4. Simplify: $$ \left(\frac{2 x^{3} y^{2}}{x^{2}}\right)^{-2} $$

Question

UpStudy AI · Accepted Answer

To simplify the expression $$ \left(\frac{2 x^{3} y^{2}}{x^{2}}\right)^{-2} $$, follow these steps:

1. **Simplify the Inner Fraction:**

$$
   \frac{2 x^{3} y^{2}}{x^{2}} = 2 x^{3-2} y^{2} = 2 x y^{2}
   $$

2. **Apply the Negative Exponent:**

$$
   \left(2 x y^{2}\right)^{-2} = \frac{1}{\left(2 x y^{2}\right)^2}
   $$

3. **Expand the Denominator:**

$$
   \left(2 x y^{2}\right)^2 = 2^2 \cdot x^2 \cdot \left(y^{2}\right)^2 = 4 x^{2} y^{4}
   $$

4. **Combine the Results:**

$$
   \frac{1}{4 x^{2} y^{4}}
   $$

So, the simplified form of the expression is:

$$
\boxed{\dfrac{1}{4\,x^{2}\,y^{4}}}
$$
$$ \frac{1}{4x^{2}y^{4}} $$

4. Simplify: \( \left(\frac{2 x^{3} y^{2}}{x^{2}}\right)^{-2} \)