Compute the exact value of the decimal power: (0.04)^3. Present the place-value reasoning clearly.

Introduction / Context:Raising a decimal to a power can be done cleanly by treating the number as a fraction with a power of 10 in the denominator. This avoids confusion and keeps track of zeros precisely.

Given Data / Assumptions:

• Compute (0.04)^3.
• 0.04 = 4/100 = 4 × 10^-2.

Concept / Approach:(a × 10^-b)^3 = a^3 × 10^-3b. Here a = 4 and b = 2. Alternatively, use fraction arithmetic: (4/100)^3 = 64/1,000,000.

Step-by-Step Solution:(0.04)^3 = (4/100)^3 = 4^3 / 100^3.4^3 = 64 and 100^3 = 1,000,000.Thus (0.04)^3 = 64 / 1,000,000 = 0.000064.

Verification / Alternative check:Using powers of ten: 0.04 = 4 × 10^-2 ⇒ (0.04)^3 = 4^3 × 10^-6 = 64 × 10^-6 = 0.000064.

Why Other Options Are Wrong:

• 0.064, 0.0064, 0.00064: Each is off by one or more decimal places.
• 0.0000064: One extra zero, indicates miscounting place values.

Common Pitfalls:Losing track of zeros when cubing denominators; mixing up squaring and cubing of 100.

Final Answer:0.000064