`{:("Column A"," ","Column B"),((3cdot 3cdot 3cdot3)/(9 cdot 9 cdot 9 cdot 9),,((1)/(3))^4):}`

Options

A. If column A is larger
B. If column B is larger
C. If the columns are equal
D. If there is not enough information to decide

Answer

To solve the given problem, we need to evaluate both Column A and Column B separately and then compare their values.

**Step 1: Evaluate Column A**

Column A is given as:
\[ \frac{3 \cdot 3 \cdot 3 \cdot 3}{9 \cdot 9 \cdot 9 \cdot 9} \]

First, we can simplify the expression in the numerator and the denominator.

The numerator can be simplified as:
\[ 3 \cdot 3 \cdot 3 \cdot 3 = 3^4 \]

The denominator can be simplified as:
\[ 9 \cdot 9 \cdot 9 \cdot 9 = 9^4 \]

Now, we know that \(9\) can be expressed as \(3^2\), so:
\[ 9^4 = (3^2)^4 = 3^{2 \cdot 4} = 3^8 \]

Now we can rewrite Column A:
\[ \frac{3^4}{3^8} \]

Using the property of exponents \(\frac{a^m}{a^n} = a^{m-n}\), we have:
\[ 3^{4-8} = 3^{-4} \]

Now, we can express \(3^{-4}\) as:
\[ \frac{1}{3^4} \]

Calculating \(3^4\):
\[ 3^4 = 81 \]

Thus, Column A simplifies to:
\[ \frac{1}{81} \]

**Step 2: Evaluate Column B**

Column B is given as:
\[ \left(\frac{1}{3}\right)^4 \]

Calculating this gives:
\[ \left(\frac{1}{3}\right)^4 = \frac{1^4}{3^4} = \frac{1}{3^4} \]

Again, we know that \(3^4 = 81\), so:
\[ \frac{1}{3^4} = \frac{1}{81} \]

**Step 3: Compare Column A and Column B**

Now we have:
- Column A = \(\frac{1}{81}\)
- Column B = \(\frac{1}{81}\)

Since both columns are equal, we conclude that:
\[ \text{Column A} = \text{Column B} \]

**Final Answer:**
Both columns are equal.

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