To simplify the expression \( 4a^3b \cdot (3ab)^{-2} \), follow these steps:
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Simplify \( (3ab)^{-2} \). Recall that \( x^{-n} = \frac{1}{x^n} \): \[ (3ab)^{-2} = \frac{1}{(3ab)^2} \]
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Calculate \( (3ab)^2 \): \[ (3ab)^2 = 3^2 \cdot a^2 \cdot b^2 = 9a^2b^2 \] Thus, \[ (3ab)^{-2} = \frac{1}{9a^2b^2} \]
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Substitute back into the expression: \[ 4a^3b \cdot (3ab)^{-2} = 4a^3b \cdot \frac{1}{9a^2b^2} \]
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Multiply the terms: \[ = \frac{4a^3b}{9a^2b^2} \]
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Simplify the expression by cancelling terms:
- \( \frac{a^3}{a^2} = a^{3-2} = a^1 = a \)
- \( \frac{b}{b^2} = b^{1-2} = b^{-1} = \frac{1}{b} \)
Thus, \[ = \frac{4a}{9b} \]
So, the simplified expression is: \[ \frac{4a}{9b} \]