22. $$ \left(5 x^{-3} y^{-2}\right)^{-(x)} $$ 240. Expand by using the distributive property. Write the answer with all variables in numerator. $$ \frac{x^{-4} y}{p^{2}}\left(\frac{y^{-1}}{x^{4}}+\frac{2 x^{-4} p^{-2}}{y^{-1}}\right) $$

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Simplify the expression by following steps:
- step0: Calculate:
  $$\left(5x^{-3}y^{-2}\right)^{-x}$$
- step1: Simplify:
  $$\left(\frac{5}{x^{3}y^{2}}\right)^{-x}$$
- step2: Expand the expression:
  $$\frac{5^{-x}}{\left(x^{3}y^{2}\right)^{-x}}$$
- step3: Express with a positive exponent:
  $$\frac{\frac{1}{5^{x}}}{\frac{1}{\left(x^{3}y^{2}\right)^{x}}}$$
- step4: Multiply by the reciprocal:
  $$\frac{1}{5^{x}}\times \left(x^{3}y^{2}\right)^{x}$$
- step5: Multiply the terms:
  $$\frac{\left(x^{3}y^{2}\right)^{x}}{5^{x}}$$
Expand the expression $$ \frac{x^{-4} y}{p^{2}}\left(\frac{y^{-1}}{x^{4}}+\frac{2 x^{-4} p^{-2}}{y^{-1}}\right) $$
Simplify the expression by following steps:
- step0: Simplify:
  $$\frac{x^{-4}y}{p^{2}}\times \left(\frac{y^{-1}}{x^{4}}+\frac{2x^{-4}p^{-2}}{y^{-1}}\right)$$
- step1: Rewrite the expression:
  $$\frac{x^{-4}y}{p^{2}}\times \left(\frac{1}{x^{4}y}+\frac{2x^{-4}p^{-2}}{y^{-1}}\right)$$
- step2: Rewrite the expression:
  $$\frac{x^{-4}y}{p^{2}}\times \left(\frac{1}{x^{4}y}+\frac{2y}{x^{4}p^{2}}\right)$$
- step3: Add the terms:
  $$\frac{x^{-4}y}{p^{2}}\times \frac{p^{2}+2y^{2}}{x^{4}yp^{2}}$$
- step4: Rewrite the expression:
  $$\frac{y}{p^{2}x^{4}}\times \frac{p^{2}+2y^{2}}{x^{4}yp^{2}}$$
- step5: Reduce the fraction:
  $$\frac{1}{p^{2}x^{4}}\times \frac{p^{2}+2y^{2}}{x^{4}p^{2}}$$
- step6: Multiply the terms:
  $$\frac{p^{2}+2y^{2}}{p^{2}x^{4}\times x^{4}p^{2}}$$
- step7: Multiply the terms:
  $$\frac{p^{2}+2y^{2}}{p^{4}x^{8}}$$
Let's break down the solutions step by step.

### Problem 22: Simplifying $$ \left(5 x^{-3} y^{-2}\right)^{-(x)} $$

1. **Understanding the expression**: The expression $$ \left(5 x^{-3} y^{-2}\right)^{-(x)} $$ involves raising a product to a negative exponent. According to the properties of exponents, we can rewrite this as:
   $$
   \frac{1}{\left(5 x^{-3} y^{-2}\right)^{x}}
   $$

2. **Applying the exponent**: Now, we can apply the exponent $$ x $$ to each part of the product:
   $$
   = \frac{1}{5^{x} \cdot (x^{-3})^{x} \cdot (y^{-2})^{x}} = \frac{1}{5^{x} \cdot x^{-3x} \cdot y^{-2x}}
   $$

3. **Rewriting with positive exponents**: We can rewrite $$ x^{-3x} $$ and $$ y^{-2x} $$ as:
   $$
   = \frac{1}{5^{x} \cdot \frac{1}{x^{3x}} \cdot \frac{1}{y^{2x}}} = \frac{x^{3x} y^{2x}}{5^{x}}
   $$

Thus, the final result is:
$$
\frac{(x^{3}y^{2})^{x}}{5^{x}}
$$

### Problem 240: Expanding the expression

1. **Understanding the expression**: We need to expand:
   $$
   \frac{x^{-4} y}{p^{2}}\left(\frac{y^{-1}}{x^{4}}+\frac{2 x^{-4} p^{-2}}{y^{-1}}\right)
   $$

2. **Distributing the term**: We will distribute $$ \frac{x^{-4} y}{p^{2}} $$ to both terms inside the parentheses:
   - First term: 
     $$
     \frac{x^{-4} y}{p^{2}} \cdot \frac{y^{-1}}{x^{4}} = \frac{x^{-4} y \cdot y^{-1}}{p^{2} x^{4}} = \frac{x^{-4} y^{0}}{p^{2} x^{4}} = \frac{1}{p^{2} x^{8}}
     $$
   - Second term:
     $$
     \frac{x^{-4} y}{p^{2}} \cdot \frac{2 x^{-4} p^{-2}}{y^{-1}} = \frac{2 x^{-4} y \cdot x^{-4} p^{-2} \cdot y^{1}}{p^{2}} = \frac{2 x^{-8} y^{2}}{p^{4}}
     $$

3. **Combining the results**: Now we combine both results:
   $$
   \frac{1}{p^{2} x^{8}} + \frac{2 y^{2}}{p^{4} x^{8}} = \frac{p^{2} + 2y^{2}}{p^{4} x^{8}}
   $$

Thus, the final result is:
$$
\frac{p^{2}+2y^{2}}{p^{4}x^{8}}
$$

### Summary of Results
- For Problem 22: 
  $$
  \frac{(x^{3}y^{2})^{x}}{5^{x}}
  $$
- For Problem 240: 
  $$
  \frac{p^{2}+2y^{2}}{p^{4}x^{8}}
  $$
For Problem 22: $$ \frac{(x^{3}y^{2})^{x}}{5^{x}} $$ For Problem 240: $$ \frac{p^{2}+2y^{2}}{p^{4}x^{8}} $$

Expand by using the distributive property. Write the answer with all variables in
numerator.

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