worksheet 1.1

1. simplify the following

   1. $(x^5{)}^7$
      $=x^{35}$
   2. $y^9\cdot y^4$
      $=y^{13}$
   3. $(3a^7{)}^4$
      $=81a^{28}$
   4. $7x^{11}\cdot 8x^{15}$
      $=56x^{26}$
   5. $(25x^{\frac{1}{3}}{)}^{\frac{3}{2}}$
      $=125x^{\frac{1}{2}}$
   6. $14a^{\frac{3}{5}}\cdot 12a^{-\frac{1}{9}}$
      $=168a^{\frac{22}{45}}$

2. simplify the following

   1. $\frac{2x}{3}+\frac{3x}{2}$
      $=\frac{4x+9x}{6}$
      $=\frac{13x}{6}$
   2. $\frac{3a}{2}-\frac{a}{4}$
      $=\frac{5a}{4}$
   3. $\frac{3x}{4}-\frac{y}{6}-\frac{x}{3}$
      $=\frac{9x-2y-4x}{12}$
      $=\frac{5x-2y}{12}$
   4. $\frac{3}{x}+\frac{2}{y}$
      $=\frac{3y+2x}{xy}$
   5. $\frac{3}{x-2}+\frac{2}{x+1}$
      $=\frac{3(x+1)+2(x-2)}{(x+1)(x-2)}$
      $=\frac{3x+3+2x-4}{(x+1)(x-2)}$
      $=\frac{5x-1}{(x+1)(x-2)}$
   6. $\frac{3}{x-1}+\frac{2}{(x-1)(x+4)}$
      $=\frac{3(x+4)+2}{(x-1)(x+4)}$
      $=\frac{3x+14}{(x-1)(x+4)}$
   7. $\frac{3}{x-2}-\frac{2}{x+2}+\frac{4}{x^2-4}$
      $=\frac{3(x+2)-2(x-2)}{(x-2)(x+2)}+\frac{4}{(x-2)(x+2)}$
      $=\frac{3x+6-2x+4+4}{(x-2)(x+2)}$
      $=\frac{x+14}{(x-2)(x+2)}$
   8. $\frac{5}{x+2}+\frac{3}{x^2+5x+6}-\frac{2-x}{x+3}$
      $=\frac{5(x+3)+3-(x+2)(2-x)}{(x+3)(x+2)}$
      $=\frac{5x+18-(-x^2+4)}{(x+3)(x+2)}$
      $=\frac{5x+18+x^2-4}{(x+3)(x+2)}$
      $=\frac{x^2+5x+14}{(x+3)(x+2)}$
   9. $\frac{x-y}{2}-\frac{1}{x-y}$
      $=\frac{(x-y{)}^2-2}{2(x-y)}$
      $=\frac{x^2-2xy-y^2-2}{2x-2y}$
   10. $\frac{3}{x-1}-\frac{4x}{1-x}$
       $=\frac{3+4x}{x-1}$
   11. $\frac{2}{x^2+3x+2}+\frac{3}{x^2+7x+10}$
       $=\frac{2}{(x+1)(x+2)}+\frac{3}{(x+2)(x+5)}$
       $=\frac{2(x+5)+3(x+1)}{(x+1)(x+2)(x+5)}$
       $=\frac{5x+13}{(x+1)(x+2)(x+5)}$
   12. $\frac{x}{2x^2+x-3}-\frac{5}{2x^2-5x-12}$
       $=\frac{x}{(2x+3)(x-1)}-\frac{5}{(2x+3)(x-4)}$
       $=\frac{x(x-4)-5(x-1)}{(2x+3)(x-1)(x-4)}$
       $=\frac{x^2-9x+5}{(2x+3)(x-1)(x-4)}$
   13. $\frac{5}{x+3}+\frac{4x}{3x-2}+\frac{2}{2-3x}$
       $=\frac{5}{x+3}+\frac{4x-2}{3x-2}$
       $=\frac{15x-10+(x+3)(4x-2)}{(x+3)(3x-2)}$
       $=\frac{15x-10+4x^2-2x+12x-6}{(x+3)(3x-2)}$
       $=\frac{4x^2+25x-16}{(x+3)(3x-2)}$
   14. $\frac{5-x}{4-x}+\frac{3x+1}{x+4}-\frac{2}{x-4}$
       $=\frac{x-7}{x-4}+\frac{3x+1}{x+4}$
       $=\frac{x^2-3x-28+3x^2-11x-4}{(x-4)(x+4)}$
       $=\frac{4x^2-14x-32}{(x-4)(x+4)}$
   15. $\frac{x-3}{3x^2+5x-2}+\frac{4-2x}{-3x^2+16x-5}$
       $=\frac{x-3}{3x^2+5x-2}+\frac{2x-4}{3x^2-16x+5}$
       $=\frac{x-3}{(3x-1)(x+3)}+\frac{2x-4}{(3x-1)(x-5)}$
       $=\frac{x^2-8x+15+2x^2-x-12}{(3x-1)(x+3)(x-5)}$
       $=\frac{3x^2-9x+3}{(3x-1)(x+3)(x-5)}$

