c{4\\sqrt{3}}{3}}} = 1 \\times \\htmlData{anchor=recip-val}{\\frac{3}{4\\sqrt{3}}}$"}],"highlights":[{"color":"amber","style":"box","anchor":"recip-val"}],"connections":[{"to":"recip-val","from":"den-val","color":"amber","label":"flip","style":"solid"}]},"order":2,"title":"Calculate the reciprocal","description":"The original expression is $1$ divided by our simplified denominator. Dividing by a fraction is the same as multiplying by its reciprocal (flipping it)."},{"type":"text","order":3,"title":"Final simplification","content":"To reach the required form, we simplify $\\frac{3}{4\\sqrt{3}}$. Since $3 = \\sqrt{3} \\times \\sqrt{3}$, we can cancel a factor of $\\sqrt{3}$ from the top and bottom:\n\n$$\\frac{3}{4\\sqrt{3}} = \\frac{\\sqrt{3} \\times \\sqrt{3}}{4\\sqrt{3}} = \\frac{\\sqrt{3}}{4}$$\n\nWriting the expression in this final form secures the last two marks M1 A1.","calculator":[{"gdc":{"expression":"1 / (1/√(3) + √(3))","instructions":"Use the fraction template to enter the complex fraction."},"ti84":{"expression":"1 / (1/√(3) + √(3))","instructions":"Enter the original calculation and the final result separately to compare their decimal values."},"desmos":{"expression":"1 / (1/sqrt(3) + sqrt(3))","instructions":"Type the expression to see the decimal result, then compare it to sqrt(3)/4."},"description":"You can verify that the original expression and the final answer are numerically identical using your calculator."}]}]}],"generatedAt":"2026-06-08T02:14:04.899Z","modelVersion":"gemini-3-flash-preview","explanationVersion":"v2"},"firstAttemptFullMarks":{"percent":47},"walkthroughVideo":null},{"id":"23b126e3-2362-4c57-bed2-652327aa332b","revisionId":"2e3e870e-9fef-48a9-86b3-3be0e534f239","questionNumber":null,"questionType":"LA","specification":"","difficulty":1,"difficultyBand":"medium","totalMarks":3,"sourceTitle":null,"calculatorUse":"non_calc","isCalculation":true,"parts":[{"id":"69cf98f8-3aa3-4af9-a833-44b6868cd286","label":"a","content":"Simplify $\\frac{5 + \\sqrt{10}}{\\sqrt{5}}$","markscheme":"- Multiplies numerator and denominator by $\\sqrt{5}$: $\\frac{5\\sqrt{5} + \\sqrt{50}}{5}$ M1\n- Simplifies $\\sqrt{50}$ to $5\\sqrt{2}$: $\\frac{5\\sqrt{5} + 5\\sqrt{2}}{5}$ M1\n- Final answer $\\sqrt{5} + \\sqrt{2}$ A1","marks":3,"order":0,"markingRubric":{"totalMarks":3,"points":[{"id":"p1","marks":1,"description":"Multiplies both the numerator and the denominator by $\\sqrt{5}$.","code":"M"},{"id":"p2","marks":1,"description":"Simplifies the surd in the numerator (e.g., $\\sqrt{50} = 5\\sqrt{2}$).","code":"M"},{"id":"p3","marks":1,"description":"Provides the final simplified expression $\\sqrt{5} + \\sqrt{2}$.","code":"A"}],"finalAnswers":["\\sqrt{5} + \\sqrt{2}"],"generatedAt":"2026-05-06T14:02:05.450Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"rationalise_denominator","label":"Rationalise the denominator","steps":[{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\frac{5+\\sqrt{10}}{\\sqrt{5}}\\times\\frac{\\htmlData{anchor=top_multiplier}{\\color{@sky}{\\sqrt{5}}}}{\\htmlData{anchor=bottom_multiplier}{\\color{@amber}{\\sqrt{5}}}}$"},{"latex":"$$\\frac{\\htmlData{anchor=new_numerator}{\\color{@sky}{5\\sqrt{5}+\\sqrt{50}}}}{\\htmlData{anchor=new_denominator}{\\color{@amber}{5}}}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"top_multiplier"},{"color":"amber","style":"text-color","anchor":"bottom_multiplier"},{"color":"sky","style":"text-color","anchor":"new_numerator"},{"color":"amber","style":"text-color","anchor":"new_denominator"}],"connections":[{"to":"new_numerator","from":"top_multiplier","color":"sky","label":"multiply numerator"}]},"order":1,"title":"Multiply top and bottom","description":"Rationalising the denominator means rewriting the fraction so there is no surd on the bottom. Multiplying by square root of 5 over square root of 5 is multiplying by 1, so the fraction keeps its value. This earns the first method mark, M1."},{"type":"hyper_latex",

Surds

Question 1

Surds Questions

Surds

Question 1

Surds Questions