(0.04)-1.5 = ?

A negative exponent means take the reciprocal of the base and make the exponent positive: a-n = 1/an. A fractional exponent p/q means take the q-th root of the base, then raise it to the power p: ap/q = (q-th root of a)p.

1. Convert the decimal base to a fraction: 0.04 = 4/100 = 1/25.

2. Apply the negative exponent rule to flip the base and make the exponent positive: (1/25)-1.5 = 251.5 = 253/2.

3. Here q = 2, so apply the fractional exponent rule as a square root: take the square root of 25 first, then cube it: 253/2 = (251/2)3 = (√25)3 = 53.

4. Evaluate the cube: 53 = 125.

Cross-check: evaluate (0.04)1.5 directly and invert it: (0.04)1.5 = 0.04 × √0.04 = 0.04 × 0.2 = 0.008, so (0.04)-1.5 = 1/0.008 = 125 — confirming the value 125.