worksheet 1.6

1. simplify the following

   1. $(3x^4{)}^4$
      $=81x^{16}$

   2. $13x^{\frac{2}{3}}\cdot 5x^{\frac{1}{5}}$
      $=65x^{\frac{13}{15}}$

   3. $(x^5{)}^{\frac{2}{7}}\cdot x^{\frac{-1}{9}}$
      $=x^{\frac{10}{7}}\cdot x^{\frac{-1}{9}}$
      $=x^{\frac{83}{63}}$

2. simplify the following

   1. $\frac{3}{x-4}+\frac{5x}{4-x}$
      $=\frac{3-5x}{x-4}$
   2. $\frac{7}{x+1}-\frac{8x}{5x-2}$
      $=\frac{35x-14-8x^2-8x}{(x+1)(5x-2)}$
      $=\frac{-8x^2+27x-14}{(x+1)(5-2)}$
   3. $\frac{3x+2}{2x^2+5x-3}-\frac{4x-3}{6x^2+3x-3}$
      $=\frac{3x+2}{(2x-1)(x+3)}-\frac{4x-3}{(2x-1)(3x+3)}$
      $=\frac{(3x+2)(3x+3)-(4x-3)(x+3)}{(2x-1)(x+3)(3x+3)}$
      $=\frac{9x^2+15x+6-(4x^2+8x-9)}{(2x-1)(x+3)(3x+3)}$
      $=\frac{5x^2+7x+15}{(2x-1)(x+3)(3x+3)}$

3. solve for x

   1. $\frac{4x}{3}+\frac{x}{6}=7$
      $9x=42$
      $x=\frac{14}{3}$
   2. $\frac{3x-1}{4}-\frac{5-2x}{3}=1$
      $\frac{9x-3-20+8x}{12}=1$
      $17x-23=12$
      $x=\frac{35}{17}$
   3. $\frac{4}{7x}-\frac{3}{2x}=\frac{3}{4}$
      $\frac{8-21}{14x}=\frac{3}{4}$
      $-52=42x$
      $x=\frac{-52}{42}$
      $x=\frac{-26}{21}$

4. express as a single fraction

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

   2. $(3-2x{)}^{\frac{1}{2}}-(3-2x{)}^{\frac{-3}{2}}$
      $=(3-2x{)}^{\frac{1}{2}}-\frac{1}{(3-2x{)}^{\frac{3}{2}}}$
      $=\frac{9-4x^2-12x-1}{(3-2x{)}^{\frac{3}{2}}}$

5. solve for x

   1. $3a-x=\frac{bx}{c}+3$
      $\frac{bx}{c}+x=3a-3$
      $\frac{x(b+c)}{c}=3a-3$
      $x=\frac{3ac-3c}{b+c}$
   2. $u=\frac{-2x-3}{bx}$
      $ubx=-2x-3$
      $x(ub+2)=-3$
      $x=\frac{-3}{ub+2}$
   3. $\frac{a}{x}+\frac{b}{2}=\frac{5}{a}$
      $\frac{2a+bx}{2x}=\frac{5}{a}$
      $2a^2+abx=10x$
      $x(10-ab)=2a^2$
      $x=\frac{2a^2}{10-ab}$

6. solve for x and y

   1. $4ax+2by=5$ and $3ax+by=-8$
      $6ax+2by=-16$
      $x=\frac{-21}{2a}$
      $by=-8+\frac{63}{2}=\frac{47}{2}$
      $y=\frac{47}{2b}$

   2. $ax-2by=1$ and $x+y=7$
      $ax+ay=7a$
      $-2by-ay=1-7a$
      $ax-2by=1$
      $y=7-x$
      $ax-2b(7-x)=1$
      $ax-14b+2bx=1$
      $(a+2b)x=14b+1$
      $x=\frac{14b+1}{a+2b}$
      $y=\frac{7a-1}{a+2b}$

7. expand the following

   1. $(x+2y{)}^3$
      $=x^3+3x^2(2y)+3x(2y{)}^2+y^3$
      $=x^3+6x^{2}y+12x{y}^2+y^3$
   2. $(3x-y{)}^3$
      $=(3x{)}^3-3(3x{)}^2(y)+3(3x)(y{)}^2-y^3$
      $=27x^3-27x^{2}y+9x{y}^2-y^3$
   3. $(\frac{2}{x}+3{)}^4$
      $=(\frac{2}{x}{)}^4+4(\frac{2}{x}{)}^3(3)+6(\frac{2}{x}{)}^2(3{)}^2+4(\frac{2}{x})(3{)}^3+3^4$
      $=\frac{8}{x^4}+\frac{96}{x^3}+\frac{108}{x^2}+\frac{216}{x}+81$

8. if $2(x+2{)}^2+3x+4=a{x}^2+bx+c$, find the values of a, b, and c
   $=2(x^2+4x+4)+3x+4$
   $=2x^2+11x+12$
   a=2
   b=11
   c=12

9. express $(x+1{)}^3$ in the form $(x+2{)}^3+a(x+2{)}^2+b(x+2)+c$
   let u=x+2
   $(x+1{)}^3=u^3+a{u}^2+bu+c$
   $(u-1{)}^3=u^3+a{u}^2+bu+c$
   $u^3-3u^2+3x-1=u^3+a{u}^2+bu+c$
   a=-3
   b=3
   c=-1