(0.00032)0.6 is:

The power-of-a-power rule states that (am)n = amn: when a number is already written as a power of a simpler base, a fractional index can be applied directly to that base's own exponent instead of separately computing a root and a power.

1. Write the index 0.6 as the fraction 3/5.

2. Express 0.00032 as a power of 0.2: 0.2 × 0.2 × 0.2 × 0.2 × 0.2 = 0.00032, so 0.00032 = 0.25.

3. Apply the power-of-a-power rule: (0.25)3/5 = 0.25 × 3/5 = 0.23.

4. Compute 0.23 = 0.2 × 0.2 × 0.2 = 0.008.

As an independent check, reverse the computation: the cube root of 0.008 is 0.2 (since 0.23 = 0.008), and raising that 0.2 to the fifth power returns 0.25 = 0.00032 — confirming the original expression was correctly reduced.