To evaluate whether the equation \( x \cdot 3^{-5} = \frac{81 \sqrt{4}}{3^{3-4} 3^{-6}} \) is true or false, we need to simplify both sides of the equation.
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Left Side: \( x \cdot 3^{-5} \)
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Right Side:
- First, simplify \( \sqrt{4} = 2 \).
- Then, the right side becomes: \[ \frac{81 \cdot 2}{3^{3-4} \cdot 3^{-6}} = \frac{162}{3^{-1} \cdot 3^{-6}} = \frac{162}{3^{-7}}. \]
- Using the property \( \frac{a}{b^{-n}} = a \cdot b^n \), we rewrite this as: \[ 162 \cdot 3^{7} = 162 \cdot 3^{7}. \]
Now we compare both sides:
- Left Side: \( x \cdot 3^{-5} \)
- Right Side: \( 162 \cdot 3^{7} \)
Since these two expressions are not equal (as \( 3^{-5} \) is not equal to \( 162 \cdot 3^{7} \)), the equation is False.
The correct response is: False. There is an error in calculation using the Quotient Property.
(Note: In reality, evaluating detailed calculations helps clarify the answer better, but the question fundamentally lies in equality evaluation based on simplified terms.)