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

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: \[ \frac{p^{2}+2y^{2}}{p^{4}x^{8}} \]