1. **Problem 38:** Simplify $ (4x^{-1})(x^{3})^{-2} $. 2. Use the power of a power rule: $ (x^{3})^{-2} = x^{3 \times (-2)} = x^{-6} $. 3. Multiply the terms: $ 4x^{-1} \times x^{-6} = 4x^{-1 + (-6)} = 4x^{-7} $. 4. Final answer: $ \boxed{4x^{-7}} $. 1. **Problem 40:** Simplify $ \frac{(3a^{-1}b^{2})^{-2}}{(2a^{2}b^{-1})^{-3}} $. 2. Apply power of a product: $ (3a^{-1}b^{2})^{-2} = 3^{-2}a^{2}b^{-4} = \frac{1}{9}a^{2}b^{-4} $. 3. Similarly, $ (2a^{2}b^{-1})^{-3} = 2^{-3}a^{-6}b^{3} = \frac{1}{8}a^{-6}b^{3} $. 4. Divide the two: $ \frac{\frac{1}{9}a^{2}b^{-4}}{\frac{1}{8}a^{-6}b^{3}} = \frac{1}{9}a^{2}b^{-4} \times \frac{8}{1}a^{6}b^{-3} = \frac{8}{9}a^{2+6}b^{-4-3} = \frac{8}{9}a^{8}b^{-7} $. 5. Final answer: $ \boxed{\frac{8}{9}a^{8}b^{-7}} $. 1. **Problem 42:** Simplify $ (3x^{-1})^{2} (4y^{-1})^{3} (2z)^{-2} $. 2. Apply power to each term: $ (3x^{-1})^{2} = 3^{2}x^{-2} = 9x^{-2} $. 3. $ (4y^{-1})^{3} = 4^{3}y^{-3} = 64y^{-3} $. 4. $ (2z)^{-2} = 2^{-2}z^{-2} = \frac{1}{4}z^{-2} $. 5. Multiply all: $ 9x^{-2} \times 64y^{-3} \times \frac{1}{4}z^{-2} = 9 \times 64 \times \frac{1}{4} x^{-2} y^{-3} z^{-2} = 144 x^{-2} y^{-3} z^{-2} $. 6. Final answer: $ \boxed{144 x^{-2} y^{-3} z^{-2}} $. 1. **Problem 44:** Simplify $ (5u^{2} v^{-3})^{-1} \cdot 3(2u^{2} v^{2})^{-2} $. 2. First term: $ (5u^{2} v^{-3})^{-1} = 5^{-1} u^{-2} v^{3} = \frac{1}{5} u^{-2} v^{3} $. 3. Second term inside parentheses: $ (2u^{2} v^{2})^{-2} = 2^{-2} u^{-4} v^{-4} = \frac{1}{4} u^{-4} v^{-4} $. 4. Multiply second term by 3: $ 3 \times \frac{1}{4} u^{-4} v^{-4} = \frac{3}{4} u^{-4} v^{-4} $. 5. Multiply both terms: $ \frac{1}{5} u^{-2} v^{3} \times \frac{3}{4} u^{-4} v^{-4} = \frac{3}{20} u^{-2-4} v^{3-4} = \frac{3}{20} u^{-6} v^{-1} $. 6. Final answer: $ \boxed{\frac{3}{20} u^{-6} v^{-1}} $. 1. **Problem 46:** Simplify $ \left[ \left( \frac{x^{-2} y^{-3} z^{-4}}{x^{-2} y^{-1} z^{2}} \right)^{-2} \right]^{3} $. 2. Simplify inside the fraction: $ \frac{x^{-2} y^{-3} z^{-4}}{x^{-2} y^{-1} z^{2}} = x^{-2 - (-2)} y^{-3 - (-1)} z^{-4 - 2} = x^{0} y^{-2} z^{-6} = y^{-2} z^{-6} $. 3. Raise to power $-2$: $ (y^{-2} z^{-6})^{-2} = y^{4} z^{12} $. 4. Raise the result to power 3: $ (y^{4} z^{12})^{3} = y^{12} z^{36} $. 5. Final answer: $ \boxed{y^{12} z^{36}} $. 1. **Problem 48:** Simplify $ \left[ \left( \frac{2^{2} x^{-2} y^{0}}{3^{2} x^{-3} y^{-2}} \right)^{-2} \right]^{-2} $. 