Mathematics • Year 9 • Unit 1 • Lesson 19
Fractional Indices — Mixed Challenge
Pull together every fractional-index idea from Lesson 19 — and combine them with the integer-index laws (product, quotient, power-of-a-power) from L01-L13. Includes a "find the mistake" and an open-ended challenge that demands all the laws at once.
1. Mixed problems — choose the right rule
Each question uses a different combination of fractional indices and the index laws from Lessons 1-19. Decide which rule applies before you start writing. Show your working. 3 marks each
1.1 Evaluate $81^{1/2}$.
1.2 Evaluate $64^{2/3}$ using "root first, then power".
1.3 Write $\sqrt[4]{x^7}$ in index form, then simplify $\dfrac{\sqrt[4]{x^7}}{\sqrt{x}}$, leaving the answer in index form.
1.4 Simplify $\sqrt{x} \cdot \sqrt[3]{x}$, leaving the answer in index form with a single fractional index.
1.5 Evaluate $\sqrt{144 x^4}$, leaving your answer in index form.
1.6 Simplify $\dfrac{x^{2/3} \cdot x^3}{x^{1/2}}$, giving the answer with a single fractional index.
2. Find the mistake
Another student has tried to simplify $x^{1/2} \cdot x^{1/3}$. Their working is shown below. Exactly one line contains a mistake. Spot it, explain why it's wrong, then re-do the working correctly. 3 marks
Student's working — simplify $x^{1/2} \cdot x^{1/3}$:
Line 1: Product rule $\to$ add the indices: $\dfrac{1}{2} + \dfrac{1}{3}$.
Line 2: Add tops and bottoms: $\dfrac{1 + 1}{2 + 3} = \dfrac{2}{5}$.
Line 3: So $x^{1/2} \cdot x^{1/3} = x^{2/5}$.
Line 4: Check: $x^{2/5} = \sqrt[5]{x^2}$.
(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? Fractions add with a common denominator, not by adding tops and bottoms. Try $\dfrac{1}{2} + \dfrac{1}{3}$ with common denominator $6$.3. Open-ended challenge — the same target, two ways
This question has more than one valid answer — there are several different ways to land on the target. 4 marks
3.1 Find two different expressions using fractional indices that simplify to $\mathbf{x^2}$. The two expressions must:
(i) be different from each other;
(ii) not just be $x^2 \cdot x^0$ or $(x^2)^1$;
(iii) each use at least one fractional index;
(iv) include at least one example that uses the product rule and one that uses the power-of-a-power rule.
For each expression you find:
- Write it down.
- Show the working that confirms it equals $x^2$.
- Briefly say which index law(s) you used.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — $81^{1/2}$
$81^{1/2} = \sqrt{81} = \mathbf{9}$ (since $9^2 = 81$).
1.2 — $64^{2/3}$ (root first)
$\sqrt[3]{64} = 4$ (since $4^3 = 64$). Then $4^2 = \mathbf{16}$.
1.3 — $\sqrt[4]{x^7} = x^{7/4}$; then $\dfrac{\sqrt[4]{x^7}}{\sqrt{x}}$
$\dfrac{x^{7/4}}{x^{1/2}} = x^{7/4 - 1/2}$. Common denominator $4$: $\dfrac{7}{4} - \dfrac{2}{4} = \dfrac{5}{4}$. Answer: $\mathbf{x^{5/4}}$.
1.4 — $\sqrt{x} \cdot \sqrt[3]{x}$
$x^{1/2} \cdot x^{1/3}$. Common denominator $6$: $\dfrac{1}{2} = \dfrac{3}{6}$ and $\dfrac{1}{3} = \dfrac{2}{6}$. Sum: $\dfrac{3}{6} + \dfrac{2}{6} = \dfrac{5}{6}$. Answer: $\mathbf{x^{5/6}}$.
1.5 — $\sqrt{144 x^4}$
$\sqrt{144 x^4} = (144 x^4)^{1/2} = 144^{1/2} \cdot (x^4)^{1/2} = 12 \cdot x^{4 \times 1/2} = \mathbf{12 x^2}$ (power-of-a-product, then root and power-of-a-power).
1.6 — $\dfrac{x^{2/3} \cdot x^3}{x^{1/2}}$
Step 1 — product rule on numerator: $x^{2/3} \cdot x^3 = x^{2/3 + 3} = x^{2/3 + 9/3} = x^{11/3}$.
Step 2 — quotient rule with denominator: $\dfrac{x^{11/3}}{x^{1/2}} = x^{11/3 - 1/2}$.
Step 3 — common denominator $6$: $\dfrac{11}{3} - \dfrac{1}{2} = \dfrac{22}{6} - \dfrac{3}{6} = \dfrac{19}{6}$.
Answer: $\mathbf{x^{19/6}}$.
2 — Find the mistake
(a) The mistake is on Line 2.
(b) The student added the fractions by adding tops and bottoms ($\dfrac{1 + 1}{2 + 3} = \dfrac{2}{5}$), but fractions don't add that way — you need a common denominator first. The correct sum is $\dfrac{1}{2} + \dfrac{1}{3} = \dfrac{3}{6} + \dfrac{2}{6} = \dfrac{5}{6}$.
(c) Corrected working:
$x^{1/2} \cdot x^{1/3} = x^{1/2 + 1/3}$
$= x^{3/6 + 2/6}$ (common denominator $6$)
$= x^{5/6}$.
Final answer: $\mathbf{x^{5/6}}$.
This is exactly the trap flagged in the lesson's "Common Pitfalls" card.
3 — Open-ended (sample solution)
We need expressions that simplify to $x^2$ using fractional indices.
Expression 1 (product rule): $x^{3/2} \cdot x^{1/2}$.
Working: $\dfrac{3}{2} + \dfrac{1}{2} = \dfrac{4}{2} = 2$, so $x^{3/2} \cdot x^{1/2} = x^2$ ✓.
Rules used: product rule (add indices), then simplify $\dfrac{4}{2}$ to $2$.
Expression 2 (power-of-a-power): $(x^{1/2})^4$.
Working: $(x^{1/2})^4 = x^{1/2 \times 4} = x^{4/2} = x^2$ ✓.
Rule used: power-of-a-power (multiply indices), then simplify $\dfrac{4}{2}$.
Other valid examples: $x^{4/3} \cdot x^{2/3}$ (product), $(x^{2/3})^3$ (power-of-a-power), $\dfrac{x^{5/2}}{x^{1/2}}$ (quotient).
Marking: 2 marks per valid expression (one for the expression itself, one for shown working confirming it equals $x^2$ with the law named). Must include one product and one power-of-a-power example for full marks.