]}],"examCountdown":{"examDate":"2027-05-26","paperLabel":"Paper 1"},"heading":"1.5.1 Laws of indices","title":"1.5.1 Laws of indices Flashcards","topbar":{"backHref":"/a-level/maths/aqa/flashcards"},"topicSidebar":"$L36"}]}],"topicId":"9ca6605d-2a51-4d7b-a5cc-a687a1a05264","walled":"$L37"}],"$L38","$L39"]