1. Simplify each expression: (a) Simplify $\sqrt[3]{252x^{5}}$: Factor $252 = 2^{2} \times 3^{2} \times 7$ and express as $\sqrt[3]{2^{2} \times 3^{2} \times 7 \times x^{5}}$. This is $\sqrt[3]{2^{2}} \times \sqrt[3]{3^{2}} \times \sqrt[3]{7} \times \sqrt[3]{x^{3} \cdot x^{2}}= (2 \times 3 \times x) \sqrt[3]{2 \times 3 \times 7 \times x^{2}} = 6x \sqrt[3]{42x^{2}}$. (b) Simplify $6x^{2} \sqrt{x}$: Write as $6x^{2}x^{\frac{1}{2}} = 6x^{2 + \frac{1}{2}} = 6x^{\frac{5}{2}}$. (c) Simplify $3x \sqrt[3]{4x^{5}}$: Express as $3x \times \sqrt[3]{4} \times \sqrt[3]{x^{3} x^{2}} = 3x \times \sqrt[3]{4} \times x \sqrt[3]{x^{2}} = 3x^{2} \sqrt[3]{4x^{2}}$. (d) Simplify $3x \sqrt[3]{4x^{2}}$: This is $3x \times \sqrt[3]{4x^{2}}$ (already simplified). 2. For $\sqrt[3]{48x^{3}} - \sqrt[3]{6x^{3}}$, write $\sqrt[3]{48x^{3}} = \sqrt[3]{48} \times x$; also $\sqrt[3]{6x^{3}} = \sqrt[3]{6} \times x$. So expression is $x \left( \sqrt[3]{48} - \sqrt[3]{6} \right)$. 3. Identify $x \sqrt[3]{3} = x \times \sqrt[3]{3}$, $x \sqrt[3]{8} = x \times 2$, $x \sqrt[3]{2}$, $x \sqrt[3]{6}$ as given. 4. Simplify $(4 \times 10^{-4})(3 \times 10^{-3}) / (6 \times 10^{-5})$: multiply numerator $4 \times 3 = 12$ and powers $10^{-4} \times 10^{-3} = 10^{-7}$, so numerator is $12 \times 10^{-7}$. Dividing by denominator $6 \times 10^{-5}$ equals $\frac{12}{6} \times 10^{-7 + 5} = 2 \times 10^{-2}$. 5. Simplify $(2xy^{-5})(-7x^{-4}y^{10}) = -14 x^{1 - 4} y^{-5 + 10} = -14 x^{-3} y^{5}$. 6. Evaluate $\frac{234}{15 + \frac{12}{3 - \frac{x + 3}{5}}}$ at $x = -7$: Calculate denominator inside: $x + 3 = -7 + 3 = -4$, so $\frac{-4}{5} = -0.8$. Then $3 - (-0.8) = 3 + 0.8 = 3.8$. Then $\frac{12}{3.8} = \approx 3.1579$. Add $15 + 3.1579 = 18.1579$. Divide $234 / 18.1579 \approx 12.89$, in fraction form $\frac{3835}{19}$ approximately equals 201.84, re-checking we need exact values: $3 - \frac{-4}{5} = \frac{15}{5} + \frac{4}{5} = \frac{19}{5}$, $\frac{12}{19/5} = 12 \times \frac{5}{19} = \frac{60}{19}$, So denominator is $15 + \frac{60}{19} = \frac{285}{19} + \frac{60}{19} = \frac{345}{19}$, Then $234 \div \frac{345}{19} = 234 \times \frac{19}{345} = \frac{4446}{345}$. 7. Calculate $(\frac{|x|}{1+|x|}) + (\frac{|y|}{1+|y|})$ for $x = -1.5$, $y = -0.75$: $|x|=1.5$, $\frac{1.5}{1+1.5} = \frac{1.5}{2.5} = \frac{3}{5}$; $|y|=0.75$, $\frac{0.75}{1+0.75} = \frac{3/4}{7/4} = \frac{3}{7}$; Sum: $\frac{3}{5} + \frac{3}{7} = \frac{21}{35} + \frac{15}{35} = \frac{36}{35}$. 8. Compute $\frac{|a-b|}{1 + |a-b|}$ for $a=3.5$, $b=7.5$: $|a-b| = |3.5-7.5|=4$, Value is $\frac{4}{1+4} = \frac{4}{5}$. 9. Simplify $(1 + \frac{x+y}{x-y}) \div \frac{4xy}{2x^{2} - 2xy}$: First part: $1 + \frac{x+y}{x-y} = \frac{x-y}{x-y} + \frac{x+y}{x-y} = \frac{(x-y)+(x+y)}{x-y} = \frac{2x}{x-y}$. Second part denominator: $\frac{4xy}{2x^{2}-2xy} = \frac{4xy}{2x(x-y)} = \frac{4xy}{2x(x-y)} = \frac{2y}{x-y}$. Divide first by second: $\frac{2x}{x-y} \div \frac{2y}{x-y} = \frac{2x}{x-y} \times \frac{x-y}{2y} = \frac{2x}{2y} = \frac{x}{y}$. 10. Simplify $\frac{5x}{x^{2}-4} - \frac{3}{x+2} - \frac{x}{x-2}$: Factor denominator: $x^{2} - 4 = (x+2)(x-2)$, Common denominator is $(x+2)(x-2)$, Rewrite terms: $\frac{5x}{(x+2)(x-2)} - \frac{3(x-2)}{(x+2)(x-2)} - \frac{x(x+2)}{(x+2)(x-2)} = \frac{5x - 3x + 6 - x^{2} - 2x}{(x+2)(x-2)} = \frac{-x^{2} + 0x + 6}{x^{2} - 4} = \frac{-x^{2} + 6}{x^{2}-4}$. 11. Simplify $\frac{6y-3}{4} \div \frac{-4y + 2}{12}$: Rewrite division as multiplication by reciprocal: $= \frac{6y-3}{4} \times \frac{12}{-4y+2} = \frac{6y-3}{4} \times \frac{12}{-4y + 2}$. Factor numerator and denominator: $6y - 3 = 3(2y - 1)$, $-4y + 2 = -2(2y -1)$. Expression becomes $\frac{3(2y -1)}{4} \times \frac{12}{-2(2y -1)}$. Cancel $(2y -1)$ terms: $= \frac{3}{4} \times \frac{12}{-2} = \frac{3}{4} \times -6 = -\frac{18}{4} = -\frac{9}{2}$. Final answers checked: (a) $6x \sqrt[3]{42x^{2}}$ (b) $6x^{5/2}$ (c) $3x^{2} \sqrt[3]{4x^{2}}$ (d) $3x \sqrt[3]{4x^{2}}$ 48. $x ( \sqrt[3]{48} - \sqrt[3]{6} )$ 49. (a) $2 \times 10^{-2}$ 50. (c) $-14x^{-3}y^{5}$ 51. (c) $\frac{4446}{345}$ 52. (a) $\frac{36}{35}$ 53. (a) $\frac{4}{5}$ 54. (c) $\frac{x}{y}$ 55. (d) $\frac{-x^{2}+6}{x^{2}-4}$ 56. none of the provided answers equals $-\frac{9}{2}$