Simplify $\frac{3^{-5n}}{9^{1-n}} \times 27^{n + 1}$

Analyze the question: We need to rewrite everything with base 3 and then use the laws of indices.

Concept:

• (a^m)^n = a^(mn)
• a^m × a^n = a^(m+n)
• a^m ÷ a^n = a^(m−n)

Solution:

Write each term as a power of 3:

• 9^(1−n) = (3²)^(1−n) = 3^(2−2n)
• 27^(n+1) = (3³)^(n+1) = 3^(3n+3)

Now substitute:

3^(−5n) ÷ 3^(2−2n) × 3^(3n+3)

Add and subtract the powers:

Exponent = −5n − (2 − 2n) + (3n + 3) = −5n − 2 + 2n + 3n + 3 = (−5n + 2n + 3n) + (−2 + 3) = 0n + 1 = 1

So the expression simplifies to 3¹ = 3.

Check with n = 0: 3⁰ / 9¹ × 27¹ = 1/9 × 27 = 3 ✓

Answer: The result is 3, which is option D.

Tip: In index questions with different bases like 3, 9 and 27, always convert everything to the same base first. Then simply combine the exponents. If the n terms cancel, as they did here, the answer is a constant. Testing n = 0 is a quick way to check your work.