$$ ( \frac { 2 } { 3 } x ^ { 3 } - \frac { 1 } { y ^ { 2 } } ) ( \frac { 1 } { 2 } x ^ { 2 } - \frac { 4 } { 3 } x y - \frac { 1 } { 2 } y ^ { 2 } ) ( y
eq 0 $$

Question

$$ ( \frac { 2 } { 3 } x ^ { 3 } - \frac { 1 } { y ^ { 2 } } ) ( \frac { 1 } { 2 } x ^ { 2 } - \frac { 4 } { 3 } x y - \frac { 1 } { 2 } y ^ { 2 } ) ( y 
eq 0 $$

UpStudy AI · Accepted Answer

Certainly! Let's expand and simplify the given expression step by step.

**Given Expression:**
$$
\left( \frac{2}{3}x^{3} - \frac{1}{y^{2}} \right) \left( \frac{1}{2}x^{2} - \frac{4}{3}xy - \frac{1}{2}y^{2} \right) \quad \text{where } y \neq 0
$$

**Step 1: Distribute Each Term in the First Parenthesis with Each Term in the Second Parenthesis**

We'll use the distributive property (also known as the FOIL method for binomials) to multiply each term in the first polynomial by each term in the second polynomial.

$$
\begin{aligned}
&\left( \frac{2}{3}x^{3} \right) \cdot \left( \frac{1}{2}x^{2} \right) + \left( \frac{2}{3}x^{3} \right) \cdot \left( -\frac{4}{3}xy \right) + \left( \frac{2}{3}x^{3} \right) \cdot \left( -\frac{1}{2}y^{2} \right) \
&\quad + \left( -\frac{1}{y^{2}} \right) \cdot \left( \frac{1}{2}x^{2} \right) + \left( -\frac{1}{y^{2}} \right) \cdot \left( -\frac{4}{3}xy \right) + \left( -\frac{1}{y^{2}} \right) \cdot \left( -\frac{1}{2}y^{2} \right)
\end{aligned}
$$

**Step 2: Simplify Each Term**

1. **First Term:**
   $$
   \frac{2}{3}x^{3} \cdot \frac{1}{2}x^{2} = \frac{2}{3} \cdot \frac{1}{2} \cdot x^{3+2} = \frac{1}{3}x^{5}
   $$

2. **Second Term:**
   $$
   \frac{2}{3}x^{3} \cdot \left( -\frac{4}{3}xy \right) = \frac{2}{3} \cdot -\frac{4}{3} \cdot x^{3+1}y = -\frac{8}{9}x^{4}y
   $$

3. **Third Term:**
   $$
   \frac{2}{3}x^{3} \cdot \left( -\frac{1}{2}y^{2} \right) = \frac{2}{3} \cdot -\frac{1}{2} \cdot x^{3}y^{2} = -\frac{1}{3}x^{3}y^{2}
   $$

4. **Fourth Term:**
   $$
   -\frac{1}{y^{2}} \cdot \frac{1}{2}x^{2} = -\frac{1}{2} \cdot \frac{x^{2}}{y^{2}}
   $$

5. **Fifth Term:**
   $$
   -\frac{1}{y^{2}} \cdot \left( -\frac{4}{3}xy \right) = \frac{4}{3} \cdot \frac{x}{y}
   $$

6. **Sixth Term:**
   $$
   -\frac{1}{y^{2}} \cdot \left( -\frac{1}{2}y^{2} \right) = \frac{1}{2}
   $$

**Step 3: Combine All Simplified Terms**

Putting it all together:
$$
\frac{1}{3}x^{5} - \frac{8}{9}x^{4}y - \frac{1}{3}x^{3}y^{2} - \frac{1}{2}\frac{x^{2}}{y^{2}} + \frac{4}{3}\frac{x}{y} + \frac{1}{2}
$$

**Final Expanded Form:**
$$
\boxed{
\frac{1}{3}x^{5} - \frac{8}{9}x^{4}y - \frac{1}{3}x^{3}y^{2} - \frac{1}{2}\frac{x^{2}}{y^{2}} + \frac{4}{3}\frac{x}{y} + \frac{1}{2}
}
$$

This is the fully expanded and simplified form of the given expression.
$$ \frac{1}{3}x^{5} - \frac{8}{9}x^{4}y - \frac{1}{3}x^{3}y^{2} - \frac{1}{2}\frac{x^{2}}{y^{2}} + \frac{4}{3}\frac{x}{y} + \frac{1}{2} $$

\( ( \frac { 2 } { 3 } x ^ { 3 } - \frac { 1 } { y ^ { 2 } } ) ( \frac { 1 } { 2 } x ^ { 2 } - \frac { 4 } { 3 } x y - \frac { 1 } { 2 } y ^ { 2 } ) ( y \neq 0 \)

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