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

2x^-3y^-3 / (2xy^3 cdot (2x^-3)^4)
Transcript text: $\frac{2 x^{-3} y^{-3}}{2 x y^{3} \cdot\left(2 x^{-3}\right)^{4}}$
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Solution

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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.
Step 1: Simplify the Numerator

The numerator of the given expression is: 2x3y3 2 x^{-3} y^{-3}

Step 2: Simplify the Denominator

The denominator of the given expression is: 2xy3(2x3)4 2 x y^{3} \cdot (2 x^{-3})^{4}

First, simplify (2x3)4(2 x^{-3})^{4}: (2x3)4=24(x3)4=16x12 (2 x^{-3})^{4} = 2^{4} \cdot (x^{-3})^{4} = 16 x^{-12}

So, the denominator becomes: 2xy316x12=32x11y3 2 x y^{3} \cdot 16 x^{-12} = 32 x^{-11} y^{3}

Step 3: Combine and Simplify the Expression

Now, combine the simplified numerator and denominator: 2x3y332x11y3 \frac{2 x^{-3} y^{-3}}{32 x^{-11} y^{3}}

Simplify the fraction: 232x3x11y3y3=116x8y6 \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: x816y6 \frac{x^{8}}{16 y^{6}}

Final Answer

x816y6 \boxed{\frac{x^{8}}{16 y^{6}}}

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