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2.","code":"B"}],"finalAnswers":["2"],"generatedAt":"2026-05-06T15:25:34.014Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"fourth_root","label":"Evaluate the fourth root","steps":[{"type":"text","order":0,"title":"Interpret the index","content":"A power of $\\frac14$ means the **fourth root**. So find the positive number which, when multiplied by itself four times, gives $16$.","highlights":[{"text":"$$16^{\\frac{1}{4}}$","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"\\htmlData{anchor=power}{2^4}=16"},{"latex":"16^{\\frac14}=\\htmlData{anchor=answer}{\\color{@green}{2}}"}],"highlights":[{"color":"green","style":"text-color-with-background","anchor":"answer"}],"connections":[{"to":"answer","from":"power","color":"green","label":"take fourth root"}]},"order":1,"title":"State the fourth root","description":"Since 2 multiplied by itself four times is 16, the fourth root is 2. This earns the B1 mark."}]}],"generatedAt":"2026-07-21T04:12:32.732Z","modelVersion":"gpt-5.6-terra","explanationVersion":"v3"},"options":[]},{"id":"cf9c38ab-8dad-4aa8-a04c-f3e4181d701e","label":"b","content":"Simplify $(16x^{12})^{\\frac{3}{4}}$.","markscheme":"- $16^{3/4} = (16^{1/4})^3 = 2^3 = 8$ B1\n- $(x^{12})^{3/4} = x^{12 \\times 3/4} = x^9$ B1\n- Final answer $8x^9$","marks":2,"order":1,"markingRubric":{"totalMarks":2,"points":[{"id":"p1","marks":1,"description":"Correctly evaluates the coefficient as 8.","code":"B"},{"id":"p2","marks":1,"description":"Correctly evaluates the power of x as $x^9$.","code":"B"}],"finalAnswers":["8x^9"],"generatedAt":"2026-05-06T15:25:34.014Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":null,"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":null,"walkthroughVideo":null},{"id":"c5442434-93bc-4a2d-990f-da6ba1f80ceb","revisionId":"c83f6c55-3b14-4a0f-9862-8614b8041f9d","questionNumber":"7","questionType":"LA","specification":"","difficulty":1,"difficultyBand":"easy","totalMarks":4,"sourceTitle":"Indices and Surds","calculatorUse":null,"isCalculation":true,"parts":[{"id":"07fcac0c-a1a2-4beb-96f0-28ef49a38804","label":"a","content":"Find the value of $8^{\\frac{4}{3}}$.","markscheme":"- $8^{1/3} = 2$ M1\n- $2^4 = 16$ A1","marks":2,"order":0,"markingRubric":{"totalMarks":2,"points":[{"id":"p1","marks":1,"description":"Finds the cube root of 8.","code":"M"},{"id":"p2","marks":1,"description":"Final answer 16.","code":"A"}],"finalAnswers":["16"],"generatedAt":"2026-05-06T15:25:34.014Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":null,"options":[]},{"id":"1383f45c-5354-459e-a2c6-f0b19b30217a","label":"b","content":"Simplify $\\frac{15x^{\\frac{4}{3}}}{3x}$.","markscheme":"- $15/3 = 5$ B1\n- $x^{4/3} \\div x^1 = x^{4/3 - 1} = x^{1/3}$ B1\n- Final answer $5x^{1/3}$","marks":2,"order":1,"markingRubric":{"totalMarks":2,"points":[{"id":"p1","marks":1,"description":"Correctly simplifies the coefficient to 5.","code":"B"},{"id":"p2","marks":1,"description":"Correctly simplifies the power of x to $x^{1/3}$.","code":"B"}],"finalAnswers":["5x^{1/3}"],"generatedAt":"2026-05-06T15:25:34.014Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":null,"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":null,"walkthroughVideo":null},{"id":"b69cdd17-b9fd-4b1d-b454-970d7fd5db12","revisionId":"dd16812c-38bc-486f-a97e-7c389b4bc7d6","questionNumber":"8","questionType":"LA","specification":"","difficulty":1,"difficultyBand":"easy","totalMarks":4,"sourceTitle":"Indices