Simplify 125$^{−\frac{1}{3}}$ × 49$^{−\frac{1}{2}}$ × 10$^0$

Analyze the question: We need to simplify a product of three terms with negative and zero indices.

Concept:

• A negative index means a reciprocal: a^(−n) = 1/aⁿ
• A fractional index means a root: a^(1/n) = ⁿ√a
• Any non-zero number to the power 0 equals 1

Solution:

1. 125^(−1/3) = 1 / 125^(1/3) = 1 / ∛125 = 1/5, since 5³ = 125
2. 49^(−1/2) = 1 / 49^(1/2) = 1 / √49 = 1/7
3. 10⁰ = 1

Multiply them:

(1/5) × (1/7) × 1 = 1/35

Tip: Handle each term separately: first deal with the negative sign by taking the reciprocal, then take the root, and remember that anything to the power 0 is 1. Don't confuse 1/35 with 35, which is what you'd get if you ignored the negative signs.