Mathematics • Year 10 • Unit 1 • Lesson 14
Surds and Operations — Mixed Challenge
Pull together every idea from Lesson 14: simplify surds by extracting perfect squares, combine like surds, multiply with the product rule, and rationalise denominators. Spot a plausible Year 10 mistake from the Misconceptions card, then design your own equivalent-surd-expression set.
1. Mixed problems — choose the right tool
Each question uses a different idea from Lesson 14. Decide which rule applies before writing. Show your working. 3 marks each
1.1 Simplify √200 and √98 to simplest surd form.
1.2 Simplify 4√7 + 3√7 and explain why this works but √3 + √5 does NOT simplify to a single surd.
1.3 Simplify √8 × √2 using the product rule. (The answer is a whole number.)
1.4 Rationalise 5 / √5 and simplify completely.
1.5 Simplify 2√12 + 3√3 by first reducing √12 to a like surd of √3.
1.6 Rationalise 4 / √8 and give your final answer in simplest form. (You will need to simplify the resulting surd at the end.)
2. Find the mistake
Another Year 10 student tried to evaluate a Pythagoras-style expression. Their working is shown below. Exactly one line contains a mistake the lesson's Misconceptions card warns about. Spot it, explain why, then re-do correctly. 3 marks
Student's working — evaluate √(9 + 16):
Line 1: √(9 + 16) = √9 + √16
Line 2: = 3 + 4
Line 3: = 7
(a) Which line contains the mistake?
(b) Explain in one or two sentences why that line is wrong.
(c) Write out the corrected working in full, including the corrected final answer.
Stuck? The lesson's Misconceptions card states clearly: the square root of a sum is NOT the sum of the square roots. You must add first, then take the root.3. Open-ended challenge — design three equivalent surd expressions
This question has many valid answers. Be creative but show every number. 4 marks
3.1 Design three different surd expressions that all simplify to the same final answer 4√3. Your three expressions must collectively include:
(i) one expression that uses the simplify a single surd technique (e.g. √48, since √48 = √(16 × 3) = 4√3);
(ii) one expression that uses the combine like surds technique (e.g. 7√3 − 3√3);
(iii) one expression that uses the rationalise a denominator technique (e.g. 12 / √3).
For each expression, show the simplification step-by-step.
Bonus: design a real-world geometry problem (e.g. the diagonal of a rectangle, or one side of a 30°-60°-90° triangle) whose answer is exactly 4√3 units, and state the problem and its answer.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — √200 and √98
√200 = √(100 × 2) = 10√2.
√98 = √(49 × 2) = 7√2.
1.2 — Like vs unlike surds
4√7 + 3√7 = (4 + 3)√7 = 7√7 — works because both terms are like surds (same surd part √7).
√3 + √5 cannot be simplified to a single surd because √3 and √5 are unlike surds (different numbers under the root). The Misconceptions card explicitly warns against writing √3 + √5 = √8.
1.3 — √8 × √2
Product rule: √8 × √2 = √(8 × 2) = √16 = 4.
1.4 — Rationalise 5 / √5
(5 / √5) × (√5 / √5) = 5√5 / 5 = √5.
1.5 — 2√12 + 3√3
Simplify √12 first: √12 = √(4 × 3) = 2√3.
So 2√12 = 2 × 2√3 = 4√3.
Combine: 4√3 + 3√3 = 7√3.
1.6 — Rationalise 4 / √8
(4 / √8) × (√8 / √8) = 4√8 / 8 = √8 / 2.
Simplify √8 = 2√2.
So √8 / 2 = (2√2) / 2 = √2.
2 — Find the mistake
(a) The mistake is on Line 1 (Lines 2 and 3 inherit it).
(b) The square root of a sum is NOT the sum of the square roots. You must add the numbers under the radical first, then take the root. The Misconceptions card calls this out using exactly this example.
(c) Corrected working:
√(9 + 16) = √25 = 5.
3 — Open-ended challenge (sample solution)
Target: 4√3.
(i) Simplify a single surd: √48 = √(16 × 3) = √16 × √3 = 4√3. ✓
(ii) Combine like surds: 7√3 − 3√3 = (7 − 3)√3 = 4√3. ✓
(iii) Rationalise a denominator: 12 / √3 = (12 / √3) × (√3 / √3) = 12√3 / 3 = 4√3. ✓
Bonus — real-world problem: A rhombus has diagonals 8 cm and 4√3 cm. Find the length of one diagonal.
Or: an equilateral triangle of side 8 cm has perpendicular height = (side × √3) / 2 = (8√3) / 2 = 4√3 cm.
Marking: 1 mark for a valid "simplify single surd" expression; 1 mark for a valid "combine like surds" expression; 1 mark for a valid "rationalise denominator" expression; 1 mark for a valid real-world geometry problem whose answer is 4√3. Any valid set scores full marks.