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Rationalize the denominator of \frac{5}{2+\sqrt{6}}$. The answer can be written as $\frac{A\sqrt{B}+C}{D}$, where $A$, $B$, $C$, and $D$ are integers, $D$ is positive, and $B$ is not divisible by the square of any prime. If the greatest common divisor of $A$, $C$, and $D$ is 1, find $A+B+C+D$.

 May 22, 2019

Best Answer 

 #1
avatar+8965 
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Rationalize the denominator of \(\frac{5}{2+\sqrt{6}}\). The answer can be written as \(\frac{A\sqrt{B}+C}{D}\), where A, B, C, and D are integers, D is positive, and B is not divisible by the square of any prime. If the greatest common divisor of A, C, and D is 1, find A+B+C+D.

__________

 

\(\frac{5}{2+\sqrt{6}}\,=\,\frac{5}{2+\sqrt6}\cdot\frac{2-\sqrt6}{2-\sqrt6}\,=\,\frac{10-5\sqrt6}{4-6}\,=\,\frac{10-5\sqrt6}{-2} \,=\,\frac{5\sqrt6-10}{2} \)

 

Now it is in the form  \(\frac{A\sqrt{B}+C}{D}\)   and...

 

A, B, C, and D  are integers

 

D  is positive

 

B  is not divisible by the square of any prime

 

the GCF of  A, C, and D  =  the GCF  of  5, -10, and 2  =  1

 

A + B + C + D  =  5 + 6 + -10 + 2  =  3

 May 22, 2019
 #1
avatar+8965 
+3
Best Answer

Rationalize the denominator of \(\frac{5}{2+\sqrt{6}}\). The answer can be written as \(\frac{A\sqrt{B}+C}{D}\), where A, B, C, and D are integers, D is positive, and B is not divisible by the square of any prime. If the greatest common divisor of A, C, and D is 1, find A+B+C+D.

__________

 

\(\frac{5}{2+\sqrt{6}}\,=\,\frac{5}{2+\sqrt6}\cdot\frac{2-\sqrt6}{2-\sqrt6}\,=\,\frac{10-5\sqrt6}{4-6}\,=\,\frac{10-5\sqrt6}{-2} \,=\,\frac{5\sqrt6-10}{2} \)

 

Now it is in the form  \(\frac{A\sqrt{B}+C}{D}\)   and...

 

A, B, C, and D  are integers

 

D  is positive

 

B  is not divisible by the square of any prime

 

the GCF of  A, C, and D  =  the GCF  of  5, -10, and 2  =  1

 

A + B + C + D  =  5 + 6 + -10 + 2  =  3

hectictar May 22, 2019

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