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

Question

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

UpStudy AI · Accepted Answer

To expand the expression:

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

we'll use the distributive property (also known as the FOIL method for binomials). This involves multiplying each term in the first parenthesis by each term in the second parenthesis.

### Step 1: Identify the Terms

Let’s denote the terms as:

- **First Expression:**
  - $$ A = \frac{2}{3} x^{3} $$
  - $$ B = -\frac{1}{y^{2}} $$

- **Second Expression:**
  - $$ C = \frac{1}{2} x^{2} $$
  - $$ D = -\frac{4}{3} x y $$
  - $$ E = -\frac{1}{2} y^{2} $$

### Step 2: Multiply Each Term

Now, multiply each term from the first expression with each term in the second expression:

1. **$$ A \times C $$:**
   $$
   \frac{2}{3} x^{3} \times \frac{1}{2} x^{2} = \frac{2}{3} \times \frac{1}{2} x^{3+2} = \frac{1}{3} x^{5}
   $$

2. **$$ A \times D $$:**
   $$
   \frac{2}{3} x^{3} \times \left(-\frac{4}{3} x y\right) = \frac{2}{3} \times \left(-\frac{4}{3}\right) x^{3+1} y = -\frac{8}{9} x^{4} y
   $$

3. **$$ A \times E $$:**
   $$
   \frac{2}{3} x^{3} \times \left(-\frac{1}{2} y^{2}\right) = \frac{2}{3} \times \left(-\frac{1}{2}\right) x^{3} y^{2} = -\frac{1}{3} x^{3} y^{2}
   $$

4. **$$ B \times C $$:**
   $$
   -\frac{1}{y^{2}} \times \frac{1}{2} x^{2} = -\frac{1}{2} \frac{x^{2}}{y^{2}} = -\frac{1}{2} x^{2} y^{-2}
   $$

5. **$$ B \times D $$:**
   $$
   -\frac{1}{y^{2}} \times \left(-\frac{4}{3} x y\right) = \frac{4}{3} \frac{x y}{y^{2}} = \frac{4}{3} \frac{x}{y} = \frac{4}{3} x y^{-1}
   $$

6. **$$ B \times E $$:**
   $$
   -\frac{1}{y^{2}} \times \left(-\frac{1}{2} y^{2}\right) = \frac{1}{2} y^{0} = \frac{1}{2}
   $$

### Step 3: Combine All Terms

Putting all the multiplied terms together:

$$
\begin{align*}
& \frac{1}{3} x^{5} - \frac{8}{9} x^{4} y - \frac{1}{3} x^{3} y^{2} - \frac{1}{2} x^{2} y^{-2} + \frac{4}{3} x y^{-1} + \frac{1}{2}
\end{align*}
$$

### 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 form of the given expression.
$$ \frac{1}{3} x^{5} - \frac{8}{9} x^{4} y - \frac{1}{3} x^{3} y^{2} - \frac{1}{2} \frac{x^{2}}{y^{2}} + \frac{4}{3} \frac{x}{y} + \frac{1}{2} $$

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

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