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Obtaining both simplified expansions earns M1."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$(7+4\\sqrt{3})-(7-4\\sqrt{3})$"},{"latex":"$$7+4\\sqrt{3}-7+4\\sqrt{3}$"}]},"order":2,"title":"Remove the bracket","description":"The subtraction sign changes both signs in the second bracket: positive seven becomes negative seven, while negative four root three becomes positive four root three."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$7+4\\sqrt{3}-7+4\\sqrt{3}$"},{"latex":"$$4\\sqrt{3}+4\\sqrt{3}$"},{"latex":"$$8\\sqrt{3}$"}]},"order":3,"title":"Collect like terms","description":"First, seven minus seven cancels. Then combine the two identical surd terms to get eight root three. 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This equals $1$, so the value is unchanged; equivalently, the numerator and denominator are each multiplied by the non-zero factor $\\sqrt{x}$. This earns M1."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\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}],"subjectCode":"igcse_maths__cie","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":"b4d9f117-99dd-4545-8c12-15910ffad122","topicSidebar":"$L3a","topicSlug":"grade-7-7-1-surds","topicTitle":"Surds"}]}],"$L3b","$L3c"]}]