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\\htmlData{anchor=multiplier}{\\color{@sky}{\\dfrac{\\sqrt{x}}{\\sqrt{x}}}}$"},{"latex":"$$\\dfrac{x\\sqrt{x}}{\\htmlData{anchor=root_product}{\\color{@amber}{\\sqrt{x}\\times\\sqrt{x}}}}$"},{"latex":"$$\\dfrac{x\\sqrt{x}}{\\htmlData{anchor=denominator_x}{\\color{@amber}{x}}}$"}],"highlights":[{"color":"sky","style":"text-color-with-background","anchor":"multiplier"},{"color":"amber","style":"text-color","anchor":"root_product"},{"color":"amber","style":"text-color","anchor":"denominator_x"}],"connections":[{"to":"denominator_x","from":"root_product","color":"amber","label":"root times root = x"}]},"order":1,"title":"Multiply by the root","description":"Multiply the fraction by $\\dfrac{\\sqrt{x}}{\\sqrt{x}}$. This equals $1$, so the value is unchanged; equivalently, the numerator and denominator are each multiplied by the non-zero factor $\\sqrt{x}$. This earns M1."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\dfrac{\\htmlData{anchor=num_x}{\\color{@orange}{x}}\\sqrt{x}}{\\htmlData{anchor=den_x}{\\color{@orange}{x}}}$"},{"latex":"$$\\dfrac{\\htmlData{anchor=cancelled_num_x}{\\color{@orange}{\\cancel{x}}}\\sqrt{x}}{\\htmlData{anchor=cancelled_den_x}{\\color{@orange}{\\cancel{x}}}}$"},{"latex":"$$\\boxed{\\htmlData{anchor=answer}{\\color{@green}{\\sqrt{x}}}}$"}],"highlights":[{"color":"orange","style":"text-color","anchor":"num_x"},{"color":"orange","style":"text-color","anchor":"den_x"},{"color":"orange","style":"text-color","anchor":"cancelled_num_x"},{"color":"orange","style":"text-color","anchor":"cancelled_den_x"}]},"order":2,"title":"Cancel the common factor","description":"Cancel the common factor $x$ in the numerator and denominator. The remaining factor is the rationalised answer, earning the accuracy mark (A1)."}]}],"generatedAt":"2026-07-20T16:37:53.342Z","modelVersion":"gpt-5.6-sol","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":{"percent":60},"walkthroughVideo":null},{"id":"a79070ee-d294-4a3d-a2d3-fbcad7907073","revisionId":"feb5a4a6-0b9d-485a-84e9-9f8ea5676e67","questionNumber":"12","questionType":"LA","specification":"","difficulty":2,"difficultyBand":"easy","totalMarks":2,"sourceTitle":"Surds","calculatorUse":null,"isCalculation":false,"parts":[{"id":"48ad923c-083f-41e5-90dd-d1bddeadf984","label":"0","content":"Rationalise the denominator $\\frac{1 + \\sqrt{5}}{\\sqrt{2}}$","markscheme":"- Award method mark for multiplying numerator and denominator by $\\sqrt{2}$: $\\frac{(1 + \\sqrt{5})\\sqrt{2}}{2}$ M1\n- Final answer $\\frac{\\sqrt{2} + \\sqrt{10}}{2}$ A1","marks":2,"order":0,"markingRubric":{"totalMarks":2,"points":[{"id":"p1","marks":1,"description":"Multiplies numerator and denominator by $\\sqrt{2}$.","code":"M"},{"id":"p2","marks":1,"description":"Final answer of $\\frac{\\sqrt{2} + \\sqrt{10}}{2}$.","code":"A"}],"finalAnswers":["\\frac{\\sqrt{2} + \\sqrt{10}}{2}"],"generatedAt":"2026-05-07T02:09:58.160Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"multiply_by_surd","label":"Multiply by the denominator surd","steps":[{"type":"text","order":0,"title":"Choose the multiplier","content":"To rationalise a denominator means to remove the surd from it. Multiply by $\\dfrac{\\sqrt{2}}{\\sqrt{2}}$; this is equal to $1$, so it does not change the value of the fraction.","highlights":[{"text":"Rationalise the denominator","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\dfrac{1+\\sqrt{5}}{\\sqrt{2}}\\times \\htmlData{anchor=factor}{\\color{@sky}{\\dfrac{\\sqrt{2}}{\\sqrt{2}}}}$"},{"latex":"$$\\htmlData{anchor=whole_fraction}{\\dfrac{(1+\\sqrt{5})\\sqrt{2}}{\\htmlData{anchor=den_product}{\\color{@amber}{\\sqrt{2}\\times\\sqrt{2}}}}}$"},{"latex":"$$\\dfrac{(1+\\sqrt{5})\\sqrt{2}}{\\htmlData{anchor=den_two}{\\color{@amber}{2}}}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"factor"},{"color":"amber","style":"text-color","anchor":"den_product"},{"color":"amber","style":"text-color","anchor":"den_two"}],"connections":[{"to":"whole_fraction","from":"factor","color":"sky","label":"multiply both parts"},{"to":"den_two","from":"den_product","color":"amber","label":"equals 2"}]},"order":1,"title":"Multiply both parts","description":"Multiplying the numerator and denominator by the square root of 2 earns the method mark, M1."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\dfrac{\\htmlData{anchor=bracket}{\\color{@orange}{(1+\\sqrt{5})\\sqrt{2}}}}{2}$"},{"latex":"$$\\dfrac{\\htmlData{anchor=expanded}{\\color{@orange}{\\sqrt{2}+\\sqrt{5}\\sqrt{2}}}}{2}$"}],"highlights":[{"color":"orange","style":"text-color","anchor":"bracket"},{"color":"orange","style":"text-color","anchor":"expanded"}],"connections":[{"to":"expanded","from":"bracket","color":"orange","label":"expand the bracket"}]},"order":2,"title":"Expand the numerator","description":"Distribute the square root of 2 across both terms in the bracket."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\dfrac{\\sqrt{2}+\\htmlData{anchor=surd_product}{\\color{@green}{\\sqrt{5}\\sqrt{2}}}}{2}$"},{"latex":"$$\\boxed{\\dfrac{\\sqrt{2}+\\htmlData{anchor=surd_ten}{\\color{@green}{\\sqrt{10}}}}{2}}$"}],"highlights":[{"color":"green","style":"text-color","anchor":"surd_product"},{"color":"green","style":"text-color-with-background","anchor":"surd_ten"}],"connections":[{"to":"surd_ten","from":"surd_product","color":"green","label":"multiply the surds"}]},"order":3,"title":"Simplify the surds","description":"Multiply the surds in the numerator. The boxed final answer earns A1."}]}],"generatedAt":"2026-07-20T16:38:20.049Z","modelVersion":"gpt-5.6-sol","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":{"percent":69},"walkthroughVideo":null},{"id":"3ef7bcea-af90-4560-9d02-414c974f9ba5","revisionId":"3b2f06fa-0b1c-41d1-bfef-7be64f582c33","questionNumber":"19","questionType":"LA","specification":"","difficulty":2,"difficultyBand":"easy","totalMarks":2,"sourceTitle":"Surds","calculatorUse":null,"isCalculation":true,"parts":[{"id":"7e968742-157e-49e1-b1a2-d9c88ecaadd4","label":"0","content":"Simplify fully $(\\sqrt{a} + \\sqrt{b})(\\sqrt{a} - \\sqrt{b})$","markscheme":"- Award method mark for expansion: $a - \\sqrt{ab} + \\sqrt{ab} - b$ M1\n- Final answer $a - b$ A1","marks":2,"order":0,"markingRubric":{"totalMarks":2,"points":[{"id":"p1","marks":1,"description":"Expands brackets to get at least 3 terms correct.","code":"M"},{"id":"p2","marks":1,"description":"Final answer of $a - b$.","code":"A"}],"finalAnswers":["a - b"],"generatedAt":"2026-05-07T02:09:58.160Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"expand","label":"Expand and cancel","steps":[{"type":"text","order":0,"title":"Multiply every term","content":"Expand by multiplying each term in the first bracket by **each** term in the second bracket. In particular, multiply the two square roots together using $\\sqrt{a}\\sqrt{b}=\\sqrt{ab}$."