Problem 22 - College Algebra Essentials

Question

Evaluate each exponential expression. $$\frac{3^{4}}{3^{7}}$$

Step-by-Step Solution

Answer

The evaluation of \(\frac{3^{4}}{3^{7}}\) gives \(\frac{1}{27}\).

Step 1: Understand the Problem

We are given two exponential terms \(3^{4}\) and \(3^{7}\) with the same base 3. We have to divide \(3^{4}\) by \(3^{7}\).

Step 2: Apply Laws of Exponents

When dividing terms with the same base, we subtract the exponents. Therefore, we rewrite the problem as \(3^{4-7}\).

Step 3: Evaluate the Expression

Subtracting the exponents gives \(3^{-3}\). This could also be written as \(\frac{1}{3^{3}}\).

Step 4: Solve \(\frac{1}{3^{3}}\)

Finally, once we solve \(\frac{1}{3^{3}}\), we get \(\frac{1}{27}\) as the result.