How To Multiply Surds

In order to multiply surds, you should first know these rules:

{ a }^{ m }\cdot { a }^{ n }={ a }^{ m+n }\\ \\ { a }^{ m }\div { a }^{ n }={ a }^{ m-n }

You should also know that:

{ a }^{ \frac { 1 }{ 2 }  }=\sqrt [ 2 ]{ { a }^{ 1 } } =\sqrt { a }

So, knowing these rules, what would you get if you multiplied: \sqrt { 3 } \cdot \sqrt { 3 } ?

Well, \sqrt { 3 } \cdot \sqrt { 3 } ={ 3 }^{ \frac { 1 }{ 2 }  }\cdot { 3 }^{ \frac { 1 }{ 2 }  }={ 3 }^{ \frac { 1 }{ 2 } +\frac { 1 }{ 2 }  }={ 3 }^{ 1 }=3.

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Now how about \sqrt { 3 } \cdot \left( -\sqrt { 3 }  \right) ?

\sqrt { 3 } \cdot \left( -\sqrt { 3 }  \right) =\sqrt { 3 } \cdot \left( -1 \right) \cdot \sqrt { 3 } \\ \\ =\sqrt { 3 } \cdot \sqrt { 3 } \cdot \left( -1 \right) =3\cdot \left( -1 \right) =-3

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What about \left( -\sqrt { 3 }  \right) \left( -\sqrt { 3 }  \right) ?

\left( -\sqrt { 3 }  \right) \left( -\sqrt { 3 }  \right) =\left( -1 \right) \cdot \sqrt { 3 } \cdot \left( -1 \right) \cdot \sqrt { 3 } \\ \\ =\left( -1 \right) \left( -1 \right) \sqrt { 3 } \sqrt { 3 } =1\cdot 3=3

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