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Since $(\\sqrt{7})^2=7$, squaring removes the square root.","highlights":[{"text":"$$(3\\sqrt{7})^2$","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"\\left(3\\sqrt{7}\\right)^2=\\htmlData{anchor=split}{\\color{@sky}{3^2\\left(\\sqrt{7}\\right)^2}}"},{"latex":"\\htmlData{anchor=result}{\\color{@green}{9\\times7=63}}"}],"highlights":[{"color":"green","style":"text-color","anchor":"result"}],"connections":[{"to":"result","from":"split","color":"sky","label":"evaluate squares","style":"solid"}]},"order":1,"title":"Square the factors","description":"Square the coefficient, then use the fact that a square root squared gives the number inside. [B1] Award for correctly simplifying to 63."}]}],"generatedAt":"2026-07-21T04:01:01.480Z","modelVersion":"gpt-5.6-terra","explanationVersion":"v3"},"options":[]},{"id":"ae7d2ae7-b952-4ebf-bfd4-932676312b90","label":"b","content":"$$(8 + \\sqrt{5})(2 - \\sqrt{5})$","markscheme":"- Expand brackets: $16 - 8\\sqrt{5} + 2\\sqrt{5} - 5$ M1\n- Simplify to $11 - 6\\sqrt{5}$ A2","marks":3,"order":1,"markingRubric":{"totalMarks":3,"points":[{"id":"p1","marks":1,"description":"Attempts to expand brackets with at least 3 terms correct.","code":"M"},{"id":"p2","marks":1,"description":"Correct simplification of integer terms (16 - 5 = 11).","code":"A"},{"id":"p3","marks":1,"description":"Correct simplification of surd terms ($-6\\sqrt{5}$).","code":"A"}],"finalAnswers":["11 - 6\\sqrt{5}"],"generatedAt":"2026-05-06T15:25:34.014Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"expand_collect","label":"Expand and collect","steps":[{"type":"text","order":0,"title":"Multiply every term","content":"Use the distributive law: multiply each term in the first bracket by **both** terms in the second bracket. Keep the signs with the terms; in particular, the final product is negative because it is $+\n\\sqrt{5}\\times-\\sqrt{5}$.\n\nWriting all four products separately is the expansion mark (M1).","highlights":[{"text":"$$(8 + \\sqrt{5})(2 - \\sqrt{5})$","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"(8+\\sqrt{5})(2-\\sqrt{5})"},{"latex":"=\\htmlData{anchor=sixteen}{16}-\\htmlData{anchor=eightroot}{8\\sqrt{5}}+\\htmlData{anchor=tworoot}{2\\sqrt{5}}-\\htmlData{anchor=five}{5}"},{"latex":"=\\htmlData{anchor=eleven}{11}-\\htmlData{anchor=sixroot}{6\\sqrt{5}}"}],"highlights":[{"color":"sky","style":"text-color","anchor":"sixteen"},{"color":"sky","style":"text-color","anchor":"five"},{"color":"purple","style":"text-color","anchor":"tworoot"}],"connections":[{"to":"sixroot","from":"eightroot","color":"purple","label":"collect like terms"}]},"order":1,"title":"Expand the brackets","description":"Write all four products. The final product of the two surds is negative five."},{"type":"text","order":2,"title":"State the simplified form","content":"The rational terms are $16-5=11$, while the like surd terms are $-8\\sqrt{5}+2\\sqrt{5}=-6\\sqrt{5}$. Therefore\n\n$$\\boxed{11-6\\sqrt{5}}$$\n\nThis correct simplification earns the remaining A2 marks. Do not turn $\\sqrt{5}$ into a decimal: the exact surd form is required.","highlights":[{"text":"Simplify","location":"specification"}]}]}],"generatedAt":"2026-09-01T09:07:19.748Z","modelVersion":"gpt-5.6-terra+diagrams-2","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":{"methods":[{"id":"dbaeff80-2697-4154-897b-61d06add6456:expand_square","label":"(a) Square each factor","steps":[{"type":"text","order":0,"title":"Use the square rule","content":"Use $(ab)^2=a^2b^2$: the square applies to both the coefficient and the surd. Since $(\\sqrt{7})^2=7$, squaring removes the square root.","highlights":[{"text":"$$(3\\sqrt{7})^2$","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"\\left(3\\sqrt{7}\\right)^2=\\htmlData{anchor=split}{\\color{@sky}{3^2\\left(\\sqrt{7}\\right)^2}}"},{"latex":"\\htmlData{anchor=result}{\\color{@green}{9\\times7=63}}"}],"highlights":[{"color":"green","style":"text-color","anchor":"result"}],"connections":[{"to":"result","from":"split","color":"sky","label":"evaluate squares","style":"solid"}]},"order":1,"title":"Square the factors","description":"Square the coefficient, then use the fact that a square root squared gives the number inside. [B1] Award for correctly simplifying to 63."}]},{"id":"ae7d2ae7-b952-4ebf-bfd4-932676312b90:expand_collect","label":"(b) Expand and collect","steps":[{"type":"text","order":0,"title":"Multiply every term","content":"Use the distributive law: multiply each term in the first bracket by **both** terms in the second bracket. Keep the signs with the terms; in particular, the final product is negative because it is $+\n\\sqrt{5}\\times-\\sqrt{5}$.