To simplify the given expression, we need to follow these steps:
The numerator of the given expression is: 2x−3y−3 2 x^{-3} y^{-3} 2x−3y−3
The denominator of the given expression is: 2xy3⋅(2x−3)4 2 x y^{3} \cdot (2 x^{-3})^{4} 2xy3⋅(2x−3)4
First, simplify (2x−3)4(2 x^{-3})^{4}(2x−3)4: (2x−3)4=24⋅(x−3)4=16x−12 (2 x^{-3})^{4} = 2^{4} \cdot (x^{-3})^{4} = 16 x^{-12} (2x−3)4=24⋅(x−3)4=16x−12
So, the denominator becomes: 2xy3⋅16x−12=32x−11y3 2 x y^{3} \cdot 16 x^{-12} = 32 x^{-11} y^{3} 2xy3⋅16x−12=32x−11y3
Now, combine the simplified numerator and denominator: 2x−3y−332x−11y3 \frac{2 x^{-3} y^{-3}}{32 x^{-11} y^{3}} 32x−11y32x−3y−3
Simplify the fraction: 232⋅x−3x−11⋅y−3y3=116⋅x8⋅y−6 \frac{2}{32} \cdot \frac{x^{-3}}{x^{-11}} \cdot \frac{y^{-3}}{y^{3}} = \frac{1}{16} \cdot x^{8} \cdot y^{-6} 322⋅x−11x−3⋅y3y−3=161⋅x8⋅y−6
Thus, the simplified expression is: x816y6 \frac{x^{8}}{16 y^{6}} 16y6x8
x816y6 \boxed{\frac{x^{8}}{16 y^{6}}} 16y6x8
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