To simplify the expression \((4a^3b)(3ab)^{-2}\), follow these steps:
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Simplify \( (3ab)^{-2} \): The negative exponent indicates a reciprocal, so we can rewrite it as: \[ (3ab)^{-2} = \frac{1}{(3ab)^2} \] Now we square each part: \[ (3ab)^2 = 3^2 \cdot a^2 \cdot b^2 = 9a^2b^2 \] Thus, \[ (3ab)^{-2} = \frac{1}{9a^2b^2} \]
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Multiply the two parts: Now, we can multiply \( (4a^3b) \) by \( \frac{1}{9a^2b^2} \): \[ (4a^3b) \cdot \frac{1}{9a^2b^2} = \frac{4a^3b}{9a^2b^2} \]
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Simplify the fraction: In the numerator, we have \( 4a^3b \), and in the denominator, we have \( 9a^2b^2 \). We can simplify:
- For the \( a \) terms: \( \frac{a^3}{a^2} = a^{3-2} = a^1 = a \)
- For the \( b \) terms: \( \frac{b}{b^2} = b^{1-2} = b^{-1} = \frac{1}{b} \)
Putting it all together: \[ \frac{4}{9} \cdot a \cdot \frac{1}{b} = \frac{4a}{9b} \]
Thus, the simplified form of the expression \((4a^3b)(3ab)^{-2}\) is: \[ \frac{4a}{9b} \]