How To Multiply Surds Contained within Fractions

First Scenario:

\frac { 1 }{ a+\sqrt { b }  } =\frac { 1 }{ \left( a+\sqrt { b }  \right)  } \cdot \frac { \left( a-\sqrt { b }  \right)  }{ \left( a-\sqrt { b }  \right)  } \\ \\ =\frac { a-\sqrt { b }  }{ { a }^{ 2 }-a\sqrt { b } +a\sqrt { b } -b } =\frac { a-\sqrt { b }  }{ { a }^{ 2 }-b }

Second Scenario:

\frac { 1 }{ a-\sqrt { b }  } =\frac { 1 }{ \left( a-\sqrt { b }  \right)  } \cdot \frac { \left( a+\sqrt { b }  \right)  }{ \left( a+\sqrt { b }  \right)  } \\ \\ =\frac { a+\sqrt { b }  }{ { a }^{ 2 }+a\sqrt { b } -a\sqrt { b } -b } =\frac { a+\sqrt { b }  }{ { a }^{ 2 }-b }

Notice that what we’re ultimately doing in both cases is multiplying the surd within a fraction by 1. When the value of the numerator is exactly the same as the value of the denominator in a fraction, what you have is 1.

You should know that 1/1, 2/2, 3/3, (a+b)/(a+b) are all equal to 1.

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