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Since $27=9\\times3$, use $9$ as the square factor.","highlights":[{"text":"Write $5\\sqrt{27}$ in the form $k\\sqrt{3}$","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$5\\sqrt{27}=5\\color{@sky}{\\htmlData{anchor=factor}{\\sqrt{9\\times3}}}$"},{"latex":"$$5\\sqrt{27}=5\\left(\\color{@green}{\\htmlData{anchor=simplified}{3\\sqrt3}}\\right)$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"factor"},{"color":"green","style":"text-color","anchor":"simplified"}],"connections":[{"to":"simplified","from":"factor","color":"green","label":"square root of 9 is 3"}]},"order":1,"title":"Split the square root","description":"Use the largest square factor of 27. Showing the factorisation using 9 and 3, or simplifying the root to three times the square root of 3, earns M1."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$5\\left(3\\sqrt{3}\\right)=\\color{@orange}{\\htmlData{anchor=product}{(5\\times3)}}\\sqrt{3}$"},{"latex":"$$5\\left(3\\sqrt{3}\\right)=\\color{@green}{\\htmlData{anchor=coefficient}{15\\sqrt{3}}}$"}],"highlights":[{"color":"orange","style":"text-color","anchor":"product"},{"color":"green","style":"text-color-with-background","anchor":"coefficient"}]},"order":2,"title":"Multiply the coefficient","description":"Multiply the numbers outside the square root. The square root part stays unchanged; the resulting expression is the required form and earns A1."}]}],"generatedAt":"2026-07-20T16:25:22.283Z","modelVersion":"gpt-5.6-sol","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":{"percent":83},"walkthroughVideo":null},{"id":"cb761f3d-d598-47d0-8d14-da3783309823","revisionId":"a0c596c9-36ad-4172-95f0-668e9f6e6ed2","questionNumber":"4","questionType":"LA","specification":"","difficulty":1,"difficultyBand":"easy","totalMarks":2,"sourceTitle":"Surds","calculatorUse":null,"isCalculation":false,"parts":[{"id":"86109eca-c428-49db-86ce-6e23018b3f78","label":"0","content":"Write $7\\sqrt{20}$ in the form $k\\sqrt{5}$, where $k$ is an integer.","markscheme":"- Award method mark for simplifying $\\sqrt{20}$ to $2\\sqrt{5}$ or showing $7(\\sqrt{4 \\times 5})$ M1\n- Final answer $14\\sqrt{5}$ A1","marks":2,"order":0,"markingRubric":{"totalMarks":2,"points":[{"id":"p1","marks":1,"description":"Simplifies $\\sqrt{20}$ to $2\\sqrt{5}$.","code":"M"},{"id":"p2","marks":1,"description":"Final answer of $14\\sqrt{5}$.","code":"A"}],"finalAnswers":["14\\sqrt{5}"],"generatedAt":"2026-05-07T02:09:58.160Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"factor_square","label":"Factor out a square number","steps":[{"type":"text","order":0,"title":"Find a square factor","content":"To write the surd using $\\sqrt{5}$, split $20$ into a square number times $5$:\n\n$$20=4\\times5$$\n\nThe number $4$ is useful because its square root is an integer.","highlights":[{"text":"Write $7\\sqrt{20}$ in the form $k\\sqrt{5}$, where $k$ is an integer.","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"\\sqrt{20}=\\htmlData{anchor=factor}{\\color{@sky}{\\sqrt{4\\times5}}}"},{"latex":"\\sqrt{20}=\\htmlData{anchor=simplified}{\\color{@green}{2\\sqrt{5}}}"}],"highlights":[{"color":"sky","style":"text-color-with-background","anchor":"factor"},{"color":"green","style":"text-color","anchor":"simplified"}],"connections":[{"to":"simplified","from":"factor","color":"sky","label":"evaluate square factor"}]},"order":1,"title":"Simplify the square root","description":"Split 20 into 4 times 5. Then evaluate the square factor. Examiner note: simplifying root 20 to 2 root 5 earns M1."},{"type":"hyper_latex","block":{"nodes":[{"latex":"7\\sqrt{20}=7\\left(\\htmlData{anchor=substitute}{\\color{@sky}{2\\sqrt{5}}}\\right)"},{"latex":"7\\sqrt{20}=\\htmlData{anchor=answer}{\\color{@green}{14\\sqrt{5}}}"}],"highlights":[{"color":"sky","style":"text-color-with-background","anchor":"substitute"},{"color":"green","style":"text-color","anchor":"answer"}],"connections":[{"to":"answer","from":"substitute","color":"orange","label":"multiply coefficients"}]},"order":2,"title":"Multiply the coefficient","description":"Substitute the simplified surd, then multiply the two integer coefficients. Examiner note: 14 root 5 earns A1."}]}],"generatedAt":"2026-07-21T03:51:31.710Z","modelVersion":"gpt-5.6-terra","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":{"percent":80},"walkthroughVideo":null},{"id":"8e84081a-aed1-439b-8908-c1900fffcb4a","revisionId":"b3170eaa-bda3-44bd-bd96-6ebd625ecb3e","questionNumber":"5","questionType":"LA","specification":"","difficulty":1,"difficultyBand":"easy","totalMarks":2,"sourceTitle":"Surds","calculatorUse":null,"isCalculation":true,"parts":[{"id":"32a8fd73-2dea-4db9-a300-d5f9c3741925","label":"0","content":"Expand and Simplify $(2 + \\sqrt{3})(2 - \\sqrt{3})$","markscheme":"- Award method mark for expansion: $4 - 2\\sqrt{3} + 2\\sqrt{3} - 3$ M1\n- Final answer $1$ A1","marks":2,"order":0,"markingRubric":{"totalMarks":2,"points":[{"id":"p1","marks":1,"description":"Expands the brackets correctly with at least 3 terms correct, e.g., $4 - 3$.","code":"M"},{"id":"p2","marks":1,"description":"Final answer of 1.","code":"A"}],"finalAnswers":["1"],"generatedAt":"2026-05-07T02:09:58.160Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"expand_conjugates","label":"Expand the brackets","steps":[{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\htmlData{anchor=product}{(2+\\sqrt{3})(2-\\sqrt{3})}$"},{"latex":"$$\\htmlData{anchor=expanded}{4-2\\sqrt{3}+2\\sqrt{3}-3}$"}],"connections":[{"to":"expanded","from":"product","color":"sky","label":"expand"}]},"order":1,"title":"Expand each product","description":"Multiply each term in the first bracket by each term in the second, keeping each sign attached to its term. The correct expansion 4 − 2√3 + 2√3 − 3 earns the M1 method mark."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\htmlData{anchor=full_expansion}{4-2\\sqrt{3}+2\\sqrt{3}-3}$"},{"latex":"$$\\htmlData{anchor=integers}{4-3}$"}],"connections":[{"to":"integers","from":"full_expansion","color":"green","label":"cancel opposite terms","style":"solid"}]},"order":2,"title":"Cancel opposite terms","description":"The opposite surd terms cancel, leaving the integer terms."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\htmlData{anchor=integer_terms}{4-3}$"},{"latex":"$$\\htmlData{anchor=final_answer}{1}$"}],"connections":[{"to":"final_answer","from":"integer_terms","color":"purple","label":"subtract","style":"solid"}]},"order":3,"title":"Finish the calculation","description":"Now subtract the remaining integers. Writing the final answer earns the A1 mark."}]}],"generatedAt":"2026-07-21T03:58:48.364Z","modelVersion":"gpt-5.6-terra","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":{"percent":61},"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"]}]