Simplify 3 $^{n − 1}$ ×  $\frac{27^{n + 1}}{81^n}$

1. Analyze the question

We are asked to simplify the expression: $3^{n-1} \times \frac{27^{n+1}}{81^n}$

To solve this, we need to express all the bases as powers of a common base. Since 3, 27, and 81 are all powers of 3, we will convert everything to base 3.

2. Explain the concept

We use the Laws of Indices:

• $a^m \times a^n = a^{m+n}$
• $a^m \div a^n = a^{m-n}$
• $(a^m)^n = a^{m \times n}$

Conversion to base 3:

• $3 = 3^1$
• $27 = 3^3$
• $81 = 3^4$

3. Show the solution

Step 1: Substitute the base 3 equivalents into the expression. $3^{n-1} \times \frac{(3^3)^{n+1}}{(3^4)^n}$

Step 2: Simplify the powers using the power rule $(a^m)^n = a^{m \times n}$. $3^{n-1} \times \frac{3^{3(n+1)}}{3^{4n}}$ $3^{n-1} \times \frac{3^{3n+3}}{3^{4n}}$

Step 3: Combine the terms using the multiplication and division rules. First, multiply the terms in the numerator: $3^{(n-1) + (3n+3)} \div 3^{4n}$ $3^{n + 3n - 1 + 3} \div 3^{4n}$ $3^{4n + 2} \div 3^{4n}$

Next, perform the division: $3^{(4n + 2) - 4n}$ $3^{4n - 4n + 2}$ $3^2$

Step 4: Calculate the final value. $3^2 = 9$

4. Identify the answer

Comparing our result to the given options: A. $3^{2n}$ B. 9 C. $3n$ D. $3^{n+1}$

The correct option is B.

5. Tip

Whenever you see indices problems involving numbers like 9, 27, 81, 125, or 625, immediately check if they can be written as powers of a smaller prime number (like 2, 3, or 5). This makes the indices much easier to add or subtract!