Simplify $\frac {25^{\frac{2}{3}} \div  25^{\frac{1}{6}}} {(\frac{1}{5})^{\frac{7}{6}} \div (\frac{1}{5})^{\frac{1}{6}}}$

Analyze the question: We have to simplify a fraction in which both the numerator and denominator involve division of powers with the same base. The key tool is the law of indices: a^m ÷ a^n = a^(m−n).

Concept:

• a^m ÷ a^n = a^(m−n)
• 25 = 5², so 25^k = 5^(2k)
• (1/5)^k = 5^(−k)

Step 1: Numerator

25^(2/3) ÷ 25^(1/6) = 25^(2/3 − 1/6) = 25^(4/6 − 1/6) = 25^(3/6) = 25^(1/2) = 5

Step 2: Denominator

(1/5)^(7/6) ÷ (1/5)^(1/6) = (1/5)^(7/6 − 1/6) = (1/5)^(6/6) = (1/5)^1 = 1/5

Step 3: Divide

5 ÷ (1/5) = 5 × 5 = 25

Answer: The result is 25, which is option A.

Tip: When the bases are the same and you see division, subtract the powers first, because it keeps the fractions simple. Also, dividing by a fraction means multiplying by its reciprocal, so 5 ÷ 1/5 = 5 × 5.