[latexpage]

1. \(\dfrac{15m^3}{4n^2}\cdot \dfrac{\cancel{17}1m^3}{\cancel{12}4n}\cdot \dfrac{\cancel{3 }1m^4}{\cancel{34 }2n^2}\Rightarrow \dfrac{15m^{10}}{32n^5}\)
2. \(\begin{array}{l}
   \\ \\ \\ \\
   \dfrac{8x-8y}{x^3+y^3}\cdot \dfrac{x^2-xy+y^2}{x^2-y^2} \\ \\
   \Rightarrow \dfrac{8\cancel{(x-y)}}{(x+y)\cancel{(x^2-xy+y^2)}}\cdot \dfrac{\cancel{x^2-xy+y^2}}{(x+y)\cancel{(x-y)}}\Rightarrow \dfrac{8}{(x+y)^2}
   \end{array}\)
3. \(\begin{array}{l}
   \\ \\ \\ \\ \\ \\ \\ \\ \\
   \text{LCD}=6(n-3) \\ \\
   \dfrac{5(n-3)-2\cdot 6(n-3)-5\cdot 6}{6(n-3)} \\ \\
   \dfrac{5n-15-12n+36-30}{6(n-3)} \\ \\
   \dfrac{-7n-9}{6(n-3)}
   \end{array}\)
4. \(\dfrac{\left(\dfrac{x^2}{y^2}-4\right)y^3}{\left(\dfrac{x+2y}{y^3}\right)y^3} \Rightarrow \dfrac{x^2y-4y^3}{x+2y}\Rightarrow \dfrac{y(x^2-4y^2)}{x+2y}\Rightarrow \dfrac{y(x-2y)\cancel{(x+2y)}}{\cancel{(x+2y)}} \\ \)
   \(\Rightarrow y(x-2y)\)
5. \(3\cdot 5+2\sqrt{36\cdot 2}-4 \)
   \(15+2\cdot 6\sqrt{2}-4\)
   \(11+12\sqrt{2}\)
6. \(\dfrac{\sqrt{m^7n^{\cancel{3}2}}}{\sqrt{2\cancel{n}}}\cdot \dfrac{\sqrt{2}}{\sqrt{2}}\Rightarrow \dfrac{\sqrt{m^6\cdot m\cdot n^2\cdot 2}}{\sqrt{4}}\Rightarrow \dfrac{m^3n\sqrt{2m}}{2}\)
7. \(\begin{array}{l}
   \\ \\ \\ \\
   \dfrac{2-x}{1-\sqrt{3}}\cdot \dfrac{1+\sqrt{3}}{1+\sqrt{3}}\Rightarrow \dfrac{2+2\sqrt{3}-x-x\sqrt{3}}{1-3} \\ \\
   \(\Rightarrow \dfrac{2+2\sqrt{3}-x-x\sqrt{3}}{-2}\text{ or }\dfrac{x+x\sqrt{3}-2-2\sqrt{3}}{2}
   \end{array}\)
8. \(\begin{array}{rrl}
   \\ \\ \\ \\ \\
   (\sqrt{7x+8})^2&=&(x)^2 \\
   7x+8&=&x^2 \\
   0&=&x^2-7x-8 \\
   0&=&(x-8)(x+1) \\ \\
   x&=&8, \cancel{-1}
   \end{array}\)
9. \(\phantom{1}\)
   1. \(\begin{array}{rrl}
      \\ \\ \\
      \dfrac{4x^2}{4}&=&\dfrac{64}{4} \\ \\
      x^2&=&16 \\
      x&=&\pm 4
      \end{array}\)
   2. \(\begin{array}{rrl}
      \\ \\ \\
      3x^2-12x&=&0 \\
      3x(x-4)&=&0 \\ \\
      x&=&0,4
      \end{array}\)
10. \(\phantom{1}\)
    1. \(\begin{array}{rrl}
       \\
       (x-5)(x-1)&=&0 \\
       x&=&5, 1
       \end{array}\)
    2. \(\begin{array}{rrl}
       \\ \\
       x^2+10x+9&=&0 \\
       (x+9)(x+1)&=&0 \\
       x&=&-9, -1
       \end{array}\)
11. \(\phantom{1}\)
    \(\left(\dfrac{x+4}{-4}=\dfrac{8}{x}\right)(-4)(x) \\ \)
    \(\begin{array}{rrl}
    x(x+4)&=&-4(8) \\
    x^2+4x&=&-32 \\
    0&=&x^2+4x+32 \hspace{0.5in} \text{Does not factor}
    \end{array}\)
12. \(\begin{array}{rrl}
    \\ \\ \\ \\ \\ \\ \\ \\ \\
    \text{Let }u&=&x^2 \\ \\
    u^2-13u+36&=&0 \\
    u^2-4u-9u+36&=&0 \\
    u(u-4)-9(u-4)&=&0 \\
    (u-4)(u-9)&=&0 \\ \\
    (x^2-4)(x^2-9)&=&0 \\
    (x-2)(x+2)(x-3)(x+3)&=&0 \\
    x&=& \pm 2, \pm 3
    \end{array}\)
13. \(\begin{array}{rrl}
    \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\
    A&=&\dfrac{1}{2}bh \\ \\
    300&=&\dfrac{1}{2}(h+10)h \\ \\
    600&=&h^2+10h \\
    0&=&h^2+10h-600 \\
    0&=&(h-20)(h+30) \\ \\
    h&=& 20, \cancel{-30} \\ \\
    \therefore b&=&h+10=30
    \end{array}\)
14. \(\phantom{1}\)
    \(x, x+2, x+4 \\ \)
    \(\begin{array}{rrrrrrrrrrr}
    &&x(x&+&4)&=&38&+&x&+&2 \\
    x^2&+&4x&&&=&x&+&40&& \\
    &-&x&-&40&&-x&-&40&& \\
    \midrule
    x^2&+&3x&-&40&=&0&&&& \\ \\
    &&&&0&=&(x&+&8)(x&-&5) \\
    &&&&x&=&\cancel{-8},&5&&& \\
    \end{array}\)
    ∴ 5, 7, 9
15. \(\begin{array}{rrl}
    \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\
    r_st_s&=&r_ft_f \\ \\
    r(4.5\text{ h})&=&(r+150)(3.0\text{ h}) \\
    4.5r&=&\phantom{-}3.0r+450 \\
    -3.0r&&-3.0r \\
    \midrule
    1.5r&=&450 \\ \\
    r&=&\dfrac{450}{1.5}\text{ or }300\text{ km/h} \\ \\
    r_f&=&300+150 \\
    r_f&=&450\text{ km/h}
    \end{array}\)