[latexpage]

1. \(\dfrac{\cancel{15}1\cancel{m^3}}{4\cancel{n^2}}\cdot \dfrac{\cancel{13}1\cancel{n^3}}{\cancel{45}3\cancel{m^6}}\cdot \dfrac{\cancel{3}1m^{\cancel{4}}}{\cancel{39}\cancel{3}1n^{\cancel{2}}}\Rightarrow \dfrac{m}{12n}\)
2. \(\dfrac{3x^2-9x}{3x+9}\cdot \dfrac{12x}{x^2+2x-15}\Rightarrow \dfrac{3x\cancel{(x-3)}}{\cancel{3}(x+3)}\cdot \dfrac{\cancel{12}4x}{(x+5)\cancel{(x-3)}}\Rightarrow \dfrac{12x^2}{(x+3)(x+5)}\)
3. \(\phantom{1}\)
   \(\left(\dfrac{2}{x-4}-\dfrac{6}{x-3}=3\right)(x+4)(x-3) \\ \)
   \(\begin{array}{rrrrcrrrcrcrr}
   2(x&-&3)&-&6(x&+&4)&=&3(x&+&4)(x&-&3) \\
   2x&-&6&-&6x&-&24&=&3(x^2&+&x&-&12) \\
   &&&&-4x&-&30&=&3x^2&+&3x&-&36 \\
   &&&&+4x&+&30&&&+&4x&+&30 \\
   \midrule
   &&&&&&0&=&3x^2&+&7x&-&6 \\
   &&&&&&0&=&(x&+&3)(3x&-&2) \\ \\
   &&&&&&x&=&-3,&\dfrac{2}{3}&&&
   \end{array}\)
4. \(\begin{array}{l}
   \dfrac{\left(\dfrac{x^2}{y^2}-9\right)y^3}{\left(\dfrac{x+3y}{y^3}\right)y^3}\Rightarrow \dfrac{x^2y-9y^3}{x+3y}\Rightarrow \dfrac{y(x^2-9y^2)}{x+3y}\Rightarrow \dfrac{y(x-3y)\cancel{(x+3y)}}{\cancel{(x+3y)}} \\ \\
   \Rightarrow y(x-3y)
   \end{array}\)
5. \(\begin{array}{l}
   \\
   5y^2+2\cdot 7y+\sqrt{25y^2\cdot y} \\
   5y^2+14y+5y\sqrt{y}
   \end{array}\)
6. \(\dfrac{15}{3-\sqrt{5}}\cdot \dfrac{3+\sqrt{5}}{3+\sqrt{5}}\Rightarrow \dfrac{45+15\sqrt{5}}{9-5}\Rightarrow \dfrac{45+15\sqrt{5}}{4}\)
7. \(\left(\dfrac{\cancel{a^0}1b^4}{c^8d^{-12}}\right)^{\frac{1}{4}}\Rightarrow \dfrac{b^{4\cdot \frac{1}{4}}}{c^{8\cdot \frac{1}{4}d^{-12\cdot \frac{1}{4}}}}\Rightarrow \dfrac{b}{c^2d^{-3}}\Rightarrow \dfrac{bd^3}{c^2}\)
8. \(\begin{array}{ll}
   \begin{array}{rrrrl}
   \\ \\
   \sqrt{2x+9}&-&3&=&x \\
   &+&3&&\phantom{x}+3 \\
   \midrule
   (\sqrt{2x+9})^2&&&=&(x+3)^2
   \end{array}
   & \hspace{0.25in}
   \begin{array}{rrrrrcrrrr}
   \\ \\ \\ \\ \\
   &2x&+&9&=&x^2&+&6x&+&9 \\
   -&2x&-&9&&&-&2x&-&9 \\
   \midrule
   &&&0&=&x^2&+&4x&& \\
   &&&0&=&x(x&+&4)&& \\ \\
   &&&x&=&0,&-4&&&
   \end{array}
   \end{array}\)
9. \(\phantom{1}\)
   1. \(\begin{array}{rrl}
      \\ \\
      8x^2-32x&=&0 \\
      8x(x-4)&=&0 \\
      x&=&0, 4
      \end{array}\)
   2. \(\begin{array}{rrl}
      \\ \\ \\
      \dfrac{3x^2}{2}&=&\dfrac{48}{3} \\ \\
      \sqrt{x^2}&=&\sqrt{16} \\
      x&=&\pm 4
      \end{array}\)
10. \(\phantom{1}\)
    1. \(\begin{array}{rrl}
       \\ \\
       x^2-5x+4&=&0 \\
       (x-4)(x-1)&=&0 \\
       x&=&1, 4
       \end{array}\)
    2. \(\begin{array}{rrl}
       \\
       (x-3)(x-1)&=&0 \\
       x&=&1, 3
       \end{array}\)
11. \(\begin{array}{rrrrcrcrcrr}
    \\ \\ \\ \\ \\ \\ \\ \\
    2(x&+&4)&=&x(x)&&&&&& \\
    2x&+&8&=&x^2&&&&&& \\ \\
    &&0&=&x^2&-&2x&-&8&& \\
    &&0&=&x^2&-&4x&+&2x&-&8 \\
    &&0&=&x(x&-&4)&+&2(x&-&4) \\
    &&0&=&(x&-&4)(x&+&2)&& \\ \\
    &&x&=&-2,&4&&&&&
    \end{array}\)
12. \(\begin{array}{rrl}
    \\ \\ \\ \\ \\ \\ \\ \\ \\
    \text{Let }u&=&x^2 \\ \\
    u^2-48u-49&=&0 \\
    (u-49)(u+1)&=&0 \\ \\
    (x^2-49)(x^2+1)&=&0 \\
    (x-7)(x+7)(x^2+1)&=&0 \\ \\
    x^2+1&=&\text{cannot be factored} \\
    x&=&\pm 7
    \end{array}\)
13. \(\begin{array}{rrl}
    \\ \\ \\ \\ \\ \\ \\
    40&=&\dfrac{1}{2}(h-2)(h) \\ \\
    80&=&h^2-2h \\
    0&=&h^2-2h-80 \\
    0&=&(h-10)(h+8) \\
    h&=&10, \cancel{-8} \\ \\
    b&=&10-2=8
    \end{array}\)
14. \(\phantom{1}\)
    \(x, x+2, x+4 \\ \)
    \(\begin{array}{crrrrrcrr}
    x(x&+&4)&=&41&+&4(x&+&2) \\
    x^2&+&4x&=&41&+&4x&+&8 \\
    &-&4x&&&-&4x&& \\
    \midrule
    &&\sqrt{x^2}&=&\sqrt{49}&&&& \\
    &&x&=&\pm 7&&&&
    \end{array}\)
    \(\phantom{1}\)
    \(\text{numbers are }7, 9, 11\text{ or }-7,-5,-3\)
15. \(\begin{array}{rrrrrrcrrr}
    \\ \\ \\ \\ \\ \\ \\ \\
    &&&d_{\text{d}}&=&d_{\text{u}}&+&4&& \\ \\
    &2(6&+&r)&=&3(6&-&r)&+&4 \\
    &12&+&2r&=&18&-&3r&+&4 \\
    -&12&+&3r&&-12&+&3r&& \\
    \midrule
    &&&\dfrac{5r}{5}&=&\dfrac{10}{5}&&&& \\ \\
    &&&r&=&2&\text{km/h}&&&
    \end{array}\)