Simplify $(216^{{\frac{1}{3}}})^{-2}$

Analyze the question: We need to simplify (216^(1/3))^(-2). This involves a fractional index (cube root) and a negative index (reciprocal).

Concept:

• A power of 1/3 means the cube root: a^(1/3) = ∛a
• A negative power means the reciprocal: a^(-n) = 1/aⁿ
• Power of a power: (a^m)^n = a^(mn)

Solution:

Step 1: Find the cube root of 216. 6 × 6 × 6 = 216, so 216^(1/3) = 6

Step 2: Apply the power of -2. (6)^(-2) = 1/6²

Step 3: Evaluate. 1/6² = 1/36

Check using the index law: (216^(1/3))^(-2) = 216^(-2/3) = 1/(216^(2/3)) = 1/(6²) = 1/36 ✓

Answer: Option A, 1/36.

Tip: For expressions like a^(m/n), take the root first (the denominator of the fraction), then the power. Perfect cubes worth memorising for exams: 8, 27, 64, 125, 216 (that is 2³, 3³, 4³, 5³, 6³). A negative index simply flips the number, so don't let it scare you.