Question : If $P=\frac{(\sqrt7-\sqrt6)}{(\sqrt7+\sqrt6)}$, then what is the value of $(P+\frac{1}{P})?$
Option 1: 12
Option 2: 13
Option 3: 24
Option 4: 26
Correct Answer: 26
Solution :
$P=\frac{(\sqrt7-\sqrt6)}{(\sqrt7+\sqrt6)}$
$P=\frac{(\sqrt7-\sqrt6)}{(\sqrt7+\sqrt6)}\times \frac{(\sqrt7-\sqrt6)}{(\sqrt7-\sqrt6)}$
$P=7+6-2\sqrt{42} = 13-2\sqrt{42}$
$\frac{1}{P} = \frac{1}{13-2\sqrt{42}}\times \frac{13+2\sqrt{42}}{13+2\sqrt{42}}$
$P=13+2\sqrt{42}$
$P+\frac{1}{P} = 13-2\sqrt{42}+13+2\sqrt{42}=26$
Hence, the correct answer is 26.
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