2. Simplify inside fraction: $ \frac{2^{2} x^{-2} y^{0}}{3^{2} x^{-3} y^{-2}} = \frac{4 x^{-2} }{9 x^{-3} y^{-2}} = \frac{4}{9} x^{-2 - (-3)} y^{0 - (-2)} = \frac{4}{9} x^{1} y^{2} $. 3. Raise to power $-2$: $ \left( \frac{4}{9} x^{1} y^{2} \right)^{-2} = \left( \frac{4}{9} \right)^{-2} x^{-2} y^{-4} = \left( \frac{9}{4} \right)^{2} x^{-2} y^{-4} = \frac{81}{16} x^{-2} y^{-4} $. 4. Raise to power $-2$ again: $ \left( \frac{81}{16} x^{-2} y^{-4} \right)^{-2} = \left( \frac{81}{16} \right)^{-2} x^{4} y^{8} = \left( \frac{16}{81} \right)^{2} x^{4} y^{8} = \frac{256}{6561} x^{4} y^{8} $. 5. Final answer: $ \boxed{\frac{256}{6561} x^{4} y^{8}} $. 1. **Problem 50:** Simplify $ \frac{x^{-1} - y^{-1}}{x^{-1} + y^{-1}} $. 2. Rewrite terms: $ x^{-1} = \frac{1}{x} $, $ y^{-1} = \frac{1}{y} $. 3. Substitute: $ \frac{\frac{1}{x} - \frac{1}{y}}{\frac{1}{x} + \frac{1}{y}} $. 4. Find common denominator $xy$ for numerator and denominator: Numerator: $ \frac{y - x}{xy} $. Denominator: $ \frac{y + x}{xy} $. 5. Divide numerator by denominator: $ \frac{\frac{y - x}{xy}}{\frac{y + x}{xy}} = \frac{y - x}{xy} \times \frac{xy}{y + x} = \frac{y - x}{y + x} $. 6. Final answer: $ \boxed{\frac{y - x}{y + x}} $. 1. **Problem 52:** Simplify $ \frac{(uv)^{-1}}{u^{-1} + v^{-1}} $. 2. Rewrite: $ (uv)^{-1} = \frac{1}{uv} $, $ u^{-1} = \frac{1}{u} $, $ v^{-1} = \frac{1}{v} $. 3. Substitute: $ \frac{\frac{1}{uv}}{\frac{1}{u} + \frac{1}{v}} $. 4. Find common denominator $uv$ in denominator: $ \frac{1}{u} + \frac{1}{v} = \frac{v}{uv} + \frac{u}{uv} = \frac{u + v}{uv} $. 5. Divide numerator by denominator: $ \frac{\frac{1}{uv}}{\frac{u + v}{uv}} = \frac{1}{uv} \times \frac{uv}{u + v} = \frac{1}{u + v} $. 6. Final answer: $ \boxed{\frac{1}{u + v}} $. 1. **Problem 54:** Simplify $ \left[ (a^{-1} + b^{-1})(a^{-1} - b^{-1}) \right]^{-2} $. 2. Recognize product as difference of squares: $ (x + y)(x - y) = x^{2} - y^{2} $. 3. Let $ x = a^{-1} $, $ y = b^{-1} $, so product is $ a^{-2} - b^{-2} $. 4. Raise to power $-2$: $ (a^{-2} - b^{-2})^{-2} $. 5. Rewrite $ a^{-2} = \frac{1}{a^{2}} $, $ b^{-2} = \frac{1}{b^{2}} $, so expression is $ \left( \frac{1}{a^{2}} - \frac{1}{b^{2}} \right)^{-2} $. 6. Find common denominator $ a^{2} b^{2} $: $ \frac{b^{2} - a^{2}}{a^{2} b^{2}} $. 7. Expression becomes $ \left( \frac{b^{2} - a^{2}}{a^{2} b^{2}} \right)^{-2} = \left( \frac{a^{2} b^{2}}{b^{2} - a^{2}} \right)^{2} = \frac{a^{4} b^{4}}{(b^{2} - a^{2})^{2}} $. 8. Final answer: $ \boxed{\frac{a^{4} b^{4}}{(b^{2} - a^{2})^{2}}} $.