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Simplify the following expressions. Where applicable, indicate any restrictions on the variables.

1. \(5\sqrt 3 \left( {\sqrt 6 - 3} \right)\)
   
   
   Solution

   \(\begin{align}
    5\sqrt 3 \left( {\sqrt 6 - 3} \right) &= \left( {5\sqrt 3 \cdot \sqrt 6 } \right) + \left( {5\sqrt 3 \cdot \left( { - 3} \right)} \right) \\
     &= \left( {5\sqrt {3\cdot 6} } \right) + \left( {5\cdot \left( { - 3} \right)} \right)\sqrt 3  \\
     &= 5\sqrt {18} + \left( { - 15} \right)\sqrt 3  \\
     &= 5\sqrt {18} - 15\sqrt 3  \\
    \end{align}\)
   
   Remember to check if any radicals can be reduced further!
   
   \(\begin{align}
    5\sqrt {18} - 15\sqrt 3 &= 5\sqrt {9\cdot 2} - 15\sqrt 3  \\
     &= 5\cdot 3\sqrt 2 - 15\sqrt 3  \\
     &= 15\sqrt 2 - 15\sqrt 3  \\
    \end{align}\)
   
   Therefore, \(5\sqrt 3 \left( {\sqrt 6 - 3} \right) = 15\sqrt 2 - 15\sqrt 3 \).
   
   
   
2. \(\left( {8\sqrt {3v^3 } + 12} \right)\left( {5\sqrt {3v^3 } - 3} \right)\)
   
   
   Solution

   \(v \ge 0\)
   
   \(\begin{align}
    \left( {8\sqrt {3v^3 } + 12} \right)\left( {5\sqrt {3v^3 } - 3} \right) &= \left[ {8\sqrt {3v^3 } \cdot 5\sqrt {3v^3 } } \right] + \left[ {8\sqrt {3v^3 } \cdot \left( { - 3} \right)} \right] + \left[ {12\cdot 5\sqrt {3v^3 } } \right] + \left[ {12 \cdot \left( { - 3} \right)} \right] \\
     &= 40\sqrt {9v^6 } - 24\sqrt {3v^3 } + 60\sqrt {3v^3 }  - 36 \\
     &= 40\cdot 3v^3 + 36\sqrt {3v^3 } - 36 \\
     &= 120v^3 + 36v\sqrt {3v}  - 36 \\
    \end{align}\)
   
   
   
3. \(\left( {3\sqrt 5 - 1} \right)\left( {3\sqrt 5 + 1} \right)\)
   
   
   Solution

   \(\begin{align}
    \left( {3\sqrt 5 - 1} \right)\left( {3\sqrt 5 + 1} \right) &= \left( {3\sqrt 5 } \right)^2 + \left( {3\sqrt 5 \cdot 1} \right) + \left[ {3\sqrt 5 \cdot \left( { - 1} \right)} \right] + \left[ {\left( { - 1} \right) \cdot 1} \right] \\
     &= 9\sqrt {25} + 3\sqrt 5 - 3\sqrt 5 - 1 \\
     &= 9\cdot 5 - 1 \\
     &= 45 - 1 \\
     &= 44 \\
    \end{align}\)
   
   

   For further information about multiplying and dividing radicals, see pp. 284 – 286 of Pre-Calculus 11.