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# Help plz !!!!

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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

#1
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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.

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

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
+3

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