Find the value of the following expression: 1/125(-2/3) + 1/625(-3/4) +…

Concept: To simplify a term with a negative fractional exponent, first rewrite the base as a power of a smaller prime, then apply the law (xm)n = x(mn), and use x(-n) = 1/xn so that 1/x(-n) = xn.

Step-by-step:

1. Rewrite each base as a power of a prime: 125 = 53, 625 = 54, and 729 = 36.

2. Apply (xm)n = x(mn) to each denominator: 125(-2/3) = (53)(-2/3) = 5-2, 625(-3/4) = (54)(-3/4) = 5-3, and 729(-3/6) = (36)(-3/6) = 3-3.

3. Use 1/x-n = xn on each term: 1/5-2 = 52 = 25, 1/5-3 = 53 = 125, and 1/3-3 = 33 = 27.

4. Add the three simplified terms: 25 + 125 + 27 = 177.

Cross-check:

Since 125(2/3) = (125(1/3))2 = 52 = 25, it follows that 125(-2/3) = 1/25, confirming the reciprocal step; the same check applies to the other two terms.

Answer: Therefore, the value of the expression is 177.