Surds & Indices • Topic 2 of 4

Fractional Indices

A fractional index is a compact way of writing a root: a^(m/n) = ⁿ√(a^m) = (ⁿ√a)^m. The denominator is the root, the numerator is the power, and you may apply them in either order — usually root first to keep numbers small. So 16^(3/4) = (⁴√16)^3 = 2^3 = 8, far easier than (16^3)^(1/4). A negative fractional index combines two rules: 27^(−2/3) = 1/27^(2/3) = 1/(∛27)^2 = 1/9. The CAT-smart move when faced with messy numbers is to write the base as a prime power first: 8^(2/3) = (2^3)^(2/3) = 2^2 = 4. The same laws of indices (add, subtract, multiply exponents) all carry over to fractions, so 2^(1/2) × 2^(1/3) = 2^(5/6). Treat the exponents as ordinary fractions and find a common denominator.

✅ Solved examples

1. Evaluate 16^(3/4).
16 = 2^4, so 16^(3/4) = 2^(4 × 3/4) = 2^3 = 8.
2. Simplify 8^(2/3) × 2^(−1).
8^(2/3) = (2^3)^(2/3) = 2^2 = 4. Then 4 × 2^(−1) = 4/2 = 2.
3. Evaluate 27^(−2/3).
27^(2/3) = (∛27)^2 = 3^2 = 9, so 27^(−2/3) = 1/9.
4. Simplify 2^(1/2) × 2^(1/3) ÷ 2^(1/6).
Exponent = 1/2 + 1/3 − 1/6 = 3/6 + 2/6 − 1/6 = 4/6 = 2/3, giving 2^(2/3) = ∛4.

✏️ Practice — try these, take hints as needed

1. Evaluate 32^(2/5).
Write 32 as 2^5.
2^(5 × 2/5).
Exponent becomes 2.
4
2. Evaluate 81^(−3/4).
81 = 3^4.
81^(3/4) = 3^3 = 27.
Take the reciprocal.
1/27
3. Simplify (64)^(1/2) × (64)^(1/3).
Add the exponents.
1/2 + 1/3 = 5/6.
64 = 2^6, so 2^(6 × 5/6).
32
4. Find x if 4^(x) = 8.
Common base 2: 2^(2x) = 2^3.
Equate exponents.
2x = 3.
x = 3/2
5. Evaluate (125)^(2/3) ÷ (25)^(1/2).
125 = 5^3, 25 = 5^2.
Top = 5^2 = 25, bottom = 5^1 = 5.
25 ÷ 5.
5

📝 Topic test — 8 questions

Auto-graded with full solutions; saved to your dashboard. Use the calculator and formula sheet (top-right) any time.

Loading questions…
← Laws of Indices Rationalization →