+31 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 $A+B+C+D$.
+9479 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
+9479 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
