Find the value of 1/125(-2/3) + 1/625(-3/4) + 1/729(-3/6)

Concept: For a power of a power, (xm)n = xmn; and a negative exponent inverts the base, x-n = 1/xn. So 1/x-n = xn — a term with a negative fractional exponent in the denominator becomes the SAME base raised to the corresponding positive exponent once it is inverted.

Applying this:

1. Express each base as a power of a common base: 125 = 53, 625 = 54, 729 = 36.

2. Substitute so the three terms become 1/(53)(-2/3) + 1/(54)(-3/4) + 1/(36)(-3/6).

3. Apply (xm)n = xmn to each: (53)(-2/3) = 5-2, (54)(-3/4) = 5-3, (36)(-3/6) = 3-3.

4. Invert each negative power using 1/x-n = xn: 1/5-2 = 52 = 25, 1/5-3 = 53 = 125, 1/3-3 = 33 = 27.

5. Add the three whole numbers: 25 + 125 + 27 = 177.

Cross-check: Verifying independently via roots: 1252/3 = (cube root of 125)2 = 52 = 25; 6253/4 = (fourth root of 625)3 = 53 = 125; 7291/2 = square root of 729 = 27 — all three match, confirming the total.

Answer: 177.