f98072b149","label":"a","content":"Simplify $5c^2d^3 \\times 2d$","markscheme":"- $10c^2d^4$ B1","marks":1,"order":0,"markingRubric":{"totalMarks":1,"points":[{"id":"p1","marks":1,"description":"$$10c^2d^4$","code":"B"}],"finalAnswers":["10c^2d^4"],"generatedAt":"2026-05-06T01:49:23.623Z","modelVersion":"gemini-3-flash-preview"},"explanation":{"methods":[{"id":"multiply_and_add_indices","label":"Multiply coefficients and add indices","steps":[{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\htmlData{anchor=original}{5c^2d^3\\times 2d}$"},{"latex":"$$\\htmlData{anchor=working}{(5\\times 2)c^2d^{3+1}}$"},{"latex":"$$\\htmlData{anchor=answer}{10c^2d^4}$"}],"connections":[{"to":"working","from":"original","color":"sky"},{"to":"answer","from":"working","color":"green"}]},"order":1,"title":"Multiply and combine","description":"Multiply the numerical coefficients. For the repeated letter d, treat the final d as having power 1 and add the powers. The factor c squared is unchanged. Reaching 10c²d⁴ earns B1."}]}],"generatedAt":"2026-07-19T07:37:16.274Z","modelVersion":"gpt-5.6-sol","explanationVersion":"v3"},"options":[]},{"id":"b1030bd6-4b80-4037-b5be-28c8b4460189","label":"b","content":"Write $64 \\times 4^5$ as a power of 4","markscheme":"- For $4^3$ or $4^{3+5}$ M1\n- $4^8$ A1","marks":2,"order":1,"markingRubric":{"totalMarks":2,"points":[{"id":"p1","marks":1,"description":"Identifies 64 as $4^3$.","code":"M"},{"id":"p2","marks":1,"description":"$$4^8$","code":"A"}],"finalAnswers":["4^8"],"generatedAt":"2026-05-06T01:49:23.623Z","modelVersion":"gemini-3-flash-preview"},"explanation":{"methods":[{"id":"index_law","label":"Rewrite then add powers","steps":[{"type":"text","order":0,"title":"Rewrite sixty-four","content":"Write $64$ using base $4$:\n\n$$64=4^3$$\n\nThis earns **M1**. Using the same base is essential before applying an index law.","highlights":[{"text":"as a power of 4","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$64\\times 4^5=\\htmlData{anchor=same_base}{\\color{@green}{4^3\\times 4^5}}$"},{"latex":"$$4^3\\times 4^5=\\htmlData{anchor=added_index}{\\color{@green}{4^{3+5}}}$"},{"latex":"$$4^{3+5}=\\boxed{4^8}$"}],"highlights":[{"color":"green","style":"text-color","anchor":"same_base"},{"color":"green","style":"text-color-with-background","anchor":"added_index"}],"connections":[{"to":"added_index","from":"same_base","color":"green","style":"solid"}]},"order":1,"title":"Add the powers","description":"Use the product law: keep the common base and add the indices. The rewrite 64 = 4³ in the previous step earns M1; simplifying to 4⁸ earns A1."}]}],"generatedAt":"2026-07-19T15:20:02.538Z","modelVersion":"gpt-5.6-sol","explanationVersion":"v3"},"options":[]},{"id":"cfe8edb0-b666-4163-bdb3-c19c8b53b085","label":"c","content":"Simplify $p^3 \\times (p^5)^2$","markscheme":"- For $p^{10}$ or $p^{3+10}$ M1\n- $p^{13}$ A1","marks":2,"order":2,"markingRubric":{"totalMarks":2,"points":[{"id":"p1","marks":1,"description":"Simplifies $(p^5)^2$ to $p^{10}$.","code":"M"},{"id":"p2","marks":1,"description":"$$p^{13}$","code":"A"}],"finalAnswers":["p^{13}"],"generatedAt":"2026-05-06T01:49:23.623Z","modelVersion":"gemini-3-flash-preview"},"explanation":{"methods":[{"id":"index_laws","label":"Apply the index laws","steps":[{"type":"text","order":0,"title":"Choose the index laws","content":"Use the power-of-a-power law first: $(p^a)^b=p^{ab}$. Then use the product law for the same base: $p^a\\times p^b=p^{a+b}$.","highlights":[{"text":"Simplify $p^3 \\times (p^5)^2$","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$p^3\\times (p^{\\htmlData{anchor=five}{\\color{@sky}{5}}})^{\\htmlData{anchor=two}{\\color{@amber}{2}}}$"},{"latex":"$$\\htmlData{anchor=calculation}{\\color{@purple}{5\\times 2=10}}$"},{"latex":"$$p^3\\times p^{\\htmlData{anchor=ten}{\\color{@green}{10}}}$"}],"highlights":[{"color":"sky","style":"text-color-with-background","anchor":"five"},{"color":"amber","style":"text-c

Indices Questions

Indices Questions

Indices Questions

Indices Questions