Final answer:
The simplified form of (4⁻²)⁵ is 2⁻¹⁰. To find this, we recognize that 4 is 2², and when we raise a power to a power, the exponents are multiplied. Thus, the expression simplifies to 2⁻¹⁰.
Step-by-step explanation:
To simplify the expression (4⁻²)⁵, we first need to understand how integer powers work. A positive integer power, like 4³, indicates repeated multiplication: 4 * 4 * 4. Similarly, a negative power like 4⁻² represents repeated multiplication in the denominator, which is equivalent to 1/(4 * 4) or 1/16.
Next, when we raise a power to a power, we multiply the exponents. For (4⁻²)⁵, we multiply the exponent -2 by 5, giving us 4⁻¹⁰. However, since 4 is 2², we can rewrite 4⁻¹⁰ as (2²)⁻¹⁰, which simplifies to 2⁻¹⁰, because (2²)⁻¹⁰ is the same as 2⁻⁴, or 2 to the power of -20.
Therefore the simplified form of (4⁻²)⁵ is 2⁻¹⁰, which corresponds to option A. 2⁻¹⁰.