Is the equation 5^−5 ⋅ (3√5)^12 = 1/5 true or false? Enter 1 for true or 2 for false.

Answer 1

To determine whether the equation \( 5^{-5} \cdot (3\sqrt{5})^{12} = \frac{1}{5} \) is true or false, let's simplify the left-hand side.

1. Start with \( 5^{-5} \): \[ 5^{-5} = \frac{1}{5^5} \]

2. Next, simplify \( (3\sqrt{5})^{12} \): \[ (3\sqrt{5})^{12} = 3^{12} \cdot (\sqrt{5})^{12} \] Since \( \sqrt{5} = 5^{1/2} \), we have: \[ (\sqrt{5})^{12} = (5^{1/2})^{12} = 5^{12/2} = 5^6 \] So, \[ (3\sqrt{5})^{12} = 3^{12} \cdot 5^6 \]

3. Now combine \( 5^{-5} \cdot (3\sqrt{5})^{12} \): \[ 5^{-5} \cdot (3\sqrt{5})^{12} = \frac{1}{5^5} \cdot (3^{12} \cdot 5^{6}) = 3^{12} \cdot \frac{5^6}{5^5} = 3^{12} \cdot 5^{6-5} = 3^{12} \cdot 5^1 = 3^{12} \cdot 5 \]

4. Now we need to compare \( 3^{12} \cdot 5 \) with \( \frac{1}{5} \): \[ 3^{12} \cdot 5 \neq \frac{1}{5} \]

Since \( 3^{12} \) is a large positive number and multiplying it by 5 gives a value significantly greater than \( \frac{1}{5} \), we conclude that:

The given equation \( 5^{-5} \cdot (3\sqrt{5})^{12} = \frac{1}{5} \) is false.

Thus, the answer is \( \boxed{2} \).