q. $$ \frac{12 x^{3} y^{3}}{4 x y^{3}}=3 x^{3} y^{4} \quad $$ r. $$ \frac{-32 x^{4} y^{12}}{8 x^{2} y^{3}}=-4 x y^{5} $$

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UpStudy AI · Accepted Answer

Let's simplify both expressions step by step.

### For expression q:
We need to simplify $$ \frac{12 x^{3} y^{3}}{4 x y^{3}} $$.

1. **Divide the coefficients**: $$ \frac{12}{4} = 3 $$.
2. **Simplify the $$ x $$ terms**: $$ \frac{x^{3}}{x} = x^{2} $$.
3. **The $$ y^{3} $$ terms cancel out**: $$ \frac{y^{3}}{y^{3}} = 1 $$.

Putting it all together, we have:
$$
\frac{12 x^{3} y^{3}}{4 x y^{3}} = 3 x^{2}
$$

### For expression r:
Now, let's simplify $$ \frac{-32 x^{4} y^{12}}{8 x^{2} y^{3}} $$.

1. **Divide the coefficients**: $$ \frac{-32}{8} = -4 $$.
2. **Simplify the $$ x $$ terms**: $$ \frac{x^{4}}{x^{2}} = x^{2} $$.
3. **Simplify the $$ y $$ terms**: $$ \frac{y^{12}}{y^{3}} = y^{9} $$.

Putting it all together, we have:
$$
\frac{-32 x^{4} y^{12}}{8 x^{2} y^{3}} = -4 x^{2} y^{9}
$$

### Summary of Results:
- For expression q: $$ \frac{12 x^{3} y^{3}}{4 x y^{3}} = 3 x^{2} $$
- For expression r: $$ \frac{-32 x^{4} y^{12}}{8 x^{2} y^{3}} = -4 x^{2} y^{9} $$
Simplified results: - $$ \frac{12 x^{3} y^{3}}{4 x y^{3}} = 3 x^{2} $$ - $$ \frac{-32 x^{4} y^{12}}{8 x^{2} y^{3}} = -4 x^{2} y^{9} $$

q. r.

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For expression q:

For expression r:

Summary of Results:

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