},{"type":"hyper_latex","block":{"nodes":[{"latex":"(\\sqrt{a}+\\sqrt{b})(\\sqrt{a}-\\sqrt{b})"},{"latex":"=\\sqrt{a}\\cdot\\sqrt{a}-\\sqrt{a}\\sqrt{b}+\\sqrt{b}\\sqrt{a}-\\sqrt{b}\\cdot\\sqrt{b}"},{"latex":"=a-\\htmlData{anchor=expanded_middle_one}{\\sqrt{ab}}+\\htmlData{anchor=expanded_middle_two}{\\sqrt{ab}}-b"}],"highlights":[{"color":"red","style":"underline","anchor":"expanded_middle_one"},{"color":"green","style":"underline","anchor":"expanded_middle_two"}]},"order":1,"title":"Expand the brackets","description":"Keep all four products before simplifying. This full expansion earns the method mark."},{"type":"hyper_latex","block":{"nodes":[{"latex":"a\\htmlData{anchor=cancel_minus}{-\\sqrt{ab}}\\htmlData{anchor=cancel_plus}{+\\sqrt{ab}}-b"},{"latex":"=\\boxed{a-b}"}],"highlights":[{"color":"red","style":"underline","anchor":"cancel_minus"},{"color":"green","style":"underline","anchor":"cancel_plus"}]},"order":2,"title":"Cancel opposite terms","description":"The middle terms are opposites: one is subtracted and one is added, so they cancel."}]},{"id":"difference-squares","label":"Use difference of squares","steps":[{"type":"text","order":0,"title":"Spot the special pattern","content":"The two brackets have the form $\\left(x+y\\right)\\left(x-y\\right)$: the same two terms appear, but the second sign changes. This is the **difference of two squares** pattern.\n\nHere, take $x=\\sqrt{a}$ and $y=\\sqrt{b}$.","highlights":[{"text":"$$(\\sqrt{a} + \\sqrt{b})(\\sqrt{a} - \\sqrt{b})$","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"(\\sqrt{a}+\\sqrt{b})(\\sqrt{a}-\\sqrt{b})=a-\\htmlData{anchor=cancel_term}{\\sqrt{ab}}+\\sqrt{ab}-b"},{"latex":"a-b"}],"highlights":[{"color":"green","style":"underline","anchor":"cancel_term"}]},"order":1,"title":"Expand then cancel","description":"Recognise the difference-of-two-squares pattern, then write the full expansion. This expansion earns M1; the simplified answer earns A1."}]}],"generatedAt":"2026-09-01T09:12:06.780Z","modelVersion":"gpt-5.6-terra+diagrams-2","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":{"methods":[{"id":"7e968742-157e-49e1-b1a2-d9c88ecaadd4:expand","label":"Expand and cancel","steps":[{"type":"text","order":0,"title":"Multiply every term","content":"Expand by multiplying each term in the first bracket by **each** term in the second bracket. In particular, multiply the two square roots together using $\\sqrt{a}\\sqrt{b}=\\sqrt{ab}$."},{"type":"hyper_latex","block":{"nodes":[{"latex":"(\\sqrt{a}+\\sqrt{b})(\\sqrt{a}-\\sqrt{b})"},{"latex":"=\\sqrt{a}\\cdot\\sqrt{a}-\\sqrt{a}\\sqrt{b}+\\sqrt{b}\\sqrt{a}-\\sqrt{b}\\cdot\\sqrt{b}"},{"latex":"=a-\\htmlData{anchor=expanded_middle_one}{\\sqrt{ab}}+\\htmlData{anchor=expanded_middle_two}{\\sqrt{ab}}-b"}],"highlights":[{"color":"red","style":"underline","anchor":"expanded_middle_one"},{"color":"green","style":"underline","anchor":"expanded_middle_two"}]},"order":1,"title":"Expand the brackets","description":"Keep all four products before simplifying. This full expansion earns the method mark."},{"type":"hyper_latex","block":{"nodes":[{"latex":"a\\htmlData{anchor=cancel_minus}{-\\sqrt{ab}}\\htmlData{anchor=cancel_plus}{+\\sqrt{ab}}-b"},{"latex":"=\\boxed{a-b}"}],"highlights":[{"color":"red","style":"underline","anchor":"cancel_minus"},{"color":"green","style":"underline","anchor":"cancel_plus"}]},"order":2,"title":"Cancel opposite terms","description":"The middle terms are opposites: one is subtracted and one is added, so they cancel."}]},{"id":"7e968742-157e-49e1-b1a2-d9c88ecaadd4:difference-squares","label":"Use difference of squares","steps":[{"type":"text","order":0,"title":"Spot the special pattern","content":"The two brackets have the form $\\left(x+y\\right)\\left(x-y\\right)$: the same two terms appear, but the second sign changes. This is the **difference of two squares** pattern.