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

Video Solution

Solution Steps

00:00 Select the appropriate sign

00:04 We'll simplify each side using the formula for the multiplication of powers

00:10 We'll apply this formula to our exercise, each time for one multiplication

00:15 We'll maintain the base and add together the exponents

00:26 We'll apply the same formula in order to simplify the right side

00:38 Note that the side with the larger exponent is always greater

00:41 This is the solution

Step-by-Step Solution

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

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 $3^{-5}\times3^{17}\times3^{-7} > 3^2\times3^1\times3$ .