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

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

Sure, let's **expand and simplify** the given expression step by step.

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

### Step 1: Distribute Each Term in the First Parenthesis to Each Term in the Second Parenthesis

We'll multiply each term in the first parenthesis by each term in the second parenthesis.

$$
\begin{align*}
\left( \frac{2}{3} x^{3} \right) \left( \frac{1}{2} x^{2} \right) &= \frac{2}{3} \times \frac{1}{2} \times x^{3} \times x^{2} = \frac{1}{3} x^{5} \
\left( \frac{2}{3} x^{3} \right) \left( -\frac{4}{3} x y \right) &= \frac{2}{3} \times -\frac{4}{3} \times x^{3} \times x y = -\frac{8}{9} x^{4} y \
\left( \frac{2}{3} x^{3} \right) \left( -\frac{1}{2} y^{2} \right) &= \frac{2}{3} \times -\frac{1}{2} \times x^{3} \times y^{2} = -\frac{1}{3} x^{3} y^{2} \
\left( -\frac{1}{y^{2}} \right) \left( \frac{1}{2} x^{2} \right) &= -\frac{1}{y^{2}} \times \frac{1}{2} x^{2} = -\frac{1}{2} \frac{x^{2}}{y^{2}} \
\left( -\frac{1}{y^{2}} \right) \left( -\frac{4}{3} x y \right) &= -\frac{1}{y^{2}} \times -\frac{4}{3} x y = \frac{4}{3} \frac{x y}{y^{2}} = \frac{4}{3} \frac{x}{y} \
\left( -\frac{1}{y^{2}} \right) \left( -\frac{1}{2} y^{2} \right) &= -\frac{1}{y^{2}} \times -\frac{1}{2} y^{2} = \frac{1}{2} \
\end{align*}
$$

### Step 2: Combine All the Terms

Putting all the expanded terms 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 Simplified Expression:

$$
\boxed{ \frac{1}{3} x^{5} - \frac{8}{9} x^{4} y - \frac{1}{3} x^{3} y^{2} + \frac{4}{3} \frac{x}{y} - \frac{1}{2} \frac{x^{2}}{y^{2}} + \frac{1}{2} }
$$

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

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

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