\n\nWriting all four products separately is the expansion mark (M1).","highlights":[{"text":"$$(8 + \\sqrt{5})(2 - \\sqrt{5})$","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"(8+\\sqrt{5})(2-\\sqrt{5})"},{"latex":"=\\htmlData{anchor=sixteen}{16}-\\htmlData{anchor=eightroot}{8\\sqrt{5}}+\\htmlData{anchor=tworoot}{2\\sqrt{5}}-\\htmlData{anchor=five}{5}"},{"latex":"=\\htmlData{anchor=eleven}{11}-\\htmlData{anchor=sixroot}{6\\sqrt{5}}"}],"highlights":[{"color":"sky","style":"text-color","anchor":"sixteen"},{"color":"sky","style":"text-color","anchor":"five"},{"color":"purple","style":"text-color","anchor":"tworoot"}],"connections":[{"to":"sixroot","from":"eightroot","color":"purple","label":"collect like terms"}]},"order":1,"title":"Expand the brackets","description":"Write all four products. The final product of the two surds is negative five."},{"type":"text","order":2,"title":"State the simplified form","content":"The rational terms are $16-5=11$, while the like surd terms are $-8\\sqrt{5}+2\\sqrt{5}=-6\\sqrt{5}$. Therefore\n\n$$\\boxed{11-6\\sqrt{5}}$$\n\nThis correct simplification earns the remaining A2 marks. Do not turn $\\sqrt{5}$ into a decimal: the exact surd form is required.","highlights":[{"text":"Simplify","location":"specification"}]}]}],"generatedAt":"2026-09-01T09:07:20.061Z","modelVersion":"gpt-5.6-terra+diagrams-2","explanationVersion":"v3"},"firstAttemptFullMarks":{"percent":58},"walkthroughVideo":null},{"id":"3b0de506-c100-485a-88ac-172998e8ef9c","revisionId":"84b3eea6-0c89-4744-ae2b-a3caceac3e47","questionNumber":"2","questionType":"LA","specification":"Given that $32\\sqrt{2} = 2^a$, find the value of $a$.","difficulty":1,"difficultyBand":"easy","totalMarks":3,"sourceTitle":"Indices and Surds","calculatorUse":null,"isCalculation":true,"parts":[{"id":"940bd32f-3871-4c4b-b000-aef58fa7ce80","label":"0","content":"Given that $32\\sqrt{2} = 2^a$, find the value of $a$.","markscheme":"- $32 = 2^5$ and $\\sqrt{2} = 2^{1/2}$ M1\n- $2^5 \\times 2^{1/2} = 2^{5.5}$ or $2^{11/2}$ M1\n- $a = 11/2$ A1","marks":3,"order":0,"markingRubric":{"totalMarks":3,"points":[{"id":"p1","marks":1,"description":"Expresses 32 as $2^5$ or $\\sqrt{2}$ as $2^{1/2}$.","code":"M"},{"id":"p2","marks":1,"description":"Uses index laws to add powers ($5 + 0.5$).","code":"M"},{"id":"p3","marks":1,"description":"Final answer $a = 11/2$ or $5.5$.","code":"A"}],"finalAnswers":["11/2","5.5"],"generatedAt":"2026-05-06T15:25:34.014Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"rewrite_in_base_two","label":"Rewrite in base 2","steps":[{"type":"text","order":0,"title":"Rewrite each factor","content":"Write both factors as powers of the same base, $2$. This earns the first method mark (M1): $32=2^5$ and $\\sqrt{2}=2^{1/2}$.","highlights":[{"text":"$$32\\sqrt{2} = 2^a$","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"\\htmlData{anchor=start}{2^5\\times2^{1/2}}"},{"latex":"2^{\\htmlData{anchor=addition}{5+\\frac12}}"},{"latex":"\\htmlData{anchor=combined}{2^{11/2}}"}],"highlights":[{"color":"sky","style":"text-color-with-background","anchor":"addition"},{"color":"green","style":"text-color","anchor":"combined"}],"connections":[{"to":"addition","from":"start","color":"sky","label":"add exponents"},{"to":"combined","from":"addition","color":"green","label":"simplify"}]},"order":1,"title":"Multiply the powers","description":"When powers with the same base are multiplied, add their exponents. This is the second method mark (M1)."},{"type":"text","order":2,"title":"Match the exponent","content":"The expression is now $2^{11/2}$, and it is given as $2^a$. Since the bases are the same, the exponents must be equal:\n\n$$a=\\frac{11}{2}$$\n\nThis is the accuracy mark (A1).","highlights":[{"text":"find the value of $a$","location":"specification"}]}]}],"generatedAt":"2026-07-21T04:11:03.902Z","modelVersion":"gpt-5.6-terra","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":null,"walkthroughVideo":null},{"id":"b954cb10-663c-4cfb-9ae3-062b9959de14","revisionId":"8a3e4673-d308-4024-9dae-f7a4ea90d8ce","questionNumber":"3","questionType":"LA","specification":"","difficulty":1,"difficultyBand":"easy","totalMarks":3,"sourceTitle":"Indices and Surds","calculatorUse":null,"isCalculation":true,"parts":[{"id":"45b2ca37-6983-4b60-a342-730755193abd","label":"a","content":"Write down the value of $125^{\\frac{1}{3}}$.","markscheme":"- $125^{1/3} = 5$ B1","marks":1,"order":0,"markingRubric":{"totalMarks":1,"points":[{"id":"p1","marks":1,"description":"Final answer 5.","code":"B"}],"finalAnswers":["5"],"generatedAt":"2026-05-06T15:25:34.014Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"cube_root","label":"Evaluate the cube root","steps":[{"type":"text","order":0,"title":"Recognise the cube root","content":"The power $\\frac{1}{3}$ means **cube root**: find the number which, when multiplied by itself three times, gives $125$.