Mathematical Programming (Manipulation) Tips

Commutative means that the order of elements contained within an expression do not alter a  certain mathematical result.

For example:

2\cdot 3=3\cdot 2=6\\ \\ 1\cdot 2\cdot 3=2\cdot 1\cdot 3=3\cdot 1\cdot 2=6\\ \\ Or:\\ \\ 1+2=2+1=3\\ \\ 1+2+3=3+2+1=2+1+3=6

Knowing this, you can say that:

a\cdot b\cdot c=c\cdot a\cdot b=b\cdot a\cdot c=c\cdot b\cdot a\\ \\ Or:\\ \\ a+b+c=c+a+b=b+a+c=c+b+a

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Associative means that you can alter the grouping of certain elements in an expression without changing its result.

\left( 2\cdot 3 \right) \cdot 4=2\cdot \left( 3\cdot 4 \right) =24\\ \\ Or:\\ \\ \left( 2+3 \right) +4=2+\left( 3+4 \right) =9

Knowing this you could say that:

\left( a\cdot b \right) \cdot c=a\cdot \left( b\cdot c \right) \\ \\ Or:\\ \\ \left( a+b \right) +c=a+\left( b+c \right)

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Warning:

Commutative and Associative manipulation should not be used when subtracting  and dividing.

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