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

  1. \(15\sqrt 5 - 8\sqrt {125} \)

    \(\begin{align}
     15\sqrt 5 - 8\sqrt {125} &= 15\sqrt 5 - 8\sqrt {25\cdot 5}  \\
      &= 15\sqrt 5 - 8\cdot 5\sqrt 5  \\
      &= 15\sqrt 5 - 40\sqrt 5  \\
      &= \left( {15 - 40} \right)\sqrt 5  \\
      &= -25\sqrt 5  \\
     \end{align}\)


  2. \(\sqrt {8s^3 } - \sqrt {50s^3 } + \sqrt {108} + \sqrt {147} \)

    \(s \ge 0\)

    \(\begin{align}
     \sqrt {8s^3 } - \sqrt {50s^3 } + \sqrt {108} + \sqrt {147} &= \sqrt {4\cdot s^2\cdot 2s} - \sqrt {25 \cdot s^2 \cdot 2s} + \sqrt {36\cdot 3} + \sqrt {49\cdot 3}  \\
      &= 2s\sqrt {2s} - 5s\sqrt {2s} + 6\sqrt 3 + 7\sqrt 3  \\
      &= \left( {2s - 5s} \right)\sqrt {2s} + \left( {6 + 7} \right)\sqrt 3  \\
      &= -3s\sqrt {2s} + 13\sqrt 3  \\
     \end{align}\)


  3. \(\sqrt[3]{{54}} + \sqrt[3]{{250}} - \sqrt[3]{2}\)

    \(\begin{align}
     \sqrt[3]{{54}} + \sqrt[3]{{250}} - \sqrt[3]{2} &= \sqrt[3]{{27\cdot 2}} + \sqrt[3]{{125\cdot 2}} - \sqrt[3]{2} \\
      &= 3\sqrt[3]{2} + 5\sqrt[3]{2} - \sqrt[3]{2} \\
      &= \left( {3 + 5 - 1} \right)\sqrt[3]{2} \\
      &= 7\sqrt[3]{2} \\
     \end{align}\)

    In many cases, radical expressions will contain like radicals when the terms are fully simplified. If you are unsure where to start, look for radicals that are already in simplest form, such as \(\sqrt[3]{2}\); they might be a clue.



    For further information about adding and subtracting radicals, see pp. 276 – 277 of Pre-Calculus 11.