To determine if the equation \( 5^{-5} \cdot (3\sqrt{5})^{12} = 15 \) is true, let's simplify the left-hand side.
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We start with \( 5^{-5} \): \[ 5^{-5} = \frac{1}{5^5} \]
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Now simplify \( (3\sqrt{5})^{12} \): \[ \sqrt{5} = 5^{1/2} \implies 3\sqrt{5} = 3 \cdot 5^{1/2} \] Therefore, \[ (3\sqrt{5})^{12} = (3 \cdot 5^{1/2})^{12} = 3^{12} \cdot (5^{1/2})^{12} = 3^{12} \cdot 5^6 \]
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Combining this with \( 5^{-5} \): \[ 5^{-5} \cdot (3\sqrt{5})^{12} = \frac{1}{5^5} \cdot (3^{12} \cdot 5^6) = 3^{12} \cdot \frac{5^6}{5^5} = 3^{12} \cdot 5^{6-5} = 3^{12} \cdot 5^1 = 3^{12} \cdot 5 \]
Now we need to check if \( 3^{12} \cdot 5 = 15 \).
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Calculate \( 3^{12} \): \( 3^{12} = 531441 \)
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Now multiply: \[ 3^{12} \cdot 5 = 531441 \cdot 5 = 2657205 \]
This means \( 3^{12} \cdot 5 \) is much greater than 15. Thus, we can conclude that:
\[ 5^{-5} \cdot (3\sqrt{5})^{12} \neq 15 \]
Therefore, the equation \( 5^{-5} \cdot (3\sqrt{5})^{12} = 15 \) is false.
The answer is 2 for false.
2 answers