Mathematics • Year 10 • Unit 1 • Lesson 13
Fractional Indices — Mixed Challenge
Pull together every idea from Lesson 13: a^(1/n) = ⁿ√a, a^(m/n) = (ⁿ√a)^m, negative indices via reciprocals, and combining fractional indices with the full set of index laws. Spot a plausible Year 10 mistake, then design your own pair of equivalent expressions.
1. Mixed problems — choose the right rule
Each question uses a different idea from Lesson 13. Decide which rule applies before writing. Show your working. 3 marks each
1.1 Evaluate: (a) 16^(1/4), (b) 27^(2/3), (c) 32^(3/5).
1.2 Write each in fractional-index form: (a) ⁵√(x²), (b) ⁷√(y³), and write y^(3/7) in radical form to double-check (b).
1.3 Evaluate 16^(−3/4) using the negative-index rule (reciprocal first).
1.4 Simplify x^(5/6) ÷ x^(1/3), leaving the answer with one fractional index.
1.5 Evaluate (27/8)^(2/3) by splitting the fraction (a^n / b^n).
1.6 Verify the lesson's claim that the two methods for a^(m/n) agree, using 8^(2/3): compute both (³√8)² and ³√(8²) and check they match.
2. Find the mistake
Another Year 10 student tried to evaluate 16^(3/4). 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 16^(3/4):
Line 1: 16^(3/4) means 16 times (3/4)
Line 2: = 16 × (3/4)
Line 3: = 12
(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 says: a fractional index is NOT multiplication and is NOT division. It means "n-th root then m-th power".3. Open-ended challenge — design three equivalent expressions
This question has many valid answers. Be creative but show every number. 4 marks
3.1 Design three different expressions that all simplify to the same whole number between 5 and 50. They must collectively use:
(i) one expression of the form a^(m/n) with m ≥ 2 and n ≥ 2 (e.g. 64^(2/3));
(ii) one expression of the form a^(−m/n) (i.e. a negative fractional index);
(iii) one expression that combines a fractional index with another index law (e.g. x^(1/2) × x^(3/2), or (a^(1/3))⁶).
For each expression, show the evaluation step-by-step and state your common final answer.
Bonus: the variable expression in (iii) should evaluate to the same numerical answer when x is chosen to be a specific positive integer that you state.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Three evaluations
(a) 16^(1/4) = ⁴√16 = 2.
(b) 27^(2/3) = (³√27)² = 3² = 9.
(c) 32^(3/5) = (⁵√32)³ = 2³ = 8.
1.2 — Radical ↔ fractional index
(a) ⁵√(x²) = x^(2/5).
(b) ⁷√(y³) = y^(3/7).
Check: y^(3/7) written back as a radical → ⁷√(y³) ✓.
1.3 — 16^(−3/4)
Reciprocal first: 16^(−3/4) = 1 / 16^(3/4).
16^(3/4) = (⁴√16)³ = 2³ = 8.
Answer: 1/8.
1.4 — x^(5/6) ÷ x^(1/3)
Division law (subtract indices): 5/6 − 1/3 = 5/6 − 2/6 = 3/6 = 1/2.
Answer: x^(1/2) (= √x).
1.5 — (27/8)^(2/3)
(27/8)^(2/3) = 27^(2/3) / 8^(2/3) = (³√27)² / (³√8)² = 3² / 2² = 9 / 4.
Answer: 9/4.
1.6 — Verify the two methods for 8^(2/3)
Method 1: (³√8)² = 2² = 4.
Method 2: ³√(8²) = ³√64 = 4.
Both methods give 4 — the two equivalent definitions in the lesson are confirmed.
2 — Find the mistake
(a) The mistake is on Line 1 (and the rest of the working inherits it).
(b) A fractional index is not multiplication. The exponent (3/4) means "take the 4th root, then cube" — denominator gives the root, numerator gives the power. The student has confused index notation with multiplying by the fraction.
(c) Corrected working:
16^(3/4) = (⁴√16)³
= 2³
= 8.
This is the exact misconception called out in Lesson 13's "Common Mistakes to Avoid" card.
3 — Open-ended challenge (sample solution)
Target value: 8. Three expressions that all equal 8:
(i) a^(m/n) form: 16^(3/4) = (⁴√16)³ = 2³ = 8. ✓
(ii) Negative fractional index: (1/64)^(−1/2) = 64^(1/2) = √64 = 8. ✓
(iii) Combining with another index law: x^(1/2) × x^(3/2). Adding indices: 1/2 + 3/2 = 4/2 = 2. So the expression = x². Pick x = √8: then x² = 8. (Or pick x = 2√2, since (2√2)² = 4 × 2 = 8.) ✓
Common final answer: 8.
Marking: 1 mark for a valid a^(m/n) expression; 1 mark for a valid negative-fractional-index expression; 1 mark for combining a fractional index with another index law and a worked x-value; 1 mark for all three evaluating to the same target value in [5, 50]. Any valid set meeting the constraints scores full marks.