Simplify; $\frac{3^{n - 1} \times 27^{n + 1}}{81^{n}}$

Step 1: Analyze the question We need to simplify an expression with powers of 3. The trick is to write every number as a power of the same base, 3.

Step 2: Concept Laws of indices:

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

Step 3: Solution Write the bases as powers of 3:

• 27 = 3³, so 27^(n+1) = 3^(3(n+1)) = 3^(3n+3)
• 81 = 3⁴, so 81^n = 3^(4n)

Numerator: 3^(n−1) × 3^(3n+3) = 3^((n−1)+(3n+3)) = 3^(4n+2)

Divide by the denominator: 3^(4n+2) ÷ 3^(4n) = 3^(4n+2−4n) = 3² = 9

Step 4: Answer The expression simplifies to 9, which is option B.

Tip: When you see different bases like 3, 27 and 81, convert them all to the same base first (3, 9, 27, 81 are all powers of 3). Then the problem reduces to adding and subtracting powers. Also, when the answer is a constant like 9, the n terms must cancel, which is a good check on your working.