3. simplify the following

   1. $\frac{x^2}{2x}\cdot \frac{4y^3}{x}$
      $=2y^3$
   2. $\frac{3x^2}{4y}\cdot \frac{y^2}{6x}$
      $=\frac{xy}{8}$
   3. $\frac{x^2}{2y}÷\frac{3xy}{6}$
      $=\frac{x}{y^2}$
   4. $\frac{4-x}{3a}\cdot \frac{a^2}{4-x}$
      $=\frac{a}{3}$
   5. $\frac{(x-1{)}^2}{x^2+3x-4}$
      $=\frac{x-1}{x+4}$
   6. $\frac{x^2-x-6}{x-3}$
      $=x+2$
   7. $\frac{x-2}{x}÷\frac{x^2-4}{2x^2}$
      $=\frac{2x-4}{x-4}$
   8. $\frac{x+2}{x^2-3x}÷\frac{4x+8}{x^2-4x+3}$
      $=\frac{x+2}{x(x-3)}\cdot \frac{(x-3)(x-1)}{4(x+2)}$
      $=\frac{x-1}{x}$
   9. $\frac{2x}{x-1}÷\frac{4x^2}{x^2-1}$
      $=\frac{2x}{x-1}\cdot \frac{(x+1)(x-1)}{4x^2}$
      $=\frac{x+1}{2}$
   10. $\frac{x^2-9}{x+2}\cdot \frac{3x+6}{x-3}÷\frac{9}{x}$
       $=\frac{(x+3)(x-3)}{x+2}\cdot \frac{3(x+2)}{x-3}\cdot \frac{x}{9}$
       $=\frac{x^2+3x}{3}$
   11. $\frac{3x}{9x-6}÷\frac{6x^2}{x-2}\cdot \frac{2}{x+5}$
       $=\frac{3x}{9x-6}\cdot \frac{x-2}{6x^2}\cdot \frac{2}{x+5}$
       $=\frac{x-2}{x(9x^2+39x-30)}$
   12. $\frac{2x^2+3x-5}{x^2+x}÷\frac{2x^2+7x+5}{x^2+2x+1}$
       $=\frac{(2x+5)(x-1)}{x(x+1)}\cdot \frac{(x+1)(x+1)}{(2x+5)(x+1)}$
       $=\frac{x-1}{x}$
   13. $\frac{x+2}{3x+5}\cdot \frac{6x+10}{x^2-4}÷\frac{8x}{4-2x}$
       $=\frac{x+2}{3x+5}\cdot \frac{2(3x+5)}{(x+2)(x-2)}\cdot \frac{4-2x}{8x}$
       $=\frac{4-2x}{4x(x-2)}$
   14. $\frac{x^2-9}{4-x^2}÷\frac{3+2x-x^2}{x^2+x-6}$
       $=\frac{(x+3)(x-3)}{(2-x)(2+x)}\cdot \frac{(x+3)(x-2)}{(x+3)(-x-1)}$
       $=\frac{(x+3)(x-3)(x-2)}{(2-x)(x+2)(-x-1)}$
   15. $\frac{2x^2+x-3}{x^2+2x}÷\frac{x^2-4x+3}{x^2}÷\frac{2x^2+3x}{x^2-x-6}$
       $=\frac{(2x+3)(x-1)}{x(x+2)}\cdot \frac{x^2}{(x-3)(x-1)}\cdot \frac{(x-3)(x+2)}{x(2x+3)}$
       $=1$