and Surds","calculatorUse":null,"isCalculation":true,"parts":[{"id":"a021cfbf-192e-405f-a817-23200225ed9d","label":"a","content":"Express $\\sqrt{108}$ in the form $a\\sqrt{3}$, where $a$ is an integer.","markscheme":"- $\\sqrt{108} = \\sqrt{36} \\times \\sqrt{3} = 6\\sqrt{3}$ B1","marks":1,"order":0,"markingRubric":{"totalMarks":1,"points":[{"id":"p1","marks":1,"description":"Final answer $6\\sqrt{3}$.","code":"B"}],"finalAnswers":["6\\sqrt{3}"],"generatedAt":"2026-05-06T15:25:34.014Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"simplify_surd","label":"Simplify the surd","steps":[{"type":"text","order":0,"title":"Find the square factor","content":"The useful square factor is $36$, because $108=36\\times3$ and $36$ is a square.","highlights":[{"text":"Express $\\sqrt{108}$ in the form $a\\sqrt{3}$, where $a$ is an integer.","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"\\sqrt{108}=\\sqrt{36\\times3}"},{"latex":"\\sqrt{108}=\\sqrt{36}\\times\\sqrt{3}"},{"latex":"\\sqrt{108}=6\\sqrt{3}"}]},"order":1,"title":"Take out the square root","description":"Write 108 as a square number multiplied by 3, then simplify the square root.\n\nB1: Correctly writes √108 = √36 × √3 = 6√3."}]}],"generatedAt":"2026-07-21T04:24:39.183Z","modelVersion":"gpt-5.6-terra","explanationVersion":"v3"},"options":[]},{"id":"99429fa5-af22-464c-83f8-e9e3caba686e","label":"b","content":"Express $(2 - \\sqrt{3})^2$ in the form $b + c\\sqrt{3}$, where $b$ and $c$ are integers to be found.","markscheme":"- $(2 - \\sqrt{3})(2 - \\sqrt{3}) = 4 - 2\\sqrt{3} - 2\\sqrt{3} + 3$ M1\n- $4 + 3 = 7$ A1\n- $-2\\sqrt{3} - 2\\sqrt{3} = -4\\sqrt{3}$ A1\n- Final answer $7 - 4\\sqrt{3}$","marks":3,"order":1,"markingRubric":{"totalMarks":3,"points":[{"id":"p1","marks":1,"description":"Attempts to expand the squared bracket with at least 3 terms correct.","code":"M"},{"id":"p2","marks":1,"description":"Correct integer part (7).","code":"A"},{"id":"p3","marks":1,"description":"Correct surd part ($-4\\sqrt{3}$).","code":"A"}],"finalAnswers":["7 - 4\\sqrt{3}"],"generatedAt":"2026-05-06T15:25:34.014Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"direct_expansion","label":"Expand the square","steps":[{"type":"text","order":0,"title":"Rewrite the square","content":"A square means multiply the expression by itself. Keep the brackets so that every term is multiplied by every term:\n\n$$(2-\\sqrt{3})^2=(2-\\sqrt{3})(2-\\sqrt{3}).$$\n\nThis is the starting point for the expansion.","highlights":[{"text":"Express $(2 - \\sqrt{3})^2$ in the form $b + c\\sqrt{3}$","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"(2-\\sqrt{3})(2-\\sqrt{3})"},{"latex":"\\htmlData{anchor=four_term_expansion}{4-2\\sqrt{3}-2\\sqrt{3}+3}"}],"highlights":[{"color":"purple","style":"underline","anchor":"four_term_expansion"}]},"order":1,"title":"Multiply every pair","description":"Use the first bracket to multiply both terms in the second bracket. This complete four-term expansion earns the M1."