is equal to

Concept: For any nonzero base a, a0 = 1 (zero-exponent law), and a-1 = 1/a (reciprocal law), so (a/b)-1 = b/a. Also, (a/b)n = an/bn (power of a quotient). This expression is a direct simplification using these three laws on the numerator and on each of the three terms in the denominator.

Application:

1. Numerator: (0.6)0 = 1 by the zero-exponent law, and (0.1)-1 = 1/0.1 = 10 by the reciprocal law. So the numerator is 1 - 10 = -9.

2. First denominator term: (3/23)-1 = (3/8)-1 = 8/3, using the reciprocal-of-a-fraction rule.

3. Second denominator term: (3/2)3 = 33/23 = 27/8, using the power-of-a-quotient rule.

4. Multiply the two terms: (8/3) x (27/8) = 27/3 = 9, since the 8s cancel.

5. Third denominator term: (-1/3)-1 = -3, again by the reciprocal law, carrying the negative sign through.

6. Denominator total: 9 + (-3) = 6.

7. Full expression: numerator / denominator = -9/6 = -3/2.

Cross-check: Using decimals, 8/3 approx 2.667 and 27/8 = 3.375, and their product is approx 9.0, matching Step 4; adding the third term gives 9 + (-3) = 6, so -9/6 = -1.5, which equals -3/2 - confirming the simplification independently.