[latexpage]

1. \(\dfrac{5\cancel{m^3}}{\cancel{4}n^{\cancel{2}}}\cdot \dfrac{\cancel{13}\cancel{n^3}}{\cancel{3}\cancel{m^3}}\cdot \dfrac{\cancel{12}m^4}{\cancel{26}2\cancel{n^2}}\Rightarrow \dfrac{5m^4}{2n}\)
2. \(\dfrac{\cancel{3x}\cancel{(x+3)}}{\cancel{3}\cancel{(x+3)}}\cdot \dfrac{6x(x+3)}{(x+6)(x-3)}\Rightarrow \dfrac{6x^2(x+3)}{(x+6)(x-3)}\)
3. \(\begin{array}{l}
   \\ \\ \\ \\ \\ \\ \\ \\
   \left(\dfrac{5x}{x+3}-\dfrac{5x}{x-3}+\dfrac{90}{x^2-9}\right)(x-3)(x+3) \\ \\
   \Rightarrow \dfrac{5x(x-3)-5x(x+3)+90}{(x+3)(x-3)} \\ \\
   \Rightarrow \dfrac{5x^2-15x-5x^2-15x+90}{(x+3)(x-3)} \\ \\
   \Rightarrow \dfrac{-30x+90}{(x+3)(x-3)}\Rightarrow \dfrac{-30\cancel{(x-3)}}{(x+3)\cancel{(x-3)}}\Rightarrow \dfrac{-30}{x+3}
   \end{array}\)
4. \(\dfrac{\left(\dfrac{9a^2}{b^2}-25\right)(b^2)}{\left(\dfrac{3a}{b}+5}\right)(b^2)}\Rightarrow \dfrac{9a^2-25b^2}{3ab+5b^2}\Rightarrow \dfrac{(3a-5b)\cancel{(3a+5b)}}{b\cancel{(3a+5b)}}\Rightarrow \dfrac{3a-5b}{b}\)
5. \(\begin{array}{l}
   \\ \\ \\
   \sqrt{2\cdot 36\cdot d^2\cdot d}+4\sqrt{2\cdot 9\cdot d^2\cdot d}-2(7d^2) \\
   6d\sqrt{2d}+4\cdot 3d\sqrt{2d}-14d^2 \\
   6d\sqrt{2d}+12d\sqrt{2d}-14d^2 \\
   18d\sqrt{2d}-14d^2
   \end{array}\)
6. \(\dfrac{\sqrt{a^{\cancel{6}5}b^3}}{\sqrt{5\cancel{a}}}\cdot \dfrac{\sqrt{5}}{\sqrt{5}}\Rightarrow \dfrac{\sqrt{5a^5b^3}}{\sqrt{25}}\Rightarrow \dfrac{\sqrt{5\cdot a^4\cdot a\cdot b^2\cdot b}}{5}\Rightarrow \dfrac{a^2b\sqrt{5ab}}{5}\)
7. \(\dfrac{\sqrt{5}}{3+\sqrt{5}}\cdot \dfrac{3-\sqrt{5}}{3-\sqrt{5}}\Rightarrow \dfrac{3\sqrt{5}-5}{9-5}\Rightarrow \dfrac{3\sqrt{5}-5}{4}\)
8. \(\begin{array}{rrl}
   \\ \\ \\ \\ \\
   (\sqrt{4x+12})^2&=&(x)^2 \\
   4x+12&=&x^2 \\
   0&=&x^2-4x-12 \\
   0&=&(x-6)(x+2) \\ \\
   x&=&6, \cancel{-2}
   \end{array}\)
9. \(\phantom{1}\)
   1. \(\begin{array}{rrl}
      \\ \\ \\
      \dfrac{2x^2}{2}&=&\dfrac{98}{2} \\ \\
      x^2&=&49 \\
      x&=& \pm 7
      \end{array}\)
   2. \(\begin{array}{rrl}
      \\ \\
      4x^2-12x&=&0 \\
      4x(x-3)&=&0 \\
      x&=&0, 3
      \end{array}\)
10. \(\phantom{1}\)
    1. \(\begin{array}{rrl}
       \\
       (x-5)(x+4)&=&0 \\
       x&=&5, -4
       \end{array}\)
    2. \(\begin{array}{rrl}
       \\ \\
       x^2-2x-35&=&0 \\
       (x-7)(x+5)&=&0 \\
       x&=&7, -5
       \end{array}\)
11. \(\phantom{1}\)
    \(\left(\dfrac{x-3}{x+2}+\dfrac{6}{x+3}=1\right)(x+2)(x+3) \\ \\ \)
    \(\begin{array}{rcccrcrrrrrcrr}
    &(x-3)&\cdot &(x+3)&+&6(x&+&2)&=&(x&+&2)(x&+&3) \\
    &x^2&-&9&+&6x&+&12&=&x^2&+&5x&+&6 \\
    -&x^2&+&9&-&6x&-&12&&-x^2&-&6x&-&12 \\
    \midrule
    &&&&&&&0&=&-x&+&3&& \\
    &&&&&&+&x&&+x&&&& \\
    \midrule
    &&&&&&&x&=&3&&&&
    \end{array}\)
12. \(\begin{array}{rrl}
    \\ \\ \\ \\ \\
    \text{Let }u&=&x^2 \\ \\
    u^2-5u+4&=&0 \\
    (u-4)(u-1)&=&0 \\ \\
    (x^2-4)(x^2-1)&=&0 \\
    (x-2)(x+2)(x-1)(x+1)&=&0 \\
    x&=&\pm 2, \pm 1
    \end{array}\)
13. \(\begin{array}{rrrrrrr}
    \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\
    L&=&3&+&W&& \\ \\
    P&=&2L&+&2W&& \\
    46&=&2(3&+&W)&+&2W \\
    46&=&6&+&2W&+&2W \\
    -6&&-6&&&& \\
    \midrule
    40&=&4W&&&& \\ \\
    W&=&\dfrac{40}{4}&=&10&& \\ \\
    \therefore L&=&W&+&3&& \\
    L&=&10&+&3&=&13
    \end{array}\)
14. \(\phantom{1}\)
    \(x, x+2, x+4 \\ \)
    \(\begin{array}{rrcrrrrrcrr}
    &&x(x&+&2)&=&16&+&x&+&4 \\
    x^2&+&2x&&&=&20&+&x&& \\
    &-&x&-&20&&-20&-&x&& \\
    \midrule
    x^2&+&x&-&20&=&0&&&& \\ \\
    &&&&0&=&(x&+&5)(x&-&4) \\
    &&&&x&=&\cancel{-5},&4&&& \\
    \end{array}\)
    ∴ 4, 6, 8
15. \(\begin{array}{rrl}
    \\ \\ \\ \\ \\ \\ \\ \\ \\
    d&=&r\cdot t \\
    d_{\text{up}}&=&d_{\text{return}} \\ \\
    4(r-5)&=&2(r+5) \\
    4r-20&=&\phantom{-}2r+10 \\
    -2r+20&&-2r+20 \\
    \midrule
    \dfrac{2r}{2}&=&\dfrac{30}{2} \\ \\
    r&=&15
    \end{array}\)