Questions: 2x^-3y^-3 / (2xy^3 cdot (2x^-3)^4)

Solution Steps

To simplify the given expression, we need to follow these steps:

1. Simplify the numerator and the denominator separately.
2. Combine the simplified forms.
3. Apply the laws of exponents to further simplify the expression.

The numerator of the given expression is: $$2 x^{-3} y^{-3}$$

The denominator of the given expression is: $$2 x y^{3} \cdot (2 x^{-3})^{4}$$

First, simplify $(2 x^{-3})^{4}$: $$(2 x^{-3})^{4} = 2^{4} \cdot (x^{-3})^{4} = 16 x^{-12}$$

So, the denominator becomes: $$2 x y^{3} \cdot 16 x^{-12} = 32 x^{-11} y^{3}$$

Now, combine the simplified numerator and denominator: $$\frac{2 x^{-3} y^{-3}}{32 x^{-11} y^{3}}$$

Simplify the fraction: $$\frac{2}{32} \cdot \frac{x^{-3}}{x^{-11}} \cdot \frac{y^{-3}}{y^{3}} = \frac{1}{16} \cdot x^{8} \cdot y^{-6}$$

Thus, the simplified expression is: $$\frac{x^{8}}{16 y^{6}}$$

Final Answer