The simplified form of (4−1 + 8−1) ÷ [1 ÷ ((3⁄2)−2)] is

Concept

For any non-zero number a, a negative exponent reverses the base: a−n = 1/(an).

Simplify powers and fractions first; division by a fraction is multiplication by its reciprocal.

Application

1. Convert the negative powers: 4−1 = 1⁄4, 8−1 = 1⁄8, and (3⁄2)−2 = (2⁄3)2 = 4⁄9.

2. Add inside the first parentheses: 1⁄4 + 1⁄8 = 2⁄8 + 1⁄8 = 3⁄8.

3. Simplify the divisor: 1 ÷ (4⁄9) = 9⁄4.

4. Divide by multiplying by the reciprocal: 3⁄8 ÷ 9⁄4 = 3⁄8 × 4⁄9 = 12⁄72 = 1⁄6.

Cross-check

Multiplying the result by the divisor gives 1⁄6 × 9⁄4 = 9⁄24 = 3⁄8, which matches the first-parentheses value.

Therefore, the simplified value is 1⁄6.