\n\nSince $5^3=125$, the required value is $5$. This is the single accuracy mark (B1).","highlights":[{"text":"$$125^{\\frac{1}{3}}$","location":"specification"}]}]}],"generatedAt":"2026-07-21T03:13:25.656Z","modelVersion":"gpt-5.6-terra","explanationVersion":"v3"},"options":[]},{"id":"66fc2f1e-fece-4f7c-95bd-51e21fd09bc1","label":"b","content":"Find the value of $125^{-\\frac{2}{3}}$.","markscheme":"- $125^{2/3} = (125^{1/3})^2 = 5^2 = 25$ M1\n- $125^{-2/3} = \\frac{1}{25}$ A1","marks":2,"order":1,"markingRubric":{"totalMarks":2,"points":[{"id":"p1","marks":1,"description":"Attempts to square the cube root or handle the negative index.","code":"M"},{"id":"p2","marks":1,"description":"Final answer $1/25$ or $0.04$.","code":"A"}],"finalAnswers":["1/25","0.04"],"generatedAt":"2026-05-06T15:25:34.014Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"fractional_indices","label":"Evaluate the fractional index","steps":[{"type":"text","order":0,"title":"Read the index","content":"For a fractional index, the denominator gives the root and the numerator gives the power. So the positive-index part $125^{\\frac{2}{3}}$ means “cube root of $125$, then square it”. The negative sign will mean take a reciprocal afterwards.","highlights":[{"text":"Find the value of $125^{-\\frac{2}{3}}$.","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"125^{\\frac{2}{3}}=\\left(125^{\\frac{1}{3}}\\right)^2"},{"latex":"\\left(125^{\\frac{1}{3}}\\right)^2=5^2"},{"latex":"5^2=25"}]},"order":1,"title":"Evaluate the positive power","description":"First find the corresponding positive power. Examiner annotation (M1): obtaining the positive power as 25."},{"type":"hyper_latex","block":{"nodes":[{"latex":"125^{-\\frac{2}{3}}=\\frac{1}{\\htmlData{anchor=positive_power}{125^{\\frac{2}{3}}}}"},{"latex":"125^{-\\frac{2}{3}}=\\frac{1}{\\htmlData{anchor=result}{25}}"}],"connections":[{"to":"result","from":"positive_power","color":"orange","label":"substitute 25"}]},"order":2,"title":"Use the negative index","description":"A negative index means take the reciprocal. Substitute the positive power found above. Examiner annotation (A1): concluding that the value is 1/25."}]}],"generatedAt":"2026-07-21T03:21:55.207Z","modelVersion":"gpt-5.6-terra","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":null,"walkthroughVideo":null}],"subjectCode":"old_maths_alevel","summaries":[{"id":"15f982e7-d5b3-4cf1-8b89-4ae54d4e77e9","difficultyBand":"easy","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"3b0de506-c100-485a-88ac-172998e8ef9c","difficultyBand":"easy","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"b954cb10-663c-4cfb-9ae3-062b9959de14","difficultyBand":"easy","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"2b407ec6-83ed-4617-b76b-05b183116712","difficultyBand":"easy","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"ca610cd6-0cab-49b4-8ed2-a39c1834d94b","difficultyBand":"easy","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"3cc82298-ad8a-4760-8c6c-84be20084779","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"c5442434-93bc-4a2d-990f-da6ba1f80ceb","difficultyBand":"easy","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"b69cdd17-b9fd-4b1d-b454-970d7fd5db12","difficultyBand":"easy","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"a8b45331-5e14-4954-b89a-e9f591219297","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"dbad9844-8a01-44fd-9076-d9159c5c4799","difficultyBand":"easy","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"d8913a69-4d3e-4745-9a63-61c0e86790ac","difficultyBand":"easy","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"af8d4769-4b41-43c8-952c-5e0d6dc1f458","difficultyBand":"easy","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true}],"title":"Indices and Surds","topicId":"6b9bbc36-190d-4978-b6f2-5d97279e2654","topicSidebar":"$L3c","topicSlug":"legacy-old-alevel-7ade92ce-b034-418b-8028-77204cc6c610","topicTitle":"Indices and Surds"}]}],"$L3d","$L3e"]}]