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
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).