},{"type":"text","order":2,"title":"Collect like terms","content":"Now collect the rational terms and the terms containing $\\sqrt{3}$ separately. The expansion in the previous step earns the **M1**. \n\nThe rational terms give $4+3=7$, earning one accuracy mark. The two surd terms are like terms, so their coefficients add: $-2\\sqrt{3}-2\\sqrt{3}=-4\\sqrt{3}$, earning the other accuracy mark.","highlights":[{"text":"Express $(2 - \\sqrt{3})^2$ in the form $b + c\\sqrt{3}$","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"\\htmlData{anchor=combined_terms}{7+(-4\\sqrt{3})}"},{"latex":"\\boxed{7-4\\sqrt{3}}"}],"highlights":[{"color":"teal","style":"underline","anchor":"combined_terms"}]},"order":3,"title":"State the required form","description":"Put the rational term first, followed by the multiple of $\\sqrt{3}$."}]}],"generatedAt":"2026-09-01T08:58:13.963Z","modelVersion":"gpt-5.6-terra+diagrams-2","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":{"methods":[{"id":"a021cfbf-192e-405f-a817-23200225ed9d:simplify_surd","label":"(a) Simplify the surd","steps":[{"type":"text","order":0,"title":"Find the square factor","content":"The useful square factor is $36$, because $108=36\\times3$ and $36$ is a square.","highlights":[{"text":"Express $\\sqrt{108}$ in the form $a\\sqrt{3}$, where $a$ is an integer.","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"\\sqrt{108}=\\sqrt{36\\times3}"},{"latex":"\\sqrt{108}=\\sqrt{36}\\times\\sqrt{3}"},{"latex":"\\sqrt{108}=6\\sqrt{3}"}]},"order":1,"title":"Take out the square root","description":"Write 108 as a square number multiplied by 3, then simplify the square root.\n\nB1: Correctly writes √108 = √36 × √3 = 6√3."}]},{"id":"99429fa5-af22-464c-83f8-e9e3caba686e:direct_expansion","label":"(b) Expand the square","steps":[{"type":"text","order":0,"title":"Rewrite the square","content":"A square means multiply the expression by itself. Keep the brackets so that every term is multiplied by every term:\n\n$$(2-\\sqrt{3})^2=(2-\\sqrt{3})(2-\\sqrt{3}).$$\n\nThis is the starting point for the expansion.","highlights":[{"text":"Express $(2 - \\sqrt{3})^2$ in the form $b + c\\sqrt{3}$","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"(2-\\sqrt{3})(2-\\sqrt{3})"},{"latex":"\\htmlData{anchor=four_term_expansion}{4-2\\sqrt{3}-2\\sqrt{3}+3}"}],"highlights":[{"color":"purple","style":"underline","anchor":"four_term_expansion"}]},"order":1,"title":"Multiply every pair","description":"Use the first bracket to multiply both terms in the second bracket. This complete four-term expansion earns the M1."},{"type":"text","order":2,"title":"Collect like terms","content":"Now collect the rational terms and the terms containing $\\sqrt{3}$ separately. The expansion in the previous step earns the **M1**. \n\nThe rational terms give $4+3=7$, earning one accuracy mark. The two surd terms are like terms, so their coefficients add: $-2\\sqrt{3}-2\\sqrt{3}=-4\\sqrt{3}$, earning the other accuracy mark.","highlights":[{"text":"Express $(2 - \\sqrt{3})^2$ in the form $b + c\\sqrt{3}$","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"\\htmlData{anchor=combined_terms}{7+(-4\\sqrt{3})}"},{"latex":"\\boxed{7-4\\sqrt{3}}"}],"highlights":[{"color":"teal","style":"underline","anchor":"combined_terms"}]},"order":3,"title":"State the required form","description":"Put the rational term first, followed by the multiple of $\\sqrt{3}$."}]}],"generatedAt":"2026-09-01T08:58:14.236Z","modelVersion":"gpt-5.6-terra+diagrams-2","explanationVersion":"v3"},"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"]}]

Indices and Surds

Question 6

Indices and Surds Questions

Indices and Surds

Question 6

Indices and Surds Questions