\n\nHere, take $x=\\sqrt{a}$ and $y=\\sqrt{b}$.","highlights":[{"text":"$$(\\sqrt{a} + \\sqrt{b})(\\sqrt{a} - \\sqrt{b})$","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"(\\sqrt{a}+\\sqrt{b})(\\sqrt{a}-\\sqrt{b})=a-\\htmlData{anchor=cancel_term}{\\sqrt{ab}}+\\sqrt{ab}-b"},{"latex":"a-b"}],"highlights":[{"color":"green","style":"underline","anchor":"cancel_term"}]},"order":1,"title":"Expand then cancel","description":"Recognise the difference-of-two-squares pattern, then write the full expansion. This expansion earns M1; the simplified answer earns A1."}]}],"generatedAt":"2026-09-01T09:12:07.036Z","modelVersion":"gpt-5.6-terra+diagrams-2","explanationVersion":"v3"},"firstAttemptFullMarks":null,"walkthroughVideo":null}],"subjectCode":"igcse_maths__oxford_aqa","summaries":[{"id":"c484ee1f-1bf6-47f3-8890-f0e0c9be7909","difficultyBand":"easy","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":false},{"id":"0dacce83-74f3-4f99-8117-8330978282ef","difficultyBand":"easy","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":false},{"id":"72342b57-0e03-4584-aeb4-c241090951c7","difficultyBand":"easy","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":false},{"id":"cb761f3d-d598-47d0-8d14-da3783309823","difficultyBand":"easy","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":false},{"id":"8e84081a-aed1-439b-8908-c1900fffcb4a","difficultyBand":"easy","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"8f495886-5503-40d6-8287-a440cec45cef","difficultyBand":"easy","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":false},{"id":"fd3b66f0-2b80-4928-9527-ffa5c326123b","difficultyBand":"easy","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"96b5c553-923a-4ae0-a9cb-0e38ba01685d","difficultyBand":"easy","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":false},{"id":"459ea282-538c-47a6-851e-37e43fa737e7","difficultyBand":"easy","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"58da9761-01de-4848-8c89-1b88aeabaae6","difficultyBand":"easy","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":false},{"id":"c00c601a-faa1-4fe7-90a3-343c6368b2cb","difficultyBand":"easy","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":false},{"id":"a79070ee-d294-4a3d-a2d3-fbcad7907073","difficultyBand":"easy","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":false},{"id":"45b051e4-21f9-4ca0-8299-deb8d6b8daf7","difficultyBand":"medium","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"8589d66b-7925-4d40-8356-acde4f97efda","difficultyBand":"medium","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"33397bbc-7bcc-4813-8a4b-b51313185314","difficultyBand":"medium","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"a2be29b3-42b1-4a7b-b349-cded36ca8985","difficultyBand":"medium","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"18eb1e0e-a1ea-4b6b-8395-5fe2b0f1da54","difficultyBand":"medium","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"1f743cac-719e-4a41-b668-629d4a63f8e0","difficultyBand":"medium","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"3ef7bcea-af90-4560-9d02-414c974f9ba5","difficultyBand":"easy","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"e56f66f9-08d3-4347-bc34-22336ea478f4","difficultyBand":"easy","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":true}],"title":"Surds","topicId":"341c0e44-a1d2-46ae-b271-971e5e815956","topicSidebar":"$L3a","topicSlug":"grade-7-7-1-surds","topicTitle":"Surds"}]}],"$L3b","$L3c"]}]