5.) Rationalizing the Denominator

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1.) Rationalize the denominator:
 $\frac{2\sqrt{8}}{\sqrt{12}}$  
Q1:  What should we multiply to the numerator and denominator?

To rationalize the denominator, we use the radical in the denominator and multiply it to the numerator and denominator.  So we will multiply by  $\sqrt{12}$ .

1.) Rationalize the denominator:
 $\frac{2\sqrt{8}}{\sqrt{12}}$  
Q2:  What do we get after we multiply the numerator and denominator by  $\sqrt{12}$  ?

Multiplying the numerator and denominator by  $\sqrt{12}$  , we get...
 $\frac{2\sqrt{8}}{\sqrt{12}}\ \cdot\ \frac{\sqrt{12}}{\sqrt{12}}\ =\ \frac{2\sqrt{8\cdot12}}{\sqrt{12\cdot12}}\ =\ \frac{2\sqrt{96}}{12}$  

1.) Rationalize the denominator:
 $\frac{2\sqrt{8}}{\sqrt{12}}$  
Q3:  The above expression is equivalent to  $\frac{2\sqrt{96}}{12}$  .  Simplify the radical and numerator.

The  $\sqrt{96}$  simplifies into 
 $\sqrt{96}=\sqrt{2\times2\times2\times2\times2\times3}\ =\ 4\sqrt{6}$  .
So we get...
 $\ \frac{2\sqrt{96}}{12}\ =\ \frac{2\ \cdot4\sqrt{6}}{12}\ =\ \frac{8\sqrt{6}}{12}$  

1.) Rationalize the denominator:
 $\frac{2\sqrt{8}}{\sqrt{12}}$  
Q4:  The above expression is equivalent to  $\frac{8\sqrt{6}}{12}$  .  Simplify the numerator and denominator.

Values inside radicals do not interact will values outside radicals usually, so we can only work with the 8 and 12  in
 $\ \frac{8\sqrt{6}}{12}$  .
Simplifying fractions - we look for the largest number that goes into 8 and 12 (which is 4), and divide it out.  So we have
 $\ \frac{8\sqrt{6}}{12}\ =\ \frac{2\sqrt{6}}{3}$  

1.) Rationalize the denominator:
 $\frac{7\sqrt{3}}{4\sqrt{7}}$  
Q1:  What should we multiply to the numerator and denominator?