4. solve the following equations

   1. $\frac{x}{3}-2=-7$
      $\frac{x}{3}=-5$
      $x=-15$
   2. $\frac{5x-2}{4}+\frac{1}{2}=-2$
      $\frac{5x-2}{4}=-\frac{5}{2}$
      $5x-2=-10$
      $5x=-8$
      $x=\frac{-8}{5}$
   3. $\frac{5x}{4}-1=\frac{3}{7}$
      $5x=\frac{40}{7}$
      $x=\frac{8}{7}$
   4. $\frac{2x-1}{3}=\frac{5x+1}{7}$
      $14x-7=15x+3$
      $x=-10$
   5. $\frac{2x+1}{3}+7=11$
      $2x+1=12$
      $x=\frac{11}{2}$
   6. $\frac{2x+1}{3}+7=\frac{3x-1}{4}$
      $4(2x+1+\frac{7}{3})=9x-3$
      $8x+4+\frac{28}{3}=9x-3$
      $x=7+\frac{28}{3}$
      $x=\frac{49}{3}$
   7. $\frac{x+2}{5}+\frac{x-1}{2}=\frac{x+1}{3}$
      $\frac{2x+4+5x-5}{10}=\frac{x+1}{3}$
      $6x+12+15x-15=10x+10$
      $11x=13$
      $x=\frac{13}{11}$
   8. $x+5=\frac{14}{x}$
      $x^2+5x-14=0$
      $(x+7)(x-2)=0$
      $x=2\vee x=-7$
   9. $\frac{x+1}{3}=\frac{10}{x}$
      $x^2+x=30$
      $x^2+x-30=0$
      $(x+6)(x-5)=0$
      $x=-6\vee x=5$
   10. $\frac{1}{x+1}=\frac{5x-6}{6x}$
       $6x=5x^2-6x+5x-6$
       $0=5x^2-7x-6$
       $0=(5x-3)(x+2)$
       $x=\frac{3}{5}\vee x=-2$
   11. $\frac{2x}{3}+\frac{x-1}{15}=\frac{2}{3}+\frac{22}{x}$
       $\frac{11x-1}{15}=\frac{2x+66}{3x}$
       $33x^2-3x=30x+990$
       $33x^2-33x-990=0$
       $x^2-x-30=0$
       $(x+5)(x-6)=0$
       $x=-5\vee x=6$
   12. $\frac{x+1}{7}+\frac{8}{x-2}=3$
       $\frac{x^2-x-2+56}{7x-14}=3$
       $x^2-x-2+56=21x-42$
       $x^2-22x+96=0$
       $(x-6)(x-16)=0$
       $x=6\vee x=16$

5. express the following as a single fraction

   1. $\sqrt{1-x}+\frac{2}{\sqrt{1-x}}$
      $=\sqrt{1-x}+\frac{2\sqrt{1-x}}{1-x}$
      $=\frac{3\sqrt{1-x}}{1-x}$
   2. $\frac{2}{\sqrt{x-4}}+\frac{2\sqrt{x-4}}{3}$
      $=\frac{6}{3\sqrt{x-4}}+\frac{2(x-4)}{3\sqrt{x-4}}$
      $=\frac{2x-2}{3\sqrt{x-4}}$
   3. $\frac{3}{\sqrt{x+4}}+\frac{2}{\sqrt{x+4}}$
      $=\frac{5}{\sqrt{x+4}}$
   4. $\frac{3}{\sqrt{x-5}}+\sqrt{x-5}$
      $=\frac{3}{\sqrt{x-5}}+\frac{x-5}{\sqrt{x-5}}$
      $=\frac{x-2}{\sqrt{x-5}}$
   5. $\frac{3x^3}{\sqrt{2x-1}}-3x^2\sqrt{2x-1}$
      $=\frac{3x^3}{\sqrt{2x-1}}-\frac{3x^2(2x-1)}{\sqrt{2x-1}}$
      $=\frac{3x^3-6x^3+3x^2}{\sqrt{2x-1}}$
      $=\frac{-3x^3+3x^2}{\sqrt{2x-1}}$
   6. $\frac{4x^4}{3\sqrt{x+3}}+3x^2\sqrt{x+3}$
      $=\frac{4x^4}{3\sqrt{x+3}}+\frac{9x^2(x+3)}{3\sqrt{x+3}}$
      $=\frac{4x^4+9x^3+27x^2}{3\sqrt{x+3}}$

6. express the following as a single fraction

   1. $(6x-3{)}^{\frac{1}{3}}-(6x-3{)}^{-\frac{2}{3}}$
      $=(6x-3{)}^{\frac{1}{3}}-\frac{1}{(6x-3{)}^{\frac{2}{3}}}$
      $=\frac{6x-3}{(6x-3{)}^{\frac{2}{3}}}-\frac{1}{(6x-3{)}^{\frac{2}{3}}}$
      $=\frac{6x-4}{(6x-3{)}^{\frac{2}{3}}}$
   2. $(2x+3{)}^{\frac{1}{4}}-2x(2x+3{)}^{-\frac{3}{4}}$
      $=(2x+3{)}^{\frac{1}{4}}-\frac{2x}{(2x+3{)}^{\frac{3}{4}}}$
      $=\frac{2x+3-2x}{(2x+3{)}^{\frac{3}{4}}}$
      $=\frac{3}{(2x+3{)}^{\frac{3}{4}}}$
   3. $(3-x{)}^{-\frac{2}{5}}-5x(3-x{)}^{\frac{3}{5}}$
      $=\frac{1}{(3-x{)}^{\frac{2}{5}}}-\frac{5x(3-x)}{(3-x{)}^{\frac{2}{5}}}$
      $=\frac{5x^2-15x+1}{(3-x{)}^{\frac{2}{5}}}$