Mathematics • Year 10 • Unit 1 • Lesson 13
Fractional Indices — Skill Drill
Build fluency with the two PATHS rules from Lesson 13: a^(1/n) = ⁿ√a and a^(m/n) = (ⁿ√a)^m. Denominator gives the root, numerator gives the power. One step at a time, from a fully worked example through guided practice to independent problems.
1. I do — fully worked example
Read every step. Each one has a short reason on the right so you can see why, not just what.
Problem. Evaluate 16^(3/4) without a calculator.
Step 1 — Identify the rule.
a^(m/n) = (ⁿ√a)^m → denominator (n) gives the root, numerator (m) gives the power.
Reason: from the Fractional Index Rules box in Lesson 13.
Step 2 — Pull out the root first (Method 1, usually easier).
Denominator = 4 → take the 4th root of 16.
⁴√16 = 2 because 2⁴ = 16.
Reason: finding the root first keeps the numbers small. (Method 2 — power first, then root — gives the same answer but harder arithmetic.)
Step 3 — Apply the power.
Numerator = 3 → raise the root to the 3rd power.
2³ = 8.
Reason: m in the numerator is the power that follows the root.
Step 4 — Write the answer.
Answer: 16^(3/4) = 8.
2. We do — fill in the missing steps
Same structure as Section 1, but with the working faded. Fill in each blank line. 4 marks
Problem. Evaluate 64^(2/3) without a calculator.
Step 1 — Identify the rule: a^(m/n) = (__________)^m. The denominator gives the __________ and the numerator gives the __________.
Step 2 — Pull out the root. Here n = __________, so we want the __________-th root of 64.
³√64 = ________ because ________³ = 64.
Step 3 — Apply the power. Numerator m = __________, so raise to the __________-th power.
(________)² = __________
Step 4 — Write the final answer:
64^(2/3) = __________
3. You do — independent practice
Show your working in the space under each problem. The first four are foundation. The middle two are standard. The last two are extension.
Foundation — unit fractions a^(1/n)
3.1 Evaluate 25^(1/2). 1 mark
3.2 Evaluate 27^(1/3). 1 mark
3.3 Evaluate 32^(1/5). 1 mark
3.4 Evaluate 16^(1/4). 1 mark
Standard — general fractions a^(m/n)
3.5 Evaluate 81^(3/4). Show both the root and the power step. 2 marks
3.6 Evaluate 32^(3/5). Show both the root and the power step. 2 marks
Extension — push your thinking
3.7 Evaluate 125^(−1/3). 3 marks
3.8 Write each radical using a fractional index: (a) ³√(y²) (b) ⁵√(x⁷) (c) √(t³). 3 marks
How did this worksheet feel?
What I'll revisit before next class:
Section 2 — We do (faded 64^(2/3))
Step 1: a^(m/n) = (ⁿ√a)^m. Denominator gives the root; numerator gives the power.
Step 2: n = 3, so take the 3rd (cube) root. ³√64 = 4 because 4³ = 64.
Step 3: m = 2, raise to the 2nd power. (4)² = 16.
Step 4: 64^(2/3) = 16.
3.1 — 25^(1/2)
25^(1/2) = √25 = 5 because 5² = 25.
3.2 — 27^(1/3)
27^(1/3) = ³√27 = 3 because 3³ = 27.
3.3 — 32^(1/5)
32^(1/5) = ⁵√32 = 2 because 2⁵ = 32.
3.4 — 16^(1/4)
16^(1/4) = ⁴√16 = 2 because 2⁴ = 16.
3.5 — 81^(3/4)
Root: ⁴√81 = 3 (since 3⁴ = 81).
Power: 3³ = 27.
3.6 — 32^(3/5)
Root: ⁵√32 = 2 (since 2⁵ = 32).
Power: 2³ = 8.
3.7 — 125^(−1/3)
Negative index → take the reciprocal: 125^(−1/3) = 1 / 125^(1/3) = 1 / ³√125.
³√125 = 5 (since 5³ = 125).
Answer: 1/5.
3.8 — Radicals to fractional indices
(a) ³√(y²) = y^(2/3) (denominator 3 from the cube root, numerator 2 from the power).
(b) ⁵√(x⁷) = x^(7/5).
(c) √(t³) = t^(3/2).