Compare Base-3 Powers: Evaluating 3^(-5)×3^17×3^(-7) ⬜ 3^2×3^1×3

To solve this problem, we'll follow these steps:

• Step 1: Simplify the left-hand side expression.

• Step 2: Simplify the right-hand side expression.

• Step 3: Compare the results.

Let's work through each step:

Step 1: Simplifying the left-hand side expression:

The expression is $3^{-5} \times 3^{17} \times 3^{-7}$.

Use the rule of exponents for multiplication: $a^m \times a^n = a^{m+n}$.

Add the exponents: $-5 + 17 - 7 = 5$.

So, $3^{-5} \times 3^{17} \times 3^{-7} = 3^5$.

Step 2: Simplifying the right-hand side expression:

The expression is $3^2 \times 3^1 \times 3$.

This can be rewritten as $3^2 \times 3^1 \times 3^1$, since $3 = 3^1$.

Add the exponents: $2 + 1 + 1 = 4$.

So, $3^2 \times 3^1 \times 3 = 3^4$.

Step 3: Compare $3^5$ and $3^4$.

Since the base 3 is the same, compare the exponents: $5$ and $4$.

Since $5 > 4$, it follows that $3^5 > 3^4$.

Therefore, the inequality sign between the two expressions is $>$.

Thus, the correct answer to the initial problem is $3^{-5}\times3^{17}\times3^{-7} > 3^2\times3^1\times3$.