To simplify the expression \( \frac{6x^3y^3}{2x^{-3}y^{-3}} \), we can follow these steps:
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Rewrite the expression: \[ \frac{6x^3y^3}{2x^{-3}y^{-3}} = \frac{6}{2} \cdot \frac{x^3}{x^{-3}} \cdot \frac{y^3}{y^{-3}} \]
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Simplify the coefficients: \[ \frac{6}{2} = 3 \]
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Simplify the variable expressions:
- For \( x \): \[ \frac{x^3}{x^{-3}} = x^{3 - (-3)} = x^{3 + 3} = x^6 \]
- For \( y \): \[ \frac{y^3}{y^{-3}} = y^{3 - (-3)} = y^{3 + 3} = y^6 \]
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Combine the results: \[ 3x^6y^6 \]
Thus, the simplified expression is: \[ 3x^6y^6 \]
This corresponds to option C: \( 3x^6y^6 \).