1. **Problem:** Check the simplification and operations on fractions and algebraic expressions as provided. 2. **Fraction simplifications and multiplications:** - a) $\frac{5}{12} \times \frac{2}{2} = \frac{10}{24}$ correct. - b) $\frac{8}{48} = \frac{1}{6}$ correct, then $\frac{1}{6} \times \frac{4}{4} = \frac{4}{24}$ correct. - 2a) $\frac{1}{4} \times \frac{8}{8} = \frac{8}{32}$ correct. - 2b) $\frac{7}{16} \times \frac{2}{2} = \frac{14}{32}$ correct. - 2c) $\frac{12}{128} = \frac{3}{32}$ correct. - 2d) $\frac{2}{24} = \frac{1}{12}$ but user wrote $\frac{1}{4}$ which is incorrect; correct simplification is $\frac{1}{12}$. - 3a) $\frac{13}{39} = \frac{1}{3}$ correct. - 3b) $\frac{-25}{40} = \frac{-5}{8}$ correct. - 3c) $\frac{42}{91} = \frac{7}{11}$ correct. - 3d) $\frac{78}{273} = \frac{2}{7}$ correct. - 3e) $\frac{243}{249}$ should simplify to $\frac{81}{83}$ not $\frac{81}{283}$. - c) $\frac{6}{36} = \frac{1}{6}$ correct. - d) $\frac{-5}{8} \times \frac{3}{3} = \frac{-15}{24}$ correct. - e) $\frac{15}{120} = \frac{1}{8}$ correct, then $\frac{1}{8} \times \frac{3}{3} = \frac{3}{24}$ correct. 3. **Algebraic simplifications:** - $\frac{126p}{p^2} = \frac{126}{p}$ correct. - $\frac{2h}{4j} = \frac{h}{2j}$ correct. - $\frac{3ab}{6bc} = \frac{a}{2c}$ correct. - $\frac{mn}{2n} = \frac{m}{2}$ correct. - $\frac{2(1+r)}{4} = \frac{1+r}{2}$ correct. 4. **Fraction additions and subtractions:** - $\frac{1}{4} + \frac{3}{5} = \frac{5}{20} + \frac{12}{20} = \frac{17}{20}$ correct. - $\frac{5}{72} - \frac{7}{12} = \frac{5}{72} - \frac{42}{72} = \frac{-37}{72}$ correct. - LCD calculations are correct. - $\frac{6}{32} + \frac{5}{48} = \frac{9}{48} + \frac{5}{48} = \frac{14}{48} = \frac{7}{24}$ correct. - $\frac{2}{3} + \frac{5}{12} - \frac{1}{6} = \frac{8}{12} + \frac{5}{12} - \frac{2}{12} = \frac{11}{12}$ correct. - $\frac{3}{14} - \frac{5}{329} = \frac{141}{658} - \frac{10}{658} = \frac{131}{658}$ correct. - $\frac{2}{3} + \frac{6}{72} - \frac{2}{11} = \frac{22}{33} + \frac{6}{72} - \frac{21}{77} = \frac{7}{77} = \frac{1}{11}$ correct. - $\frac{6}{32} + \frac{7}{48} - \frac{3}{6} = \frac{9}{48} + \frac{7}{48} - \frac{40}{48} = \frac{-24}{48} = \frac{-1}{2}$ correct. - $\frac{7}{15} + \frac{1}{5} - \frac{1}{4} = \frac{28}{60} + \frac{12}{60} - \frac{15}{60} = \frac{25}{60} = \frac{5}{12}$ correct. 5. **Algebraic expressions:** - a) $\frac{x}{2} + \frac{x}{5} = \frac{5x}{10} + \frac{2x}{10} = \frac{7x}{10}$ correct. - b) $\frac{7}{3x} - \frac{2}{6x} = \frac{7}{3x} - \frac{1}{3x} = \frac{6}{3x} = \frac{2}{x}$ correct. - c) $\frac{a}{2n} + \frac{2a}{3n} = \frac{3a}{6n} + \frac{4a}{6n} = \frac{7a}{6n}$ correct. - d) $\frac{5}{3p} - \frac{2}{4p} = \frac{20}{12p} - \frac{6}{12p} = \frac{14}{12p} = \frac{7}{6p}$ correct. 6. **Multiplications and divisions:** - $\frac{1}{2} \times \frac{1}{6} = \frac{1}{12}$ correct. - $\frac{9}{16} \times \frac{13}{2} = \frac{117}{32}$ correct. - $\frac{2}{3} \times \frac{5}{8} = \frac{10}{24} = \frac{5}{12}$ correct. - Other multiplications and divisions checked and correct. 7. **Exponents and roots:** - $\sqrt{253} \approx 15.9$ but user wrote 15, close but approximate. - $\sqrt[3]{-125} = -5$ correct. - $\sqrt[3]{-27} = -3$ correct. - $\sqrt{49} = 7$ correct. - $128^{1/7} = 2$ correct. - Other exponent and root calculations are correct. 8. **Algebraic expansions and factorizations:** - $(a-2)(a+3) = a^2 + a - 6$ correct. - $(p+q)(1-q) = p - pq + 1 - q$ correct. - $(p+q)^2 - (p-q)^2 = 4pq$ correct. - Factoring expressions like $5m^2 + 15m = 5m(m+3)$ correct. 9. **Summary:** - Most answers are correct. - Minor errors: 2d) $\frac{2}{24}$ simplified incorrectly as $\frac{1}{4}$; correct is $\frac{1}{12}$. - 3e) $\frac{243}{249}$ simplified incorrectly; correct simplification is $\frac{81}{83}$. - Approximate root $\sqrt{253}$ is about 15.9, not exactly 15. Overall, your work is very good with minor corrections needed. Final note: Keep practicing simplification and always check for exact simplification steps.